cover           ,relativity3 ,the       
,special and ,general ,theory    by     
,albert ,einstein      ,authorised      
,translation by ,robert ,w4 ,lawson     
,,albert ,,einstein ,,reference         
,,archive  ,,relativity3 ,,the ,,special
 ,,and ,,general ,,theory  ,,by ,,albert
 ,,einstein      ,written3 #aiaf 7this  
revised edition3 #aibd7  ,source3       
,relativity3 ,the ,special and ,general 
,theory 7#aibj7  ,publisher3 ,methuen   
,co ,ltd  ,first ,published3 ,december1 
#aiaf  ,translated3 ,robert ,w4 ,lawson 
7,authorised translation7               
,transcription_/,markup3 ,brian ,basgen 
,transcription to text3 ,gregory ,b4    
,newby  ,thanks to3 ,einstein ,reference
 ,archive 7marxists4org7  ,the ,einstein
 ,reference ,archive is online at3      
http3_/_/www4marxists4org_/reference_/ar
chive_/einstein_/index4htm              
,contents     ,preface     ,part ,i3    
,the ,special ,theory of ,relativity    
,i4 ,physical ,meaning of ,geometrical  
,propositions   ,,ii4 ,the ,system of   

,co-ordinates   ,,iii4 ,space and ,time 
in ,classical ,mechanics   ,,iv4 ,the   
,galileian ,system of ,co-ordinates     
,v4 ,the ,principle of ,relativity 7in  
the ,restricted ,sense7   ,,vi4 ,the    
,theorem of the ,addition of ,velocities
 employed in ,classical ,mechanics      
,,vii4 ,the ,apparent ,incompatability  
of the ,law of ,propagation of ,light   
with the ,principle of ,relativity      
,,viii4 ,on the ,idea of ,time in       
,physics   ,,ix4 ,the ,relativity of    
,simultaneity   ,x4 ,on the ,relativity 
of the ,conception of ,distance   ,,xi4 
,the ,lorentz ,transformation   ,,xii4  
,the ,behaviour of ,measuring-,rods and 
,clocks in ,motion   ,,xiii4 ,theorem of
 the ,addition of ,velocities4 ,the     
,experiment of ,fizeau   ,,xiv4 ,the    
,heuristic ,value of the ,theory of     
,relativity   ,,xv4 ,general ,results of
 the ,theory   ,,xvi4 ,experience and   
the ,special ,theory of ,relativity     
,,xvii4 ,minkowski's ,four-dimensional  
,space     ,part ,,ii3 ,the ,general    

,theory of ,relativity   ,,xviii4       
,special and ,general ,principle of     
,relativity   ,,xix4 ,the ,gravitational
 ,field   ,,xx4 ,the ,equality of       
,inertial and ,gravitational ,mass as an
 ,argument for the ,general ,postulate  
of ,relativity   ,,xxi4 ,in ,what       
,respects are the ,foundations of       
,classical ,mechanics and of the        
,special ,theory of ,relativity         
,unsatisfactory8   ,,xxii4 ,a ,few      
,inferences from the ,general ,principle
 of ,relativity   ,,xxiii4 ,behaviour of
 ,clocks and ,measuring-,rods on a      
,rotating ,body of ,reference   ,,xxiv4 
,euclidean and non-,euclidean ,continuum
   ,,xxv4 ,gaussian ,co-ordinates       
,,xxvi4 ,the ,space-,time ,continuum of 
the ,special ,theory of ,relativity     
,considered as a ,euclidean ,continuum  
,,xxvii4 ,the ,space-,time ,continuum of
 the ,general ,theory of ,relativity is 
,not a ,euclidean ,continuum   ,,xxviii4
 ,exact ,formulation of the ,general    
,principle of ,relativity   ,,xxix4 ,the

 ,solution of the ,problem of           
,gravitation on the ,basis of the       
,general ,principle of ,relativity      
,part ,,iii3 ,considerations on the     
,universe as a ,whole   ,,xxx4          
,cosmological ,difficulties of ,newton's
 ,theory   ,,xxxi4 ,the ,possibility of 
a 8,finite0 and yet 8,unbounded0        
,universe   ,,xxxii4 ,the ,structure of 
,space ,according to the ,general       
,theory of ,relativity     ,appendices3 
,i4 ,simple ,derivation of the ,lorentz 
,transformation 7supplementary to       
section ,,xi7   ,,ii4 ,minkowski's      
,four-,dimensional ,space 78,world07    
7supplementary to section ,,xvii7       
,,iii4 ,the ,experimental ,confirmation 
of the ,general ,theory of ,relativity  
,,iv4 ,the ,structure of ,space         
,according to the ,general ,theory of   
,relativity 7supplementary to section   
,,xxxii7   ,v4 ,relativity and the      
,problem of ,space      ,note3 ,the     
fifth ,appendix was added by ,einstein  
at the time of the  fifteenth           

re-printing of this book2 and as a      
result is still under  copyright        
restrictions so cannot be added without 
the permission of the  publisher4       
,,preface      ,the present book is     
intended1 as far as possible1 to give an
 exact  insight into the theory of      
,relativity to those readers who1 from a
  general scientific and philosophical  
point of view1 are interested in  the   
theory1 but who are not conversant with 
the mathematical apparatus  of          
theoretical physics4 ,the work presumes 
a standard of education  corresponding  
to that of a university matriculation   
examination1 and1  despite the shortness
 of the book1 a fair amount of patience 
and force  of will on the part of the   
reader4 ,the author has spared himself  
no  pains in his endeavour to present   
the main ideas in the simplest and  most
 intelligible form1 and on the whole1 in
 the sequence and  connection in which  
they actually originated4 ,in the       
interest of  clearness1 it appeared to  

me inevitable that ,i should repeat     
myself  frequently1 without paying the  
slightest attention to the elegance of  
the presentation4 ,i adhered            
scrupulously to the precept of that     
brilliant theoretical physicist ,l4     
,boltzmann1 according to whom matters   
of elegance ought to be left to the     
tailor and to the cobbler4 ,i make  no  
pretence of having withheld from the    
reader difficulties which are  inherent 
to the subject4 ,on the other hand1 ,i  
have purposely treated  the empirical   
physical foundations of the theory in a 
8step-motherly0  fashion1 so that       
readers unfamiliar with physics may not 
feel like the  wanderer who was unable  
to see the forest for the trees4 ,may   
the book  bring some one a few happy    
hours of suggestive thought6            
,december1 #aiaf     ,a4 ,,einstein     
,,part ,i3 ,,the ,,special ,,theory ,,of
 ,,relativity          ,i4  ,,physical  
,,meaning ,,of ,,geometrical            
,,propositions      ,in your schooldays 

most of you who read this book made     
acquaintance  with the noble building of
 ,euclid's geometry1 and you            
remember--perhaps  with more respect    
than love--the magnificent structure1 on
 the lofty  staircase of which you were 
chased about for uncounted hours by     
conscientious teachers4 ,by reason of   
our past experience1 you would          
certainly regard everyone with disdain  
who should pronounce even the  most     
out-of-the-way proposition of this      
science to be untrue4 ,but  perhaps this
 feeling of proud certainty would leave 
you immediately if  some one were to ask
 you3 8,what1 then1 do you mean by the  
assertion  that these propositions are  
true80 ,let us proceed to give this     
question  a little consideration4       
,geometry sets out from certain         
conceptions such as 8plane10 8point10   
and 8straight line10 with which we are  
able to associate more or less  definite
 ideas1 and from certain simple         
propositions 7axioms7 which1 in  virtue 

of these ideas1 we are inclined to      
accept as 8true40 ,then1 on  the basis  
of a logical process1 the justification 
of which we feel  ourselves compelled to
 admit1 all remaining propositions are  
shown to  follow from those axioms1 i4e4
 they are proven4 ,a proposition is then
  correct 78true07 when it has been     
derived in the recognised manner from   
the axioms4 ,the question of 8truth0 of 
the individual geometrical  propositions
 is thus reduced to one of the 8truth0  
of the axioms4 ,now  it has long been   
known that the last question is not only
 unanswerable  by the methods of        
geometry1 but that it is in itself      
entirely without  meaning4 ,we cannot   
ask whether it is true that only one    
straight line  goes through two points4 
,we can only say that ,euclidean        
geometry deals  with things called      
8straight lines10 to each of which is   
ascribed the  property of being uniquely
 determined by two points situated on   
it4 ,the  concept 8true0 does not tally 

with the assertions of pure geometry1   
because by the word 8true0 we are       
eventually in the habit of  designating 
always the correspondence with a 8real0 
object2 geometry1  however1 is not      
concerned with the relation of the ideas
 involved in it  to objects of          
experience1 but only with the logical   
connection of these  ideas among        
themselves4    ,it is not difficult to  
understand why1 in spite of this1 we    
feel  constrained to call the           
propositions of geometry 8true40        
,geometrical  ideas correspond to more  
or less exact objects in nature1 and    
these  last are undoubtedly the         
exclusive cause of the genesis of those 
ideas4  ,geometry ought to refrain from 
such a course1 in order to give to its  
structure the largest possible logical  
unity4 ,the practice1 for  example1 of  
seeing in a 8distance0 two marked       
positions on a  practically rigid body  
is something which is lodged deeply in  
our habit  of thought4 ,we are          

accustomed further to regard three      
points as being  situated on a straight 
line1 if their apparent positions can be
 made to  coincide for observation with 
one eye1 under suitable choice of our   
place of observation4    ,if1 in        
pursuance of our habit of thought1 we   
now supplement the  propositions of     
,euclidean geometry by the single       
proposition that two  points on a       
practically rigid body always correspond
 to the same  distance 7line-interval71 
independently of any changes in position
 to  which we may subject the body1 the 
propositions of ,euclidean geometry     
then resolve themselves into            
propositions on the possible relative   
position of practically rigid bodies4#a 
,geometry which has been  supplemented  
in this way is then to be treated as a  
branch of physics4  ,we can now         
legitimately ask as to the 8truth0 of   
geometrical  propositions interpreted in
 this way1 since we are justified in    
asking  whether these propositions are  

satisfied for those real things we have 
associated with the geometrical ideas4  
,in less exact terms we can  express    
this by saying that by the 8truth0 of a 
geometrical proposition  in this sense  
we understand its validity for a        
construction with rule  and compasses4  
#a ,it follows that a natural object is 
associated also with a   straight line4 
,three points ,a1 ,b and ,c on a rigid  
body thus lie in   a straight line when 
the points ,a and ,c being given1 ,b is 
chosen   such that the sum of the       
distances ,,ab and ,,bc is as short as  
possible4 ,this incomplete suggestion   
will suffice for the present   purpose4 
,of course the conviction of the 8truth0
 of geometrical propositions in  this   
sense is founded exclusively on rather  
incomplete experience4 ,for  the present
 we shall assume the 8truth0 of the     
geometrical  propositions1 then at a    
later stage 7in the general theory of   
relativity7 we shall see that this      
8truth0 is limited1 and we shall        

consider the extent of its limitation4  
,,ii4  ,,the ,,system ,,of              
,,co-,,ordinates      ,on the basis of  
the physical interpretation of distance 
which has been  indicated1 we are also  
in a position to establish the distance 
between  two points on a rigid body by  
means of measurements4 ,for this purpose
  we require a 8distance0 7rod ,s7 which
 is to be used once and for all1  and   
which we employ as a standard measure4  
,if1 now1 ,a and ,b are two  points on a
 rigid body1 we can construct the line  
joining them  according to the rules of 
geometry2 then1 starting from ,a1 we can
  mark off the distance ,s time after   
time until we reach ,b4 ,the  number of 
these operations required is the        
numerical measure of the  distance ,,ab4
 ,this is the basis of all measurement  
of length4#b       #b ,here we have     
assumed that there is nothing left over 
i4e4 that   the measurement gives a     
whole number4 ,this difficulty is got   
over by   the use of divided            

measuring-rods1 the introduction of     
which does not   demand any             
fundamentally new method4      ,every   
description of the scene of an event or 
of the position of an  object in space  
is based on the specification of the    
point on a rigid  body 7body of         
reference7 with which that event or     
object coincides4  ,this applies not    
only to scientific description1 but also
 to everyday  life4 ,if ,i analyse the  
place specification 8,trafalgar ,square1
  ,london0#c ,i arrive at the following 
result4 ,the earth is the rigid  body to
 which the specification of place       
refers2 8,trafalgar ,square1  ,london0  
is a well-defined point1 to which a name
 has been assigned1 and  with which the 
event coincides in space4#d       #c    
,i have chosen this as being more       
familiar to the ,english reader than    
the 8,potzdammer ,platz1 ,berlin10 which
 is referred to in the original4  7,r4  
,w4 ,l47       #d ,it is not necessary  
here to investigate further the         

significance   of the expression        
8coincidence in space40 ,this conception
 is   sufficiently obvious to ensure    
that differences of opinion are         
scarcely likely to arise as to its      
applicability in practice4      ,this   
primitive method of place specification 
deals only with places on  the surface  
of rigid bodies1 and is dependent on the
 existence of  points on this surface   
which are distinguishable from each     
other4 ,but  we can free ourselves from 
both of these limitations without       
altering  the nature of our             
specification of position4 ,if1 for     
instance1 a cloud  is hovering over     
,trafalgar ,square1 then we can         
determine its position  relative to the 
surface of the earth by erecting a pole 
perpendicularly  on the ,square1 so that
 it reaches the cloud4 ,the length of   
the pole  measured with the standard    
measuring-rod1 combined with the        
specification of the position of the    
foot of the pole1 supplies us with  a   

complete place specification4 ,on the   
basis of this illustration1 we  are able
 to see the manner in which a refinement
 of the conception of  position has been
 developed4    7a7 ,we imagine the rigid
 body1 to which the place specification 
is  referred1 supplemented in such a    
manner that the object whose position   
we require is reached by the completed  
rigid body4    7b7 ,in locating the     
position of the object1 we make use of a
 number  7here the length of the pole   
measured with the measuring-rod7 instead
  of designated points of reference4    
7c7 ,we speak of the height of the cloud
 even when the pole which  reaches the  
cloud has not been erected4 ,by means of
 optical  observations of the cloud from
 different positions on the ground1 and 
taking into account the properties of   
the propagation of light1 we  determine 
the length of the pole we should have   
required in order to  reach the cloud4  
,from this consideration we see that it 
will be advantageous if1 in the         

description of position1 it should be   
possible by means of numerical  measures
 to make ourselves independent of the   
existence of marked  positions          
7possessing names7 on the rigid body of 
reference4 ,in the  physics of          
measurement this is attained by the     
application of the  ,cartesian system of
 co-ordinates4    ,this consists of     
three plane surfaces perpendicular to   
each other and  rigidly attached to a   
rigid body4 ,referred to a system of    
co-ordinates1  the scene of any event   
will be determined 7for the main part7  
by the  specification of the lengths of 
the three perpendiculars or             
co-ordinates 7x1 y1 z7 which can be     
dropped from the scene of the  event to 
those three plane surfaces4 ,the lengths
 of these three  perpendiculars can be  
determined by a series of manipulations 
with  rigid measuring-rods performed    
according to the rules and methods laid 
down by ,euclidean geometry4    ,in     
practice1 the rigid surfaces which      

constitute the system of  co-ordinates  
are generally not available2            
furthermore1 the magnitudes  of the     
co-ordinates are not actually determined
 by constructions with  rigid rods1 but 
by indirect means4 ,if the results of   
physics and  astronomy are to maintain  
their clearness1 the physical meaning of
  specifications of position must always
 be sought in accordance with the  above
 considerations4#e       #e ,a          
refinement and modification of these    
views does not become   necessary until 
we come to deal with the general theory 
of relativity1   treated in the second  
part of this book4      ,we thus obtain 
the following result3 ,every description
 of events in  space involves the use of
 a rigid body to which such events have 
to be  referred4 ,the resulting         
relationship takes for granted that the 
laws of  ,euclidean geometry hold for   
8distances20 the 8distance0 being       
represented physically by means of the  
convention of two marks on a  rigid     

body4      ,,iii4    ,,space ,,and      
,,time ,,in ,,classical ,,mechanics     
,the purpose of mechanics is to describe
 how bodies change their  position in   
space with 8time40 ,i should load my    
conscience with grave  sins against the 
sacred spirit of lucidity were ,i to    
formulate the aims  of mechanics in this
 way1 without serious reflection and    
detailed  explanations4 ,let us proceed 
to disclose these sins4    ,it is not   
clear what is to be understood here by  
8position0 and  8space40 ,i stand at the
 window of a railway carriage which is  
travelling uniformly1 and drop a stone  
on the embankment1 without  throwing it4
 ,then1 disregarding the influence of   
the air resistance1 ,i  see the stone   
descend in a straight line4 ,a          
pedestrian who observes the  misdeed    
from the footpath notices that the stone
 falls to earth in a  parabolic curve4  
,i now ask3 ,do the 8positions0         
traversed by the stone  lie 8in reality0
 on a straight line or on a parabola8   

,moreover1 what is  meant here by motion
 8in space08 ,from the considerations of
 the  previous section the answer is    
self-evident4 ,in the first place we    
entirely shun the vague word 8space10 of
 which1 we must honestly  acknowledge1  
we cannot form the slightest conception1
 and we replace it  by 8motion relative 
to a practically rigid body of          
reference40 ,the  positions relative to 
the body of reference 7railway carriage 
or  embankment7 have already been       
defined in detail in the preceding      
section4 ,if instead of 8body of        
reference0 we insert 8system of         
co-ordinates10 which is a useful idea   
for mathematical description1 we  are in
 a position to say3 ,the stone traverses
 a straight line relative  to a system  
of co-ordinates rigidly attached to the 
carriage1 but  relative to a system of  
co-ordinates rigidly attached to the    
ground  7embankment7 it describes a     
parabola4 ,with the aid of this example 
it  is clearly seen that there is no    

such thing as an independently  existing
 trajectory 7lit4 8path-curve0#f1 but   
only a trajectory  relative to a        
particular body of reference4       #f  
,that is1 a curve along which the body  
moves4      ,in order to have a complete
 description of the motion1 we must     
specify how the body alters its position
 with time2 i4e4 for every  point on the
 trajectory it must be stated at what   
time the body is  situated there4 ,these
 data must be supplemented by such a    
definition of  time that1 in virtue of  
this definition1 these time-values can  
be  regarded essentially as magnitudes  
7results of measurements7 capable of    
observation4 ,if we take our stand on   
the ground of classical mechanics1  we  
can satisfy this requirement for our    
illustration in the following  manner4  
,we imagine two clocks of identical     
construction2 the man at the            
railway-carriage window is holding one  
of them1 and the man on the  footpath   
the other4 ,each of the observers       

determines the position on  his own     
reference-body occupied by the stone at 
each tick of the clock  he is holding in
 his hand4 ,in this connection we have  
not taken account  of the inaccuracy    
involved by the finiteness of the       
velocity of  propagation of light4 ,with
 this and with a second difficulty      
prevailing  here we shall have to deal  
in detail later4      ,,iv4 ,,the       
,,galileian ,,system ,,of               
,,co-,,ordinates      ,as is well known1
 the fundamental law of the mechanics of
  ,galilei-,newton1 which is known as   
the law of inertia1 can be stated  thus3
 ,a body removed sufficiently far from  
other bodies continues in a  state of   
rest or of uniform motion in a straight 
line4 ,this law not  only says something
 about the motion of the bodies1 but it 
also  indicates the reference-bodies or 
systems of coordinates1 permissible  in 
mechanics1 which can be used in         
mechanical description4 ,the visible    
fixed stars are bodies for which the law

 of inertia certainly holds to  a high  
degree of approximation4 ,now if we use 
a system of co-ordinates  which is      
rigidly attached to the earth1 then1    
relative to this system1  every fixed   
star describes a circle of immense      
radius in the course of  an astronomical
 day1 a result which is opposed to the  
statement of the  law of inertia4 ,so   
that if we adhere to this law we must   
refer these  motions only to systems of 
coordinates relative to which the fixed 
stars do not move in a circle4 ,a system
 of co-ordinates of which the  state of 
motion is such that the law of inertia  
holds relative to it is  called a       
8,galileian system of co-ordinates40    
,the laws of the mechanics  of          
,galilei-,newton can be regarded as     
valid only for a ,galileian system  of  
co-ordinates4      ,v4    ,,the         
,,principle ,,of ,,relativity 7,,in     
,,the ,,restricted ,,sense7    ,in order
 to attain the greatest possible        
clearness1 let us return to  our example

 of the railway carriage supposed to be 
travelling  uniformly4 ,we call its     
motion a uniform translation 78uniform0 
because  it is of constant velocity and 
direction1 8translation0 because        
although the carriage changes its       
position relative to the embankment  yet
 it does not rotate in so doing74 ,let  
us imagine a raven flying  through the  
air in such a manner that its motion1 as
 observed from the  embankment1 is      
uniform and in a straight line4 ,if we  
were to observe  the flying raven from  
the moving railway carriage4 we should  
find that  the motion of the raven would
 be one of different velocity and       
direction1 but that it would still be   
uniform and in a straight line4         
,expressed in an abstract manner we may 
say3 ,if a mass m is moving  uniformly  
in a straight line with respect to a    
co-ordinate system ,k1  then it will    
also be moving uniformly and in a       
straight line relative  to a second     
co-ordinate system ,k provided that the 

latter is  executing a uniform          
translatory motion with respect to ,k4  
,in  accordance with the discussion     
contained in the preceding section1 it  
follows that3    ,if ,k is a ,galileian 
co-ordinate system4 then every other    
co-ordinate  system ,k is a ,galileian  
one1 when1 in relation to ,k1 it is in a
  condition of uniform motion of        
translation4 ,relative to ,k the        
mechanical laws of ,galilei-,newton hold
 good exactly as they do with  respect  
to ,k4    ,we advance a step farther in 
our generalisation when we express the  
tenet thus3 ,if1 relative to ,k1 ,k is a
 uniformly moving co-ordinate  system   
devoid of rotation1 then natural        
phenomena run their course with  respect
 to ,k according to exactly the same    
general laws as with  respect to ,k4    
,this statement is called the principle 
of relativity  7in the restricted       
sense74    ,as long as one was convinced
 that all natural phenomena were capable
 of  representation with the help of    

classical mechanics1 there was no need  
to doubt the validity of this principle 
of relativity4 ,but in view of  the more
 recent development of electrodynamics  
and optics it became  more and more     
evident that classical mechanics affords
 an insufficient  foundation for the    
physical description of all natural     
phenomena4 ,at  this juncture the       
question of the validity of the         
principle of  relativity became ripe for
 discussion1 and it did not appear      
impossible  that the answer to this     
question might be in the negative4      
,nevertheless1 there are two general    
facts which at the outset speak  very   
much in favour of the validity of the   
principle of relativity4  ,even though  
classical mechanics does not supply us  
with a sufficiently  broad basis for the
 theoretical presentation of all        
physical phenomena1  still we must grant
 it a considerable measure of 8truth10  
since it  supplies us with the actual   
motions of the heavenly bodies with a   

delicacy of detail little short of      
wonderful4 ,the principle of  relativity
 must therefore apply with great        
accuracy in the domain of  mechanics4   
,but that a principle of such broad     
generality should hold  with such       
exactness in one domain of phenomena1   
and yet should be  invalid for another1 
is a priori not very probable4    ,we   
now proceed to the second argument1 to  
which1 moreover1 we shall  return later4
 ,if the principle of relativity 7in the
 restricted sense7  does not hold1 then 
the ,galileian co-ordinate systems ,k1  
,k1 ,k1  etc41 which are moving         
uniformly relative to each other1 will  
not be  equivalent for the description  
of natural phenomena4 ,in this case we  
should be constrained to believe that   
natural laws are capable of being       
formulated in a particularly simple     
manner1 and of course only on  condition
 that1 from amongst all possible        
,galileian co-ordinate  systems1 we     
should have chosen one 7,k#j7 of a      

particular state of  motion as our body 
of reference4 ,we should then be        
justified 7because  of its merits for   
the description of natural phenomena7 in
 calling this  system 8absolutely at    
rest10 and all other ,galileian systems 
,k 8in  motion40 ,if1 for instance1 our 
embankment were the system ,k#j then our
  railway carriage would be a system ,k1
 relative to which less simple  laws    
would hold than with respect to ,k#j4   
,this diminished simplicity  would be   
due to the fact that the carriage ,k    
would be in motion  7i4e4 8really07with 
respect to ,k#j4 ,in the general laws of
 nature  which have been formulated with
 reference to ,k1 the magnitude and     
direction of the velocity of the        
carriage would necessarily play a  part4
 ,we should expect1 for instance1 that  
the note emitted by an  organpipe placed
 with its axis parallel to the direction
 of travel  would be different from that
 emitted if the axis of the pipe were   
placed perpendicular to this direction4 

,now in virtue of its motion in an orbit
 round the sun1 our earth is  comparable
 with a railway carriage travelling with
 a velocity of about  #cj kilometres per
 second4 ,if the principle of relativity
 were not valid  we should therefore    
expect that the direction of motion of  
the earth at  any moment would enter    
into the laws of nature1 and also that  
physical  systems in their behaviour    
would be dependent on the orientation in
  space with respect to the earth4 ,for 
owing to the alteration in  direction of
 the velocity of revolution of the earth
 in the course of a  year1 the earth    
cannot be at rest relative to the       
hypothetical system  ,k#j throughout the
 whole year4 ,however1 the most careful 
observations  have never revealed such  
anisotropic properties in terrestrial   
physical  space1 i4e4 a physical        
non-equivalence of different directions4
 ,this  is very powerful argument in    
favour of the principle of relativity4  
,,vi4    ,,the ,,theorem ,,of ,,the     

,,addition ,,of ,,velocities ,,employed 
,,in ,,classical  ,,mechanics    ,let us
 suppose our old friend the railway     
carriage to be travelling  along the    
rails with a constant velocity v1 and   
that a man traverses  the length of the 
carriage in the direction of travel with
 a velocity  w4 ,how quickly or1 in     
other words1 with what velocity ,w does 
the  man advance relative to the        
embankment during the process8 ,the only
  possible answer seems to result from  
the following consideration3 ,if  the   
man were to stand still for a second1 he
 would advance relative to  the         
embankment through a distance v equal   
numerically to the velocity  of the     
carriage4 ,as a consequence of his      
walking1 however1 he traverses  an      
additional distance w relative to the   
carriage1 and hence also  relative to   
the embankment1 in this second1 the     
distance w being  numerically equal to  
the velocity with which he is walking4  
,thus in  total he covers the distance  

,w  v  w relative to the embankment in  
the second considered4 ,we shall see    
later that this result1 which  expresses
 the theorem of the addition of         
velocities employed in  classical       
mechanics1 cannot be maintained2 in     
other words1 the law that  we have just 
written down does not hold in reality4  
,for the time being1  however1 we shall 
assume its correctness4      ,,vii4     
,,the ,,apparent ,,incompatibility ,,of 
,,the ,,law ,,of ,,propagation ,,of     
,,light ,,with  ,,the ,,principle ,,of  
,,relativity    ,there is hardly a      
simpler law in physics than that        
according to which  light is propagated 
in empty space4 ,every child at school  
knows1 or  believes he knows1 that this 
propagation takes place in straight     
lines  with a velocity c  #cjj1jjj      
km4_/sec4 ,at all events we know with   
great  exactness that this velocity is  
the same for all colours1 because if    
this were not the case1 the minimum of  
emission would not be observed          

simultaneously for different colours    
during the eclipse of a fixed star  by  
its dark neighbour4 ,by means of similar
 considerations based on  observations  
of double stars1 the ,dutch astronomer  
,de ,sitter was also  able to show that 
the velocity of propagation of light    
cannot depend on  the velocity of motion
 of the body emitting the light4 ,the   
assumption  that this velocity of       
propagation is dependent on the         
direction 8in  space0 is in itself      
improbable4    ,in short1 let us assume 
that the simple law of the constancy of 
the  velocity of light c 7in vacuum7 is 
justifiably believed by the child  at   
school4 ,who would imagine that this    
simple law has plunged the              
conscientiously thoughtful physicist    
into the greatest intellectual          
difficulties8 ,let us consider how these
 difficulties arise4    ,of course we   
must refer the process of the           
propagation of light 7and  indeed every 
other process7 to a rigid reference-body

 7co-ordinate  system74 ,as such a      
system let us again choose our          
embankment4 ,we shall  imagine the air  
above it to have been removed4 ,if a ray
 of light be  sent along the embankment1
 we see from the above that the tip of  
the  ray will be transmitted with the   
velocity c relative to the  embankment4 
,now let us suppose that our railway    
carriage is again  travelling along the 
railway lines with the velocity v1 and  
that its  direction is the same as that 
of the ray of light1 but its velocity of
  course much less4 ,let us inquire     
about the velocity of propagation of    
the ray of light relative to the        
carriage4 ,it is obvious that we can    
here apply the consideration of the     
previous section1 since the ray of      
light plays the part of the man walking 
along relatively to the  carriage4 ,the 
velocity ,w of the man relative to the  
embankment is  here replaced by the     
velocity of light relative to the       
embankment4 w  is the required velocity 

of light with respect to the carriage1  
and we  have    w  c - v4    ,the       
velocity of propagation ot a ray of     
light relative to the carriage  thus    
comes out smaller than c4    ,but this  
result comes into conflict with the     
principle of relativity  set forth in   
,section ,v4 ,for1 like every other     
general law of nature1  the law of the  
transmission of light in vacuo in vacuum
 must1  according to the principle of   
relativity1 be the same for the railway 
carriage as reference-body as when the  
rails are the body of reference4  ,but1 
from our above consideration1 this would
 appear to be impossible4  ,if every ray
 of light is propagated relative to the 
embankment with the  velocity c1 then   
for this reason it would appear that    
another law of  propagation of light    
must necessarily hold with respect to   
the  carriage--a result contradictory to
 the principle of relativity4    ,in    
view of this dilemma there appears to be
 nothing else for it than to  abandon   

either the principle of relativity or   
the simple law of the  propagation of   
light in vacuo4 ,those of you who have  
carefully  followed the preceding       
discussion are almost sure to expect    
that we  should retain the principle of 
relativity1 which appeals so            
convincingly to the intellect because it
 is so natural and simple4 ,the  law of 
the propagation of light in vacuo would 
then have to be  replaced by a more     
complicated law conformable to the      
principle of  relativity4 ,the          
development of theoretical physics      
shows1 however1 that  we cannot pursue  
this course4 ,the epoch-making          
theoretical  investigations of ,h4 ,a4  
,lorentz on the electrodynamical and    
optical  phenomena connected with moving
 bodies show that experience in this    
domain leads conclusively to a theory of
 electromagnetic phenomena1 of  which   
the law of the constancy of the velocity
 of light in vacuo is a  necessary      
consequence4 ,prominent theoretical     

physicists were therefore  more inclined
 to reject the principle of relativity1 
in spite of the  fact that no empirical 
data had been found which were          
contradictory to  this principle4    ,at
 this juncture the theory of relativity 
entered the arena4 ,as a  result of an  
analysis of the physical conceptions of 
time and space1 it  became evident that 
in reality there is not the least       
incompatibilitiy  between the principle 
of relativity and the law of propagation
 of  light1 and that by systematically  
holding fast to both these laws a       
logically rigid theory could be arrived 
at4 ,this theory has been called  the   
special theory of relativity to         
distinguish it from the extended        
theory1 with which we shall deal later4 
,in the following pages we shall        
present the fundamental ideas of the    
special theory of relativity4           
,,viii4    ,,on ,,the ,,idea ,,of ,,time
 ,,in ,,physics    ,lightning has struck
 the rails on our railway embankment at 

two places  ,a and ,b far distant from  
each other4 ,i make the additional      
assertion that these two lightning      
flashes occurred simultaneously4 ,if  ,i
 ask you whether there is sense in this 
statement1 you will answer my  question 
with a decided 8,yes40 ,but if ,i now   
approach you with the  request to       
explain to me the sense of the statement
 more precisely1 you  find after some   
consideration that the answer to this   
question is not  so easy as it appears  
at first sight4    ,after some time     
perhaps the following answer would occur
 to you3 8,the  significance of the     
statement is clear in itself and needs  
no further  explanation2 of course it   
would require some consideration if ,i  
were to  be commissioned to determine by
 observations whether in the actual case
  the two events took place             
simultaneously or not40 ,i cannot be    
satisfied  with this answer for the     
following reason4 ,supposing that as a  
result  of ingenious considerations an  

able meteorologist were to discover that
  the lightning must always strike the  
places ,a and ,b simultaneously1  then  
we should be faced with the task of     
testing whether or not this  theoretical
 result is in accordance with the       
reality4 ,we encounter the  same        
difficulty with all physical statements 
in which the conception  8simultaneous0 
plays a part4 ,the concept does not     
exist for the  physicist until he has   
the possibility of discovering whether  
or not it  is fulfilled in an actual    
case4 ,we thus require a definition of  
simultaneity such that this definition  
supplies us with the method by  means of
 which1 in the present case1 he can     
decide by experiment  whether or not    
both the lightning strokes occurred     
simultaneously4 ,as  long as this       
requirement is not satisfied1 ,i allow  
myself to be  deceived as a physicist   
7and of course the same applies if ,i am
 not a  physicist71 when ,i imagine that
 ,i am able to attach a meaning to the  

statement of simultaneity4 7,i would ask
 the reader not to proceed  farther     
until he is fully convinced on this     
point47    ,after thinking the matter   
over for some time you then offer the   
following suggestion with which to test 
simultaneity4 ,by measuring  along the  
rails1 the connecting line ,,ab should  
be measured up and an  observer placed  
at the mid-point ,m of the distance     
,,ab4 ,this observer  should be supplied
 with an arrangement 7e4g4 two mirrors  
inclined at  #ij7 which allows him      
visually to observe both places ,a and  
,b at  the same time4 ,if the observer  
perceives the two flashes of lightning  
at the same time1 then they are         
simultaneous4    ,i am very pleased with
 this suggestion1 but for all that ,i   
cannot  regard the matter as quite      
settled1 because ,i feel constrained to 
raise  the following objection3 8,your  
definition would certainly be right1 if 
only ,i knew that the light by means of 
which the observer at ,m  perceives the 

lightning flashes travels along the     
length ,a  ,m with  the same velocity as
 along the length ,b  ,m4 ,but an       
examination of  this supposition would  
only be possible if we already had at   
our  disposal the means of measuring    
time4 ,it would thus appear as though we
  were moving here in a logical circle40
    ,after further consideration you    
cast a somewhat disdainful glance at    
me--and rightly so--and you declare3 8,i
 maintain my previous definition        
nevertheless1 because in reality it     
assumes absolutely nothing about  light4
 ,there is only one demand to be made of
 the definition of  simultaneity1       
namely1 that in every real case it must 
supply us with an  empirical decision as
 to whether or not the conception that  
has to be  defined is fulfilled4 ,that  
my definition satisfies this demand is  
indisputable4 ,that light requires the  
same time to traverse the path  ,a  ,m  
as for the path ,b  ,m is in reality    
neither a  supposition nor a hypothesis 

about the physical nature of light1 but 
a stipulation which ,i can make of my   
own freewill in order to arrive  at a   
definition of simultaneity40    ,it is  
clear that this definition can be used  
to give an exact meaning  not only to   
two events1 but to as many events as we 
care to choose1  and independently of   
the positions of the scenes of the      
events with  respect to the body of     
reference#g 7here the railway           
embankment74 ,we  are thus led also to a
 definition of 8time0 in physics4 ,for  
this  purpose we suppose that clocks of 
identical construction are placed at    
the points ,a1 ,b and ,c of the railway 
line 7co-ordinate system7 and  that they
 are set in such a manner that the      
positions of their pointers  are        
simultaneously 7in the above sense7 the 
same4 ,under these  conditions we       
understand by the 8time0 of an event the
 reading  7position of the hands7 of    
that one of these clocks which is in the
  immediate vicinity 7in space7 of the  

event4 ,in this manner a time-value  is 
associated with every event which is    
essentially capable of  observation4    
#g ,we suppose further that1 when three 
events ,a1 ,b and ,c occur in           
different places in such a manner that1 
if ,a is simultaneous with   ,b1 and ,b 
is simultaneous with ,c 7simultaneous in
 the sense of   the above definition71  
then the criterion for the simultaneity 
of the   pair of events ,a1 ,c is also  
satisfied4 ,this assumption is a        
physical   hypothesis about the law of  
propagation of light2 it must certainly 
be   fulfilled if we are to maintain the
 law of the constancy of the   velocity 
of light in vacuo4      ,this           
stipulation contains a further physical 
hypothesis1 the validity  of which will 
hardly be doubted without empirical     
evidence to the  contrary4 ,it has been 
assumed that all these clocks go at the 
same  rate if they are of identical     
construction4 ,stated more exactly3     
,when  two clocks arranged at rest in   

different places of a reference-body are
  set in such a manner that a particular
 position of the pointers of  the one   
clock is simultaneous 7in the above     
sense7 with the same  position1 of the  
pointers of the other clock1 then       
identical 8settings0  are always        
simultaneous 7in the sense of the above 
definition74      ,,ix4    ,,the        
,,relativity ,,of ,,simultaneity    ,up 
to now our considerations have been     
referred to a particular body of        
reference1 which we have styled a       
8railway embankment40 ,we suppose a     
very long train travelling along the    
rails with the constant velocity v  and 
in the direction indicated in ,fig #a4  
,people travelling in this  train will  
with a vantage view the train as a rigid
 reference-body  7co-ordinate system72  
they regard all events in reference to  
the train4  ,then every event which     
takes place along the line also takes   
place at a  particular point of the     
train4 ,also the definition of          

simultaneity can  be given relative to  
the train in exactly the same way as    
with respect  to the embankment4 ,as a  
natural consequence1 however1 the       
following  question arises3    image#jja
      ,are two events 7e4g4 the two     
strokes of lightning ,a and ,b7 which   
are simultaneous with reference to the  
railway embankment also  simultaneous   
relatively to the train8 ,we shall show 
directly that the  answer must be in the
 negative4    ,when we say that the     
lightning strokes ,a and ,b are         
simultaneous  with respect to be        
embankment1 we mean3 the rays of light  
emitted at  the places ,a and ,b1 where 
the lightning occurs1 meet each other at
  the mid-point ,m of the length ,a  ,b 
of the embankment4 ,but the  events ,a  
and ,b also correspond to positions ,a  
and ,b on the  train4 ,let ,m be the    
mid-point of the distance ,a  ,b on the 
travelling train4 ,just when the flashes
 7as judged from the embankment7  of    
lightning occur1 this point ,m naturally

 coincides with the point  ,m but it    
moves towards the right in the diagram  
with the velocity v  of the train4 ,if  
an observer sitting in the position ,m  
in the train  did not possess this      
velocity1 then he would remain          
permanently at ,m1  and the light rays  
emitted by the flashes of lightning ,a  
and ,b  would reach him simultaneously1 
i4e4 they would meet just where he is   
situated4 ,now in reality 7considered   
with reference to the railway           
embankment7 he is hastening towards the 
beam of light coming from ,b1  whilst he
 is riding on ahead of the beam of light
 coming from ,a4  ,hence the observer   
will see the beam of light emitted from 
,b earlier  than he will see that       
emitted from ,a4 ,observers who take the
 railway  train as their reference-body 
must therefore come to the conclusion   
that the lightning flash ,b took place  
earlier than the lightning  flash ,a4   
,we thus arrive at the important result3
    ,events which are simultaneous with 

reference to the embankment are not     
simultaneous with respect to the train1 
and vice versa 7relativity of           
simultaneity74 ,every reference-body    
7co-ordinate system7 has its own        
particular time2 unless we are told the 
reference-body to which the  statement  
of time refers1 there is no meaning in a
 statement of the  time of an event4    
,now before the advent of the theory of 
relativity it had always tacitly  been  
assumed in physics that the statement of
 time had an absolute  significance1    
i4e4 that it is independent of the state
 of motion of  the body of reference4   
,but we have just seen that this        
assumption is  incompatible with the    
most natural definition of simultaneity2
 if we  discard this assumption1 then   
the conflict between the law of the     
propagation of light in vacuo and the   
principle of relativity  7developed in  
,section ,,vii7 disappears4    ,we were 
led to that conflict by the             
considerations of ,section ,,vi1 which  

are now no longer tenable4 ,in that     
section we concluded that the man in    
the carriage1 who traverses the distance
 w per second relative to the  carriage1
 traverses the same distance also with  
respect to the  embankment in each      
second of time4 ,but1 according to the  
foregoing  considerations1 the time     
required by a particular occurrence with
  respect to the carriage must not be   
considered equal to the duration of  the
 same occurrence as judged from the     
embankment 7as reference-body74  ,hence 
it cannot be contended that the man in  
walking travels the  distance w relative
 to the railway line in a time which is 
equal to  one second as judged from the 
embankment4    ,moreover1 the           
considerations of ,section ,,vi are     
based on yet a second  assumption1      
which1 in the light of a strict         
consideration1 appears to  be arbitrary1
 although it was always tacitly made    
even before the  introduction of the    
theory of relativity4      ,x4    ,,on  

,,the ,,relativity ,,of ,,the           
,,conception ,,of ,,distance    ,let us 
consider two particular points on the   
train #h travelling along  the          
embankment with the velocity v1 and     
inquire as to their distance  apart4 ,we
 already know that it is necessary to   
have a body of reference  for the       
measurement of a distance1 with respect 
to which body the  distance can be      
measured up4 ,it is the simplest plan to
 use the train  itself as reference-body
 7co-ordinate system74 ,an observer in  
the train  measures the interval by     
marking off his measuring-rod in a      
straight  line 7e4g4 along the floor of 
the carriage7 as many times as is       
necessary to take him from the one      
marked point to the other4 ,then the    
number which tells us how often the rod 
has to be laid down is the  required    
distance4       #h e4g4 the middle of   
the first and of the hundredth carriage4
      ,it is a different matter when the
 distance has to be judged from the     

railway line4 ,here the following method
 suggests itself4 ,if we call  ,a and ,b
 the two points on the train whose      
distance apart is  required1 then both  
of these points are moving with the     
velocity v  along the embankment4 ,in   
the first place we require to determine 
the  points ,a and ,b of the embankment 
which are just being passed by the  two 
points ,a and ,b at a particular time   
t--judged from the  embankment4 ,these  
points ,a and ,b of the embankment can  
be  determined by applying the          
definition of time given in ,section    
,,viii4  ,the distance between these    
points ,a and ,b is then measured by    
repeated  application of the            
measuring-rod along the embankment4     
,a priori it is by no means certain that
 this last measurement will  supply us  
with the same result as the first4 ,thus
 the length of the  train as measured   
from the embankment may be different    
from that  obtained by measuring in the 
train itself4 ,this circumstance leads  

us  to a second objection which must be 
raised against the apparently  obvious  
consideration of ,section ,,vi4 ,namely1
 if the man in the carriage  covers the 
distance w in a unit of time--measured  
from the  train1--then this distance--as
 measured from the embankment is not    
necessarily also equal to w4      ,,xi4 
,,the ,,lorentz ,,transformation    ,the
 results of the last three sections show
 that the apparent  incompatibility of  
the law of propagation of light with the
 principle  of relativity 7,section     
,,vii7 has been derived by means of a   
consideration which borrowed two        
unjustifiable hypotheses from  classical
 mechanics2 these are as follows3       
7#a7 ,the time-interval 7time7 between  
two events is independent of the        
condition of motion of the body of      
reference4      7#b7 ,the space-interval
 7distance7 between two points of a     
rigid body is  independent of the       
condition of motion of the body of      
reference4      ,if we drop these       

hypotheses1 then the dilemma of ,section
 ,,vii  disappears1 because the theorem 
of the addition of velocities derived   
in ,section ,,vi becomes invalid4 ,the  
possibility presents itself that the    
law of the propagation of light in vacuo
 may be compatible with the  principle  
of relativity1 and the question arises3 
,how have we to modify  the             
considerations of ,section ,,vi in order
 to remove the apparent  disagreement   
between these two fundamental results of
 experience8 ,this  question leads to a 
general one4 ,in the discussion of      
,section ,,vi we  have to do with places
 and times relative both to the train   
and to the  embankment4 ,how are we to  
find the place and time of an event in  
relation to the train1 when we know the 
place and time of the event  with       
respect to the railway embankment8 ,is  
there a thinkable answer to  this       
question of such a nature that the law  
of transmission of light  in vacuo does 
not contradict the principle of         

relativity8 ,in other  words3 ,can we   
conceive of a relation between place and
 time of the  individual events relative
 to both reference-bodies1 such that    
every  ray of light possesses the       
velocity of transmission c relative to  
the  embankment and relative to the     
train8 ,this question leads to a quite  
definite positive answer1 and to a      
perfectly definite transformation  law  
for the space-time magnitudes of an     
event when changing over from  one body 
of reference to another4    ,before we  
deal with this1 we shall introduce the  
following incidental  consideration4 ,up
 to the present we have only considered 
events taking  place along the          
embankment1 which had mathematically to 
assume the  function of a straight line4
 ,in the manner indicated in ,section   
,,ii we  can imagine this reference-body
 supplemented laterally and in a        
vertical direction by means of a        
framework of rods1 so that an event     
which takes place anywhere can be       

localised with reference to this        
framework4 ,similarly1 we can imagine   
the train travelling with the  velocity 
v to be continued across the whole of   
space1 so that every  event1 no matter  
how far off it may be1 could also be    
localised with  respect to the second   
framework4 ,without committing any      
fundamental  error1 we can disregard the
 fact that in reality these frameworks  
would  continually interfere with each  
other1 owing to the impenetrability of  
solid bodies4 ,in every such framework  
we imagine three surfaces  perpendicular
 to each other marked out1 and          
designated as 8co-ordinate  planes0     
78co-ordinate system074 ,a co-ordinate  
system ,k then  corresponds to the      
embankment1 and a co-ordinate system ,k 
to the  train4 ,an event1 wherever it   
may have taken place1 would be fixed in 
space with respect to ,k by the three   
perpendiculars x1 y1 z on the           
co-ordinate planes1 and with regard to  
time by a time value t4  ,relative to   

,k1 the same event would be fixed in    
respect of space and  time by           
corresponding values x1 y1 z1 t1 which  
of course are not  identical with x1 y1 
z1 t4 ,it has already been set forth in 
detail  how these magnitudes are to be  
regarded as results of physical         
measurements4    image#jjb              
,obviously our problem can be exactly   
formulated in the following  manner4    
,what are the values x1 y1 z1 t1 of an  
event with respect  to ,k1 when the     
magnitudes x1 y1 z1 t1 of the same event
 with  respect to ,k are given8 ,the    
relations must be so chosen that the law
  of the transmission of light in vacuo 
is satisfied for one and the same  ray  
of light 7and of course for every ray7  
with respect to ,k and  ,k4 ,for the    
relative orientation in space of the    
co-ordinate systems  indicated in the   
diagram 7,fig4 #b71 this problem is     
solved by means of  the equations3      
image#jjc      y  y    z  z             
image#jjd      ,this system of equations

 is known as the 8,lorentz              
transformation40#i       #i ,a simple   
derivation of the ,lorentz              
transformation is given in   ,appendix  
,i4      ,if in place of the law of     
transmission of light we had taken as   
our  basis the tacit assumptions of the 
older mechanics as to the absolute      
character of times and lengths1 then    
instead of the above we should  have    
obtained the following equations3    x  
x - vt      y  y      z  z      t  t    
,this system of equations is often      
termed the 8,galilei transformation40   
,the ,galilei transformation can be     
obtained from the ,lorentz              
transformation by substituting an       
infinitely large value for the  velocity
 of light c in the latter               
transformation4    ,aided by the        
following illustration1 we can readily  
see that1 in  accordance with the       
,lorentz transformation1 the law of the 
transmission  of light in vacuo is      
satisfied both for the reference-body ,k

 and  for the reference-body ,k4 ,a     
light-signal is sent along the positive 
x-axis1 and this light-stimulus advances
 in accordance with the  equation    x  
ct1      i4e4 with the velocity c4      
,according to the equations of the      
,lorentz  transformation1 this simple   
relation between x and t involves a     
relation between x and t4 ,in point of  
fact1 if we substitute for  x the value 
ct in the first and fourth equations of 
the ,lorentz  transformation1 we obtain3
    image#jje      from which1 by       
division1 the expression    x  ct       
immediately follows4 ,if referred to the
 system ,k1 the propagation of  light   
takes place according to this equation4 
,we thus see that the  velocity of      
transmission relative to the            
reference-body ,k is also  equal to c4  
,the same result is obtained for rays of
 light advancing  in any other direction
 whatsoever4 ,of cause this is not      
surprising1  since the equations of the 
,lorentz transformation were derived    

conformably to this point of view4      
,,xii4    ,,the ,,behaviour ,,of        
,,measuring-,,rods ,,and ,,clocks ,,in  
,,motion    ,place a metre-rod in the   
x-axis of ,k in such a manner that one  
end 7the beginning7 coincides with the  
point x  #j whilst the other  end 7the  
end of the rod7 coincides with the point
 x  #a4 ,what is the  length of the     
metre-rod relatively to the system ,k8  
,in order to learn  this1 we need only  
ask where the beginning of the rod and  
the end of  the rod lie with respect to 
,k at a particular time t of the system 
,k4 ,by means of the first equation of  
the ,lorentz transformation the  values 
of these two points at the time t  #j   
can be shown to be    image#jjf      the
 distance between the points being      
image#jjg      ,but the metre-rod is    
moving with the velocity v relative to  
,k4 ,it  therefore follows that the     
length of a rigid metre-rod moving in   
the  direction of its length with a     
velocity v is    image#jjh      of a    

metre4 ,the rigid rod is thus shorter   
when in motion than when at  rest1 and  
the more quickly it is moving1 the      
shorter is the rod4 ,for  the velocity v
  c we should have    image#jji      and
 for still greater velocities the       
square-root becomes imaginary4  ,from   
this we conclude that in the theory of  
relativity the velocity c  plays the    
part of a limiting velocity1 which can  
neither be reached nor  exceeded by any 
real body4    ,of course this feature of
 the velocity c as a limiting velocity  
also  clearly follows from the equations
 of the ,lorentz transformation1 for    
these became meaningless if we choose   
values of v greater than c4    ,if1 on  
the contrary1 we had considered a       
metre-rod at rest in the  x-axis with   
respect to ,k1 then we should have found
 that the length  of the rod as judged  
from ,k would have been    image#jaj    
this is quite in accordance with the    
principle of relativity which  forms the
 basis of our considerations4    ,a     

priori it is quite clear that we must be
 able to learn something  about the     
physical behaviour of measuring-rods and
 clocks from the  equations of          
transformation1 for the magnitudes z1 y1
 x1 t1 are  nothing more nor less than  
the results of measurements obtainable  
by  means of measuring-rods and clocks4 
,if we had based our considerations  on 
the ,galileian transformation we should 
not have obtained a  contraction of the 
rod as a consequence of its motion4     
,let us now consider a seconds-clock    
which is permanently situated at  the   
origin 7x  #j7 of ,k4 t  #j and t  #a   
are two successive  ticks of this clock4
 ,the first and fourth equations of the 
,lorentz  transformation give for these 
two ticks3    t  #j    and    image#jaa 
,as judged from ,k1 the clock is moving 
with the velocity v2 as  judged from    
this reference-body1 the time which     
elapses between two  strokes of the     
clock is not one second1 but            
image#jab      seconds1 i4e4 a somewhat 

larger time4 ,as a consequence of its   
motion  the clock goes more slowly than 
when at rest4 ,here also the velocity  c
 plays the part of an unattainable      
limiting velocity4      ,,xiii4         
,,theorem ,,of ,,the ,,addition ,,of    
,,velocities4 ,,the ,,experiment ,,of   
,,fizeau    ,now in practice we can move
 clocks and measuring-rods only with    
velocities that are small compared with 
the velocity of light2 hence we  shall  
hardly be able to compare the results of
 the previous section  directly with the
 reality4 ,but1 on the other hand1 these
 results must  strike you as being very 
singular1 and for that reason ,i shall  
now draw  another conclusion from the   
theory1 one which can easily be derived 
from the foregoing considerations1 and  
which has been most elegantly  confirmed
 by experiment4    ,in ,section ,,vi we 
derived the theorem of the addition of  
velocities in  one direction in the form
 which also results from the hypotheses 
of  classical mechanics4 ,this theorem  

can also be deduced readily from the    
,galilei transformation 7,section ,,xi74
 ,in place of the man walking inside    
the carriage1 we introduce a point      
moving relatively to the co-ordinate    
system ,k in accordance with the        
equation    x  wt    ,by means of the   
first and fourth equations of the       
,galilei  transformation we can express 
x and t in terms of x and t1  and we    
then obtain    x  7v  w7t    ,this      
equation expresses nothing else than the
 law of motion of the  point with       
reference to the system ,k 7of the man  
with reference to  the embankment74 ,we 
denote this velocity by the symbol ,w1  
and we then  obtain1 as in ,section     
,,vi1    ,w  v  w 4 4 4 4 4 4 4 7,a74   
,but we can carry out this consideration
 just as well on the basis of  the      
theory of relativity4 ,in the equation  
x  wt    we must then express x and t in
 terms of x and t1 making use  of the   
first and fourth equations of the       
,lorentz transformation4  ,instead of   

the equation 7,a7 we then obtain the    
equation    image#jac      which        
corresponds to the theorem of addition  
for velocities in one  direction        
according to the theory of relativity4  
,the question now  arises as to which of
 these two theorems is the better in    
accord with  experience4 ,on this point 
we are enlightened by a most important  
experiment which the brilliant physicist
 ,fizeau performed more than  half a    
century ago1 and which has been repeated
 since then by some of  the best        
experimental physicists1 so that there  
can be no doubt about  its result4 ,the 
experiment is concerned with the        
following question4  ,light travels in a
 motionless liquid with a particular    
velocity w4  ,how quickly does it travel
 in the direction of the arrow in the   
tube  ,t 7see the accompanying diagram1 
,fig4 #c7 when the liquid above         
mentioned is flowing through the tube   
with a velocity v8    image#jad      ,in
 accordance with the principle of       

relativity we shall certainly have  to  
take for granted that the propagation of
 light always takes place  with the same
 velocity w with respect to the liquid1 
whether the  latter is in motion with   
reference to other bodies or not4 ,the  
velocity  of light relative to the      
liquid and the velocity of the latter   
relative  to the tube are thus known1   
and we require the velocity of light    
relative to the tube4    ,it is clear   
that we have the problem of ,section    
,,vi again before us4 ,the  tube plays  
the part of the railway embankment or of
 the co-ordinate  system ,k1 the liquid 
plays the part of the carriage or of the
  co-ordinate system ,k1 and finally1   
the light plays the part of the  man    
walking along the carriage1 or of the   
moving point in the present  section4   
,if we denote the velocity of the light 
relative to the tube by  ,w1 then this  
is given by the equation 7,a7 or 7,b71  
according as the  ,galilei              
transformation or the ,lorentz          

transformation corresponds to the       
facts4 ,experiment#aj decides in favour 
of equation 7,b7 derived from  the      
theory of relativity1 and the agreement 
is1 indeed1 very exact4  ,according to  
recent and most excellent measurements  
by ,zeeman1 the  influence of the       
velocity of flow v on the propagation of
 light is  represented by formula 7,b7  
to within one per cent4       #aj       
,fizeau found      image#jae      where 
image#jaf      is the index of          
refraction of the liquid4 ,on the other 
hand1 owing to  the smallness of        
image#jag      as compared with #a1 we  
can replace 7,b7 in the first place by  
image#jah      or to the same order of  
approximation by      image#jai         
which agrees with ,fizeau's result4     
,nevertheless we must now draw attention
 to the fact that a theory of  this     
phenomenon was given by ,h4 ,a4 ,lorentz
 long before the statement of  the      
theory of relativity4 ,this theory was  
of a purely electrodynamical  nature1   

and was obtained by the use of          
particular hypotheses as to the         
electromagnetic structure of matter4    
,this circumstance1 however1 does  not  
in the least diminish the conclusiveness
 of the experiment as a  crucial test in
 favour of the theory of relativity1 for
 the  electrodynamics of                
,maxwell-,lorentz1 on which the original
 theory was  based1 in no way opposes   
the theory of relativity4 ,rather has   
the  latter been developed trom         
electrodynamics as an astoundingly      
simple  combination and generalisation  
of the hypotheses1 formerly independent 
of each other1 on which electrodynamics 
was built4      ,,xiv4    ,,the         
,,heuristic ,,value ,,of ,,the ,,theory 
,,of ,,relativity    ,our train of      
thought in the foregoing pages can be   
epitomised in the  following manner4    
,experience has led to the conviction   
that1 on the one  hand1 the principle of
 relativity holds true and that on the  
other hand  the velocity of transmission

 of light in vacuo has to be considered 
equal to a constant c4 ,by uniting these
 two postulates we obtained  the law of 
transformation for the rectangular      
co-ordinates x1 y1 z  and the time t of 
the events which constitute the         
processes of  nature4 ,in this          
connection we did not obtain the        
,galilei  transformation1 but1 differing
 from classical mechanics1 the ,lorentz 
transformation4    ,the law of          
transmission of light1 the acceptance of
 which is justified  by our actual      
knowledge1 played an important part in  
this process of  thought4 ,once in      
possession of the ,lorentz              
transformation1 however1 we  can combine
 this with the principle of relativity1 
and sum up the  theory thus3    ,every  
general law of nature must be so        
constituted that it is  transformed into
 a law of exactly the same form when1   
instead of the  space-time variables x1 
y1 z1 t of the original coordinate      
system  ,k1 we introduce new space-time 

variables x1 y1 z1 t of a  co-ordinate  
system ,k4 ,in this connection the      
relation between the  ordinary and the  
accented magnitudes is given by the     
,lorentz  transformation4 ,or in brief3 
,general laws of nature are co-variant  
with  respect to ,lorentz               
transformations4    ,this is a definite 
mathematical condition that the theory  
of relativity  demands of a natural law1
 and in virtue of this1 the theory      
becomes a  valuable heuristic aid in the
 search for general laws of nature4 ,if 
a  general law of nature were to be     
found which did not satisfy this        
condition1 then at least one of the two 
fundamental assumptions of the  theory  
would have been disproved4 ,let us now  
examine what general  results the latter
 theory has hitherto evinced4      ,,xv4
    ,,general ,,results ,,of ,,the      
,,theory    ,it is clear from our       
previous considerations that the        
7special7 theory  of relativity has     
grown out of electrodynamics and optics4

 ,in these  fields it has not           
appreciably altered the predictions of  
theory1 but it  has considerably        
simplified the theoretical structure1   
i4e4 the  derivation of laws1 and--what 
is incomparably more important--it has  
considerably reduced the number of      
independent hypotheses forming the      
basis of theory4 ,the special theory of 
relativity has rendered the             
,maxwell-,lorentz theory so plausible1  
that the latter would have been         
generally accepted by physicists even if
 experiment had decided less            
unequivocally in its favour4            
,classical mechanics required to be     
modified before it could come into  line
 with the demands of the special theory 
of relativity4 ,for the main  part1     
however1 this modification affects only 
the laws for rapid  motions1 in which   
the velocities of matter v are not very 
small as  compared with the velocity of 
light4 ,we have experience of such rapid
  motions only in the case of electrons 

and ions2 for other motions the         
variations from the laws of classical   
mechanics are too small to make         
themselves evident in practice4 ,we     
shall not consider the motion of  stars 
until we come to speak of the general   
theory of relativity4 ,in  accordance   
with the theory of relativity the       
kinetic energy of a  material point of  
mass m is no longer given by the        
well-known  expression    image#jbj     
but by the expression    image#jba      
,this expression approaches infinity as 
the velocity v approaches the  velocity 
of light c4 ,the velocity must therefore
 always remain less  than c1 however    
great may be the energies used to       
produce the  acceleration4 ,if we       
develop the expression for the kinetic  
energy in  the form of a series1 we     
obtain    image#jbb      ,when          
image#jbc      is small compared with   
unity1 the third of these terms is      
always small  in comparison with the    
second1 which last is alone considered  

in  classical mechanics4 ,the first term
 mc#b does not contain the  velocity1   
and requires no consideration if we are 
only dealing with the  question as to   
how the energy of a point-mass2 depends 
on the velocity4  ,we shall speak of its
 essential significance later4    ,the  
most important result of a general      
character to which the special  theory  
of relativity has led is concerned with 
the conception of mass4  ,before the    
advent of relativity1 physics recognised
 two conservation  laws of fundamental  
importance1 namely1 the law of the      
conservation of  energy and the law of  
the conservation of mass these two      
fundamental  laws appeared to be quite  
independent of each other4 ,by means of 
the  theory of relativity they have been
 united into one law4 ,we shall now     
briefly consider how this unification   
came about1 and what meaning is  to be  
attached to it4    ,the principle of    
relativity requires that the law of the 
conservation  of energy should hold not 

only with reference to a co-ordinate    
system  ,k1 but also with respect to    
every co-ordinate system ,k which is in 
a state of uniform motion of translation
 relative to ,k1 or1 briefly1  relative 
to every 8,galileian0 system of         
co-ordinates4 ,in contrast to  classical
 mechanics2 the ,lorentz transformation 
is the deciding factor  in the          
transition from one such system to      
another4    ,by means of comparatively  
simple considerations we are led to draw
 the  following conclusion from these   
premises1 in conjunction with the       
fundamental equations of the            
electrodynamics of ,maxwell3 ,a body    
moving  with the velocity v1 which      
absorbs#aa an amount of energy ,e#j in  
the form of radiation without suffering 
an alteration in velocity in  the       
process1 has1 as a consequence1 its     
energy increased by an amount           
image#jbd         #aa ,e#j is the energy
 taken up1 as judged from a co-ordinate 
system   moving with the body4      ,in 

consideration of the expression given   
above for the kinetic energy  of the    
body1 the required energy of the body   
comes out to be    image#jbe      ,thus 
the body has the same energy as a body  
of mass    image#jbf      moving with   
the velocity v4 ,hence we can say3 ,if a
 body takes up an  amount of energy     
,e#j1 then its inertial mass increases  
by an amount    image#jbg      the      
inertial mass of a body is not a        
constant but varies according to  the   
change in the energy of the body4 ,the  
inertial mass of a system of  bodies can
 even be regarded as a measure of its   
energy4 ,the law of the  conservation of
 the mass of a system becomes identical 
with the law of  the conservation of    
energy1 and is only valid provided that 
the system  neither takes up nor sends  
out energy4 ,writing the expression for 
the  energy in the form    image#jbh    
we see that the term mc#b1 which has    
hitherto attracted our attention1  is   
nothing else than the energy possessed  

by the body#ab before it  absorbed the  
energy ,e#j4       #ab ,as judged from a
 co-ordinate system moving with the     
body4      ,a direct comparison of this 
relation with experiment is not possible
 at  the present time 7#aibj2 see,note1 
p4 #dh71 owing to the fact that the     
changes in energy ,e#j to which we can  
subject a system are not large  enough  
to make themselves perceptible as a     
change in the inertial mass  of the     
system4    image#jbg      is too small  
in comparison with the mass m1 which was
 present before  the alteration of the  
energy4 ,it is owing to this            
circumstance that  classical mechanics  
was able to establish successfully the  
conservation  of mass as a law of       
independent validity4       ,note ,the  
equation ,e  mc#b has been thoroughly   
proved time and again   since this time4
      ,let me add a final remark of a   
fundamental nature4 ,the success of the 
,faraday-,maxwell interpretation of     
electromagnetic action at a distance    

resulted in physicists becoming         
convinced that there are no such things 
as instantaneous actions at a distance  
7not involving an intermediary  medium7 
of the type of ,newton's law of         
gravitation4    ,according to the theory
 of relativity1 action at a distance    
with the  velocity of light always takes
 the place of instantaneous action at a 
distance or of action at a distance with
 an infinite velocity of  transmission4 
,this is connected with the fact that   
the velocity c  plays a fundamental role
 in this theory4 ,in ,part ,,ii we shall
 see in  what way this result becomes   
modified in the general theory of       
relativity4      ,,xvi4    ,,experience 
,,and ,,the ,,special ,,theory ,,of     
,,relativity    ,to what extent is the  
special theory of relativity supported  
by  experience8 ,this question is not   
easily answered for the reason already  
mentioned in connection with the        
fundamental experiment of ,fizeau4 ,the 
special theory of relativity has        

crystallised out from the               
,maxwell-,lorentz theory of             
electromagnetic phenomena4 ,thus all    
facts of  experience which support the  
electromagnetic theory also support the 
theory of relativity4 ,as being of      
particular importance1 ,i mention here  
the fact that the theory of relativity  
enables us to predict the  effects      
produced on the light reaching us from  
the fixed stars4 ,these  results are    
obtained in an exceedingly simple       
manner1 and the effects  indicated1     
which are due to the relative motion of 
the earth with  reference to those fixed
 stars are found to be in accord with   
experience4 ,we refer to the yearly     
movement of the apparent position of    
the fixed stars resulting from the      
motion of the earth round the sun       
7aberration71 and to the influence of   
the radial components of the  relative  
motions of the fixed stars with respect 
to the earth on the  colour of the light
 reaching us from them4 ,the latter     

effect manifests  itself in a slight    
displacement of the spectral lines of   
the light  transmitted to us from a     
fixed star1 as compared with the        
position of  the same spectral lines    
when they are produced by a terrestrial 
source  of light 7,doppler principle74  
,the experimental arguments in favour of
  the ,maxwell-,lorentz theory1 which   
are at the same time arguments in       
favour of the theory of relativity1 are 
too numerous to be set forth  here4 ,in 
reality they limit the theoretical      
possibilities to such an  extent1 that  
no other theory than that of ,maxwell   
and ,lorentz has been  able to hold its 
own when tested by experience4    ,but  
there are two classes of experimental   
facts hitherto obtained which  can be   
represented in the ,maxwell-,lorentz    
theory only by the  introduction of an  
auxiliary hypothesis1 which in          
itself--i4e4 without  making use of the 
theory of relativity--appears           
extraneous4    ,it is known that cathode

 rays and the so-called -rays emitted by
  radioactive substances consist of     
negatively electrified particles        
7electrons7 of very small inertia and   
large velocity4 ,by examining the       
deflection of these rays under the      
influence of electric and magnetic      
fields1 we can study the law of motion  
of these particles very exactly4    ,in 
the theoretical treatment of these      
electrons1 we are faced with the        
difficulty that electrodynamic theory of
 itself is unable to give an  account of
 their nature4 ,for since electrical    
masses of one sign repel  each other1   
the negative electrical masses          
constituting the electron  would        
necessarily be scattered under the      
influence of their mutual  repulsions1  
unless there are forces of another kind 
operating between  them1 the nature of  
which has hitherto remained obscure to  
us4#ac ,if  we now assume that the      
relative distances between the          
electrical masses  constituting the     

electron remain unchanged during the    
motion of the  electron 7rigid          
connection in the sense of classical    
mechanics71 we  arrive at a law of      
motion of the electron which does not   
agree with  experience4 ,guided by      
purely formal points of view1 ,h4 ,a4   
,lorentz was  the first to introduce the
 hypothesis that the form of the        
electron  experiences a contraction in  
the direction of motion in consequence  
of  that motion4 the contracted length  
being proportional to the expression    
image#jbi      ,this1 hypothesis1 which 
is not justifiable by any               
electrodynamical  facts1 supplies us    
then with that particular law of motion 
which has  been confirmed with great    
precision in recent years4       #ac    
,the general theory of relativity       
renders it likely that the   electrical 
masses of an electron are held together 
by gravitational   forces4      ,the    
theory of relativity leads to the same  
law of motion1 without  requiring any   

special hypothesis whatsoever as to the 
structure and the  behaviour of the     
electron4 ,we arrived at a similar      
conclusion in  ,section ,,xiii in       
connection with the experiment of       
,fizeau1 the result of  which is        
foretold by the theory of relativity    
without the necessity of  drawing on    
hypotheses as to the physical nature of 
the liquid4    ,the second class of     
facts to which we have alluded has      
reference to the  question whether or   
not the motion of the earth in space can
 be made  perceptible in terrestrial    
experiments4 ,we have already remarked  
in  ,section ,v that all attempts of    
this nature led to a negative result4   
,before the theory of relativity was put
 forward1 it was difficult to  become   
reconciled to this negative result1 for 
reasons now to be  discussed4 ,the      
inherited prejudices about time and     
space did not allow  any doubt to arise 
as to the prime importance of the       
,galileian  transformation for changing 

over from one body of reference to      
another4  ,now assuming that the        
,maxwell-,lorentz equations hold for a  
reference-body ,k1 we then find that    
they do not hold for a  reference-body  
,k moving uniformly with respect to ,k1 
if we assume  that the relations of the 
,galileian transformation exist between 
the  co-ordinates of ,k and ,k4 ,it thus
 appears that1 of all ,galileian        
co-ordinate systems1 one 7,k7           
corresponding to a particular state of  
motion is physically unique4 ,this      
result was interpreted physically by    
regarding ,k as at rest with respect to 
a hypothetical ther of space4  ,on the  
other hand1 all coordinate systems ,k   
moving relatively to ,k  were to be     
regarded as in motion with respect to   
the ther4 ,to this  motion of ,k against
 the ther 78ther-drift0 relative to ,k7 
were  attributed the more complicated   
laws which were supposed to hold        
relative to ,k4 ,strictly speaking1 such
 an ther-drift ought also to  be assumed

 relative to the earth1 and for a long  
time the efforts of  physicists were    
devoted to attempts to detect the       
existence of an  ther-drift at the      
earth's surface4    ,in one of the most 
notable of these attempts ,michelson    
devised a method  which appears as      
though it must be decisive4 ,imagine two
 mirrors so  arranged on a rigid body   
that the reflecting surfaces face each  
other4  ,a ray of light requires a      
perfectly definite time ,t to pass from 
one  mirror to the other and back again1
 if the whole system be at rest with    
respect to the ther4 ,it is found by    
calculation1 however1 that a  slightly  
different time ,t is required for this  
process1 if the body1  together with the
 mirrors1 be moving relatively to the   
ther4 ,and yet  another point3 it is    
shown by calculation that for a given   
velocity v  with reference to the ther1 
this time ,t is different when the body 
is moving perpendicularly to the planes 
of the mirrors from that  resulting when

 the motion is parallel to these planes4
 ,although the  estimated difference    
between these two times is exceedingly  
small1  ,michelson and ,morley performed
 an experiment involving interference in
  which this difference should have been
 clearly detectable4 ,but the           
experiment gave a negative result--a    
fact very perplexing to physicists4     
,lorentz and ,fitz,gerald rescued the   
theory from this difficulty by  assuming
 that the motion of the body relative to
 the ther produces a  contraction of the
 body in the direction of motion1 the   
amount of  contraction being just       
sufficient to compensate for the        
difference in  time mentioned above4    
,comparison with the discussion in      
,section ,,xii  shows that also from the
 standpoint of the theory of relativity 
this  solution of the difficulty was the
 right one4 ,but on the basis of the    
theory of relativity the method of      
interpretation is incomparably more     
satisfactory4 ,according to this theory 

there is no such thing as a  8specially 
favoured0 7unique7 co-ordinate system to
 occasion the  introduction of the      
ther-idea1 and hence there can be no    
ther-drift1  nor any experiment with    
which to demonstrate it4 ,here the      
contraction  of moving bodies follows   
from the two fundamental principles of  
the  theory1 without the introduction of
 particular hypotheses2 and as the      
prime factor involved in this           
contraction we find1 not the motion in  
itself1 to which we cannot attach any   
meaning1 but the motion with  respect to
 the body of reference chosen in the    
particular case in  point4 ,thus for a  
co-ordinate system moving with the earth
 the mirror  system of ,michelson and   
,morley is not shortened1 but it is     
shortened  for a co-ordinate system     
which is at rest relatively to the sun4 
,,xvii4    ,,minkowski',s               
,,four-,,dimensional ,,space    ,the    
non-mathematician is seized by a        
mysterious shuddering when he  hears of 

8four-dimensional0 things1 by a feeling 
not unlike that  awakened by thoughts of
 the occult4 ,and yet there is no more  
common-place statement than that the    
world in which we live is a             
four-dimensional space-time continuum4  
,space is a three-dimensional continuum4
 ,by this we mean that it is  possible  
to describe the position of a point 7at 
rest7 by means of  three numbers        
7co-ordinates7 x1 y1 z1 and that there  
is an indefinite  number of points in   
the neighbourhood of this one1 the      
position of  which can be described by  
co-ordinates such as x#a1 y#a1 z#a1     
which may  be as near as we choose to   
the respective values of the            
co-ordinates  x1 y1 z1 of the first     
point4 ,in virtue of the latter property
 we  speak of a 8continuum10 and owing  
to the fact that there are three        
co-ordinates we speak of it as being    
8three-dimensional40    ,similarly1 the 
world of physical phenomena which was   
briefly called  8world0 by ,minkowski is

 naturally four dimensional in the      
space-time  sense4 ,for it is composed  
of individual events1 each of which is  
described by four numbers1 namely1 three
 space co-ordinates x1 y1 z1  and a time
 co-ordinate1 the time value t4 ,the    
8world0 is in this  sense also a        
continuum2 for to every event there are 
as many  8neighbouring0 events 7realised
 or at least thinkable7 as we care to   
choose1 the co-ordinates x#a1 y#a1 z#a1 
t#a of which differ by an  indefinitely 
small amount from those of the event x1 
y1 z1 t  originally considered4 ,that we
 have not been accustomed to regard the 
world in this sense as a                
four-dimensional continuum is due to the
 fact  that in physics1 before the      
advent of the theory of relativity1 time
  played a different and more           
independent rle1 as compared with the   
space coordinates4 ,it is for this      
reason that we have been in the habit   
of treating time as an independent      
continuum4 ,as a matter of fact1        

according to classical mechanics1 time  
is absolute1 i4e4 it is  independent of 
the position and the condition of motion
 of the system  of co-ordinates4 ,we see
 this expressed in the last equation of 
the  ,galileian transformation 7t  t74  
,the four-dimensional mode of           
consideration of the 8world0 is natural 
on  the theory of relativity1 since     
according to this theory time is robbed 
of its independence4 ,this is shown by  
the fourth equation of the  ,lorentz    
transformation3    image#jcj            
,moreover1 according to this equation   
the time difference t of two  events    
with respect to ,k does not in general  
vanish1 even when the  time difference t
 of the same events with reference to ,k
 vanishes4  ,pure 8space-distance0 of   
two events with respect to ,k results in
  8time-distance 0 of the same events   
with respect to ,k4 ,but the  discovery 
of ,minkowski1 which was of importance  
for the formal  development of the      
theory of relativity1 does not lie here4

 ,it is to be  found rather in the fact 
of his recognition that the             
four-dimensional  space-time continuum  
of the theory of relativity1 in its most
 essential  formal properties1 shows a  
pronounced relationship to the          
three-dimensional continuum of          
,euclidean geometrical space4#ad ,in    
order to give due prominence to this    
relationship1 however1 we must  replace 
the usual time co-ordinate t by an      
imaginary magnitude    image#jca        
proportional to it4 ,under these        
conditions1 the natural laws satisfying 
the demands of the 7special7 theory of  
relativity assume mathematical  forms1  
in which the time co-ordinate plays     
exactly the same role as the  three     
space co-ordinates4 ,formally1 these    
four co-ordinates correspond  exactly to
 the three space co-ordinates in        
,euclidean geometry4 ,it must  be clear 
even to the non-mathematician that1 as a
 consequence of this  purely formal     
addition to our knowledge1 the theory   

perforce gained  clearness in no mean   
measure4       #ad ,cf4 the somewhat    
more detailed discussion in ,appendix   
,,ii4      ,these inadequate remarks can
 give the reader only a vague notion of 
the  important idea contributed by      
,minkowski4 ,without it the general     
theory  of relativity1 of which the     
fundamental ideas are developed in the  
following pages1 would perhaps have got 
no farther than its long  clothes4      
,minkowski's work is doubtless difficult
 of access to anyone  inexperienced in  
mathematics1 but since it is not        
necessary to have a  very exact grasp of
 this work in order to understand the   
fundamental  ideas of either the special
 or the general theory of relativity1 ,i
  shall leave it here at present1 and   
revert to it only towards the end  of   
,part ,,ii4      ,,part ,,ii3 ,,the     
,,general ,,theory ,,of ,,relativity    
,,xviii4    ,,special ,,and ,,general   
,,principle ,,of ,,relativity    ,the   
basal principle1 which was the pivot of 

all our previous  considerations1 was   
the special principle of relativity1    
i4e4 the  principle of the physical     
relativity of all uniform motion4 ,let  
as  once more analyse its meaning       
carefully4    ,it was at all times clear
 that1 from the point of view of the    
idea it  conveys to us1 every motion    
must be considered only as a relative   
motion4 ,returning to the illustration  
we have frequently used of the          
embankment and the railway carriage1 we 
can express the fact of the  motion here
 taking place in the following two      
forms1 both of which are  equally       
justifiable3    7a7 ,the carriage is in 
motion relative to the embankment1      
7b7 ,the embankment is in motion        
relative to the carriage4      ,in 7a7  
the embankment1 in 7b7 the carriage1    
serves as the body of  reference in our 
statement of the motion taking place4   
,if it is simply  a question of         
detecting or of describing the motion   
involved1 it is in  principle immaterial

 to what reference-body we refer the    
motion4 ,as  already mentioned1 this is 
self-evident1 but it must not be        
confused  with the much more            
comprehensive statement called 8the     
principle of  relativity10 which we have
 taken as the basis of our              
investigations4    ,the principle we    
have made use of not only maintains that
 we may  equally well choose the        
carriage or the embankment as our       
reference-body for the description of   
any event 7for this1 too1 is            
self-evident74 ,our principle rather    
asserts what follows3 ,if we  formulate 
the general laws of nature as they are  
obtained from  experience1 by making use
 of    7a7 the embankment as            
reference-body1      7b7 the railway    
carriage as reference-body1      then   
these general laws of nature 7e4g4 the  
laws of mechanics or the  law of the    
propagation of light in vacuo7 have     
exactly the same form  in both cases4   
,this can also be expressed as follows3 

,for the physical  description of       
natural processes1 neither of the       
reference bodies ,k1  ,k is unique 7lit4
 8specially marked out07 as compared    
with the other4  ,unlike the first1 this
 latter statement need not of necessity 
hold a  priori2 it is not contained in  
the conceptions of 8motion0 and         
8reference-body0 and derivable from     
them2 only experience can decide  as to 
its correctness or incorrectness4    ,up
 to the present1 however1 we have by no 
means maintained the  equivalence of all
 bodies of reference ,k in connection   
with the  formulation of natural laws4  
,our course was more on the following   
,iines4 ,in the first place1 we started 
out from the assumption that  there     
exists a reference-body ,k1 whose       
condition of motion is such  that the   
,galileian law holds with respect to it3
 ,a particle left to  itself and        
sufficiently far removed from all other 
particles moves  uniformly in a straight
 line4 ,with reference to ,k 7,galileian

  reference-body7 the laws of nature    
were to be as simple as possible4  ,but 
in addition to ,k1 all bodies of        
reference ,k should be given  preference
 in this sense1 and they should be      
exactly equivalent to ,k  for the       
formulation of natural laws1 provided   
that they are in a state  of uniform    
rectilinear and non-rotary motion with  
respect to ,k2 all  these bodies of     
reference are to be regarded as         
,galileian  reference-bodies4 ,the      
validity of the principle of relativity 
was  assumed only for these             
reference-bodies1 but not for others    
7e4g4  those possessing motion of a     
different kind74 ,in this sense we speak
 of  the special principle of           
relativity1 or special theory of        
relativity4    ,in contrast to this we  
wish to understand by the 8general      
principle of  relativity0 the following 
statement3 ,all bodies of reference ,k1 
,k1  etc41 are equivalent for the       
description of natural phenomena        

7formulation of the general laws of     
nature71 whatever may be their  state of
 motion4 ,but before proceeding farther1
 it ought to be pointed  out that this  
formulation must be replaced later by a 
more abstract  one1 for reasons which   
will become evident at a later stage4   
,since the introduction of the special  
principle of relativity has been        
justified1 every intellect which strives
 after generalisation must feel  the    
temptation to venture the step towards  
the general principle of  relativity4   
,but a simple and apparently quite      
reliable consideration  seems to suggest
 that1 for the present at any rate1     
there is little  hope of success in such
 an attempt2 ,let us imagine ourselves  
transferred to our old friend the       
railway carriage1 which is travelling   
at a uniform rate4 ,as long as it is    
moving uniformly1 the occupant of  the  
carriage is not sensible of its motion1 
and it is for this reason  that he can  
without reluctance interpret the facts  

of the case as  indicating that the     
carriage is at rest1 but the embankment 
in motion4  ,moreover1 according to the 
special principle of relativity1 this   
interpretation is quite justified also  
from a physical point of view4  ,if the 
motion of the carriage is now changed   
into a non-uniform motion1  as for      
instance by a powerful application of   
the brakes1 then the  occupant of the   
carriage experiences a correspondingly  
powerful jerk  forwards4 ,the retarded  
motion is manifested in the mechanical  
behaviour  of bodies relative to the    
person in the railway carriage4 ,the    
mechanical behaviour is different from  
that of the case previously  considered1
 and for this reason it would appear to 
be impossible that  the same mechanical 
laws hold relatively to the             
non-uniformly moving  carriage1 as hold 
with reference to the carriage when at  
rest or in  uniform motion4 ,at all     
events it is clear that the ,galileian  
law does  not hold with respect to the  

non-uniformly moving carriage4 ,because 
of  this1 we feel compelled at the      
present juncture to grant a kind of     
absolute physical reality to non-uniform
 motion1 in opposition to the  general  
principle of relativity4 ,but in what   
follows we shall soon see  that this    
conclusion cannot be maintained4        
,,xix4    ,,the ,,gravitational ,,field 
8,if we pick up a stone and then let it 
go1 why does it fall to the  ground80   
,the usual answer to this question is3  
8,because it is attracted  by the       
earth40 ,modern physics formulates the  
answer rather differently  for the      
following reason4 ,as a result of the   
more careful study of  electromagnetic  
phenomena1 we have come to regard action
 at a distance  as a process impossible 
without the intervention of some        
intermediary  medium4 ,if1 for instance1
 a magnet attracts a piece of iron1 we  
cannot  be content to regard this as    
meaning that the magnet acts directly on
  the iron through the intermediate     

empty space1 but we are constrained  to 
imagine--after the manner of            
,faraday--that the magnet always calls  
into being something physically real in 
the space around it1 that  something    
being what we call a 8magnetic field40  
,in its turn this  magnetic field       
operates on the piece of iron1 so that  
the latter  strives to move towards the 
magnet4 ,we shall not discuss here the  
justification for this incidental       
conception1 which is indeed a  somewhat 
arbitrary one4 ,we shall only mention   
that with its aid  electromagnetic      
phenomena can be theoretically          
represented much more  satisfactorily   
than without it1 and this applies       
particularly to the  transmission of    
electromagnetic waves4 ,the effects of  
gravitation also  are regarded in an    
analogous manner4    ,the action of the 
earth on the stone takes place          
indirectly4 ,the earth  produces in its 
surrounding a gravitational field1 which
 acts on the  stone and produces its    

motion of fall4 ,as we know from        
experience1 the  intensity of the action
 on a body dimishes according to a quite
  definite law1 as we proceed farther   
and farther away from the earth4  ,from 
our point of view this means3 ,the law  
governing the properties of  the        
gravitational field in space must be a  
perfectly definite one1 in  order       
correctly to represent the diminution of
 gravitational action  with the distance
 from operative bodies4 ,it is something
 like this3 ,the  body 7e4g4 the earth7 
produces a field in its immediate       
neighbourhood  directly2 the intensity  
and direction of the field at points    
farther  removed from the body are      
thence determined by the law which      
governs  the properties in space of the 
gravitational fields themselves4    ,in 
contrast to electric and magnetic       
fields1 the gravitational field         
exhibits a most remarkable property1    
which is of fundamental importance  for 
what follows4 ,bodies which are moving  

under the sole influence of a           
gravitational field receive an          
acceleration1 which does not in the     
least depend either on the material or  
on the physical state of the  body4 ,for
 instance1 a piece of lead and a piece  
of wood fall in  exactly the same manner
 in a gravitational field 7in vacuo71   
when  they start off from rest or with  
the same initial velocity4 ,this law1   
which holds most accurately1 can be     
expressed in a different form in  the   
light of the following consideration4   
,according to ,newton's law of motion1  
we have    7,force7  7inertial mass7 x  
7acceleration71    where the 8inertial  
mass0 is a characteristic constant of   
the  accelerated body4 ,if now          
gravitation is the cause of the         
acceleration1  we then have    7,force7 
7gravitational mass7 x 7intensity of the
 gravitational  field71    where the    
8gravitational mass0 is likewise a      
characteristic constant  for the body4  
,from these two relations follows3      

image#jcb      ,if now1 as we find from 
experience1 the acceleration is to be   
independent of the nature and the       
condition of the body and always the    
same for a given gravitational field1   
then the ratio of the  gravitational to 
the inertial mass must likewise be the  
same for all  bodies4 ,by a suitable    
choice of units we can thus make this   
ratio equal  to unity4 ,we then have the
 following law3 ,the gravitational mass 
of a  body is equal to its inertial     
mass4    ,it is true that this important
 law had hitherto been recorded in      
mechanics1 but it had not been          
interpreted4 ,a satisfactory            
interpretation can be obtained only if  
we recognise the following fact3  ,the  
same quality of a body manifests itself 
according to  circumstances as 8inertia0
 or as 8weight0 7lit4 8heaviness074 ,in 
the  following section we shall show to 
what extent this is actually the  case1 
and how this question is connected with 
the general postulate of  relativity4   

,,xx4    ,,the ,,equality ,,of          
,,inertial ,,and ,,gravitational ,,mass 
,,as ,,an ,,argument ,,for ,,the        
,,general ,,postulate ,,of ,,relativity 
,we imagine a large portion of empty    
space1 so far removed from stars  and   
other appreciable masses1 that we have  
before us approximately the  conditions 
required by the fundamental law of      
,galilei4 ,it is then  possible to      
choose a ,galileian reference-body for  
this part of space  7world71 relative to
 which points at rest remain at rest and
 points in  motion continue permanently 
in uniform rectilinear motion4 ,as      
reference-body let us imagine a spacious
 chest resembling a room with  an       
observer inside who is equipped with    
apparatus4 ,gravitation  naturally does 
not exist for this observer4 ,he must   
fasten himself with  strings to the     
floor1 otherwise the slightest impact   
against the floor  will cause him to    
rise slowly towards the ceiling of the  
room4    ,to the middle of the lid of   

the chest is fixed externally a hook    
with  rope attached1 and now a 8being0  
7what kind of a being is immaterial to  
us7 begins pulling at this with a       
constant force4 ,the chest together     
with the observer then begin to move    
8upwards0 with a uniformly  accelerated 
motion4 ,in course of time their        
velocity will reach  unheard-of         
values--provided that we are viewing all
 this from another  reference-body which
 is not being pulled with a rope4       
,but how does the man in the chest      
regard the ,process8 ,the acceleration  
of the chest will be transmitted to him 
by the reaction of the floor of  the    
chest4 ,he must therefore take up this  
pressure by means of his legs  if he    
does not wish to be laid out full length
 on the floor4 ,he is then  standing in 
the chest in exactly the same way as    
anyone stands in a  room of a home on   
our earth4 ,if he releases a body which 
he previously  had in his land1 the     
accelertion of the chest will no longer 

be  transmitted to this body1 and for   
this reason the body will approach  the 
floor of the chest with an accelerated  
relative motion4 ,the  observer will    
further convince himself that the       
acceleration of the  body towards the   
floor of the chest is always of the same
 magnitude1  whatever kind of body he   
may happen to use for the experiment4   
,relying on his knowledge of the        
gravitational field 7as it was          
discussed in the preceding section71 the
 man in the chest will thus  come to the
 conclusion that he and the chest are in
 a gravitational  field which is        
constant with regard to time4 ,of course
 he will be  puzzled for a moment as to 
why the chest does not fall in this     
gravitational field4 just then1 however1
 he discovers the hook in the  middle of
 the lid of the chest and the rope which
 is attached to it1  and he consequently
 comes to the conclusion that the chest 
is suspended  at rest in the            
gravitational field4    ,ought we to    

smile at the man and say that he errs in
 his conclusion8 ,i  do not believe we  
ought to if we wish to remain           
consistent2 we must  rather admit that  
his mode of grasping the situation      
violates neither  reason nor known      
mechanical laws4 ,even though it is     
being accelerated  with respect to the  
8,galileian space0 first considered1 we 
can  nevertheless regard the chest as   
being at rest4 ,we have thus good       
grounds for extending the principle of  
relativity to include bodies of         
reference which are accelerated with    
respect to each other1 and as a  result 
we have gained a powerful argument for a
 generalised postulate  of relativity4  
,we must note carefully that the        
possibility of this mode of             
interpretation rests on the fundamental 
property of the gravitational  field of 
giving all bodies the same acceleration1
 or1 what comes to the  same thing1 on  
the law of the equality of inertial and 
gravitational  mass4 ,if this natural   

law did not exist1 the man in the       
accelerated  chest would not be able to 
interpret the behaviour of the bodies   
around  him on the supposition of a     
gravitational field1 and he would not be
  justified on the grounds of experience
 in supposing his reference-body  to be 
8at rest40    ,suppose that the man in  
the chest fixes a rope to the inner side
 of the  lid1 and that he attaches a    
body to the free end of the rope4 ,the  
result of this will be to stretch the   
rope so that it will hang  8vertically0 
downwards4 ,if we ask for an opinion of 
the cause of  tension in the rope1 the  
man in the chest will say3 8,the        
suspended body  experiences a downward  
force in the gravitational field1 and   
this is  neutralised by the tension of  
the rope2 what determines the magnitude 
of the tension of the rope is the       
gravitational mass of the suspended     
body40 ,on the other hand1 an observer  
who is poised freely in space  will     
interpret the condition of things thus3 

8,the rope must perforce  take part in  
the accelerated motion of the chest1 and
 it transmits this  motion to the body  
attached to it4 ,the tension of the rope
 is just  large enough to effect the    
acceleration of the body4 ,that which   
determines the magnitude of the tension 
of the rope is the inertial  mass of the
 body40 ,guided by this example1 we see 
that our extension  of the principle of 
relativity implies the necessity of the 
law of  the equality of inertial and    
gravitational mass4 ,thus we have       
obtained  a physical interpretation of  
this law4    ,from our consideration of 
the accelerated chest we see that a     
general  theory of relativity must yield
 important results on the laws of       
gravitation4 ,in point of fact1 the     
systematic pursuit of the general  idea 
of relativity has supplied the laws     
satisfied by the gravitational  field4  
,before proceeding farther1 however1 ,i 
must warn the reader  against a         
misconception suggested by these        

considerations4 ,a  gravitational field 
exists for the man in the chest1 despite
 the fact  that there was no such field 
for the co-ordinate system first chosen4
  ,now we might easily suppose that the 
existence of a gravitational field  is  
always only an apparent one4 ,we might  
also think that1 regardless  of the kind
 of gravitational field which may be    
present1 we could  always choose another
 reference-body such that no            
gravitational field  exists with        
reference to it4 ,this is by no means   
true for all  gravitational fields1 but 
only for those of quite special form4   
,it is1  for instance1 impossible to    
choose a body of reference such that1 as
  judged from it1 the gravitational     
field of the earth 7in its entirety7    
vanishes4    ,we can now appreciate why 
that argument is not convincing1 which  
we  brought forward against the general 
principle of relativity at the end  of  
,section ,,xviii4 ,it is certainly true 
that the observer in the railway        

carriage experiences a jerk forwards as 
a result of the application of  the     
brake1 and that he recognises1 in this  
the non-uniformity of motion            
7retardation7 of the carriage4 ,but he  
is compelled by nobody to refer  this   
jerk to a 8real0 acceleration           
7retardation7 of the carriage4 ,he      
might also interpret his experience     
thus3 8,my body of reference 7the       
carriage7 remains permanently at rest4  
,with reference to it1 however1  there  
exists 7during the period of application
 of the brakes7 a  gravitational field  
which is directed forwards and which is 
variable  with respect to time4 ,under  
the influence of this field1 the        
embankment  together with the earth     
moves non-uniformly in such a manner    
that their  original velocity in the    
backwards direction is continuously     
reduced40      ,,xxi4    ,,in ,,what    
,,respects ,,are ,,the ,,foundations    
,,of ,,classical ,,mechanics ,,and ,,of 
,,the  ,,special ,,theory ,,of          

,,relativity ,,unsatisfactory8    ,we   
have already stated several times that  
classical mechanics starts  out from the
 following law3 ,material particles     
sufficiently far removed  from other    
material particles continue to move     
uniformly in a straight  line or        
continue in a state of rest4 ,we have   
also repeatedly emphasised  that this   
fundamental law can only be valid for   
bodies of reference ,k  which possess   
certain unique states of motion1 and    
which are in uniform  translational     
motion relative to each other4 ,relative
 to other  reference-bodies ,k the law  
is not valid4 ,both in classical        
mechanics  and in the special theory of 
relativity we therefore differentiate   
between reference-bodies ,k relative to 
which the recognised 8laws of  nature0  
can be said to hold1 and                
reference-bodies ,k relative to which   
these laws do not hold4    ,but no      
person whose mode of thought is logical 
can rest satisfied with  this condition 

of things4 ,he asks3 8,how does it come 
that certain  reference-bodies 7or their
 states of motion7 are given priority   
over  other reference-bodies 7or their  
states of motion78 ,what is the reason  
for this preference80 ,in order to show 
clearly what ,i mean by this  question1 
,i shall make use of a comparison4    ,i
 am standing in front of a gas range4   
,standing alongside of each other  on   
the range are two pans so much alike    
that one may be mistaken for  the other4
 ,both are half full of water4 ,i notice
 that steam is being  emitted           
continuously from the one pan1 but not  
from the other4 ,i am  surprised at     
this1 even if ,i have never seen either 
a gas range or a  pan before4 ,but if ,i
 now notice a luminous something of     
bluish colour  under the first pan but  
not under the other1 ,i cease to be     
astonished1  even if ,i have never      
before seen a gas flame4 ,for ,i can    
only say that  this bluish something    
will cause the emission of the steam1 or

 at least  possibly it may do so4 ,if1  
however1 ,i notice the bluish something 
in  neither case1 and if ,i observe that
 the one continuously emits steam       
whilst the other does not1 then ,i shall
 remain astonished and  dissatisfied    
until ,i have discovered some           
circumstance to which ,i can  attribute 
the different behaviour of the two pans4
    ,analogously1 ,i seek in vain for a 
real something in classical mechanics   
7or in the special theory of relativity7
 to which ,i can attribute the          
different behaviour of bodies considered
 with respect to the reference  systems 
,k and ,k4#ae ,newton saw this objection
 and attempted to  invalidate it1 but   
without success4 ,but ,e4 ,mach         
recognised it most  clearly of all1 and 
because of this objection he claimed    
that mechanics  must be placed on a new 
basis4 ,it can only be got rid of by    
means of a  physics which is conformable
 to the general principle of relativity1
  since the equations of such a theory  

hold for every body of reference1       
whatever may be its state of motion4    
#ae ,the objection is of importance more
 especially when the state of   motion  
of the reference-body is of such a      
nature that it does not   require any   
external agency for its maintenance1    
e4g4 in the case   when the             
reference-body is rotating uniformly4   
,,xxii4    ,a ,,few ,,inferences ,,from 
,,the ,,general ,,principle ,,of        
,,relativity    ,the considerations of  
,section ,,xx show that the general     
principle of  relativity puts us in a   
position to derive properties of the    
gravitational field in a purely         
theoretical manner4 ,let us suppose1 for
  instance1 that we know the space-time 
8course0 for any natural process        
whatsoever1 as regards the manner in    
which it takes place in the  ,galileian 
domain relative to a ,galileian body of 
reference ,k4 ,by  means of purely      
theoretical operations 7i4e4 simply by  
calculation7  we are then able to find  

how this known natural process appears1 
as  seen from a reference-body ,k which 
is accelerated relatively to ,k4  ,but  
since a gravitational field exists with 
respect to this new body of  reference  
,k1 our consideration also teaches us   
how the gravitational  field influences 
the process studied4    ,for example1 we
 learn that a body which is in a state  
of uniform  rectilinear motion with     
respect to ,k 7in accordance with the   
law of  ,galilei7 is executing an       
accelerated and in general curvilinear  
motion  with respect to the accelerated 
reference-body ,k 7chest74 ,this        
acceleration or curvature corresponds to
 the influence on the moving  body of   
the gravitational field prevailing      
relatively to ,k4 ,it is  known that a  
gravitational field influences the      
movement of bodies in  this way1 so that
 our consideration supplies us with     
nothing  essentially new4    ,however1  
we obtain a new result of fundamental   
importance when we carry  out the       

analogous consideration for a ray of    
light4 ,with respect to the  ,galileian 
reference-body ,k1 such a ray of light  
is transmitted  rectilinearly with the  
velocity c4 ,it can easily be shown that
 the  path of the same ray of light is  
no longer a straight line when we       
consider it with reference to the       
accelerated chest 7reference-body  ,k74 
,from this we conclude1 that1 in        
general1 rays of light are  propagated  
curvilinearly in gravitational fields4  
,in two respects this  result is of     
great importance4    ,in the first      
place1 it can be compared with the      
reality4 ,although a  detailed          
examination of the question shows that  
the curvature of light  rays required by
 the general theory of relativity is    
only exceedingly  small for the         
gravitational fields at our disposal in 
practice1 its  estimated magnitude for  
light rays passing the sun at grazing   
incidence  is nevertheless #a4g seconds 
of arc4 ,this ought to manifest itself  

in  the following way4 ,as seen from the
 earth1 certain fixed stars appear  to  
be in the neighbourhood of the sun1 and 
are thus capable of  observation during 
a total eclipse of the sun4 ,at such    
times1 these  stars ought to appear to  
be displaced outwards from the sun by an
  amount indicated above1 as compared   
with their apparent position in the  sky
 when the sun is situated at another    
part of the heavens4 ,the  examination  
of the correctness or otherwise of this 
deduction is a  problem of the greatest 
importance1 the early solution of which 
is to  be expected of astronomers4#af   
#af ,by means of the star photographs of
 two expeditions equipped by a   ,joint 
,committee of the ,royal and ,royal     
,astronomical ,societies1 the           
existence of the deflection of light    
demanded by theory was first   confirmed
 during the solar eclipse of #bith ,may1
 #aiai4 7,cf4 ,appendix   ,,iii47       
,in the second place our result shows   
that1 according to the general  theory  

of relativity1 the law of the constancy 
of the velocity of light  in vacuo1     
which constitutes one of the two        
fundamental assumptions in  the special 
theory of relativity and to which we    
have already  frequently referred1      
cannot claim any unlimited validity4 ,a 
curvature  of rays of light can only    
take place when the velocity of         
propagation  of light varies with       
position4 ,now we might think that as a 
consequence  of this1 the special theory
 of relativity and with it the whole    
theory  of relativity would be laid in  
the dust4 ,but in reality this is not   
the  case4 ,we can only conclude that   
the special theory of relativity cannot 
claim an unlimited domain of validity2  
its results hold only so long as  we are
 able to disregard the influences of    
gravitational fields on the  phenomena  
7e4g4 of light74    ,since it has often 
been contended by opponents of the      
theory of  relativity that the special  
theory of relativity is overthrown by   

the  general theory of relativity1 it is
 perhaps advisable to make the facts  of
 the case clearer by means of an        
appropriate comparison4 ,before the     
development of electrodynamics the laws 
of electrostatics were looked  upon as  
the laws of electricity4 ,at the present
 time we know that  electric fields can 
be derived correctly from electrostatic 
considerations only for the case1 which 
is never strictly realised1 in  which   
the electrical masses are quite at rest 
relatively to each other1  and to the   
co-ordinate system4 ,should we be       
justified in saying that  for this      
reason electrostatics is overthrown by  
the field-equations of  ,maxwell in     
electrodynamics8 ,not in the least4     
,electrostatics is  contained in        
electrodynamics as a limiting case2 the 
laws of the latter  lead directly to    
those of the former for the case in     
which the fields  are invariable with   
regard to time4 ,no fairer destiny could
 be allotted  to any physical theory1   

than that it should of itself point out 
the way  to the introduction of a more  
comprehensive theory1 in which it lives 
on as a limiting case4    ,in the       
example of the transmission of light    
just dealt with1 we have  seen that the 
general theory of relativity enables us 
to derive  theoretically the influence  
of a gravitational field on the course  
of  natural processes1 the laws of which
 are already known when a  gravitational
 field is absent4 ,but the most         
attractive problem1 to the  solution of 
which the general theory of relativity  
supplies the key1  concerns the         
investigation of the laws satisfied by  
the gravitational  field itself4 ,let us
 consider this for a moment4    ,we are 
acquainted with space-time domains which
 behave 7approximately7  in a           
8,galileian0 fashion under suitable     
choice of reference-body1  i4e4 domains 
in which gravitational fields are       
absent4 ,if we now  refer such a domain 
to a reference-body ,k possessing any   

kind of  motion1 then relative to ,k    
there exists a gravitational field which
  is variable with respect to space and 
time4#ag ,the character of this  field  
will of course depend on the motion     
chosen for ,k4 ,according to  the       
general theory of relativity1 the       
general law of the gravitational  field 
must be satisfied for all gravitational 
fields obtainable in this  way4 ,even   
though by no means all gravitationial   
fields can be produced  in this way1 yet
 we may entertain the hope that the     
general law of  gravitation will be     
derivable from such gravitational fields
 of a  special kind4 ,this hope has been
 realised in the most beautiful manner4 
,but between the clear vision of this   
goal and its actual realisation it  was 
necessary to surmount a serious         
difficulty1 and as this lies deep  at   
the root of things1 ,i dare not withhold
 it from the reader4 ,we  require to    
extend our ideas of the space-time      
continuum still farther4       #ag ,this

 follows from a generalisation of the   
discussion in ,section   ,,xx4          
,,xxiii4    ,,behaviour ,,of ,,clocks   
,,and ,,measuring-,,rods ,,on ,a        
,,rotating ,,body ,,of ,,reference      
,hitherto ,i have purposely refrained   
from speaking about the physical        
interpretation of space- and time-data  
in the case of the general  theory of   
relativity4 ,as a consequence1 ,i am    
guilty of a certain  slovenliness of    
treatment1 which1 as we know from the   
special theory of  relativity1 is far   
from being unimportant and pardonable4  
,it is now  high time that we remedy    
this defect2 but ,i would mention at the
  outset1 that this matter lays no small
 claims on the patience and on  the     
power of abstraction of the reader4     
,we start off again from quite special  
cases1 which we have frequently  used   
before4 ,let us consider a space time   
domain in which no  gravitational field 
exists relative to a reference-body ,k  
whose state  of motion has been suitably

 chosen4 ,k is then a ,galileian        
reference-body as regards the domain    
considered1 and the results of the      
special theory of relativity hold       
relative to ,k4 ,let us suppose the     
same domain referred to a second body of
 reference ,k1 which is  rotating       
uniformly with respect to ,k4 ,in order 
to fix our ideas1 we  shall imagine ,k  
to be in the form of a plane circular   
disc1 which  rotates uniformly in its   
own plane about its centre4 ,an observer
 who is  sitting eccentrically on the   
disc ,k is sensible of a force which    
acts outwards in a radial direction1 and
 which would be interpreted as  an      
effect of inertia 7centrifugal force7 by
 an observer who was at rest  with      
respect to the original reference-body  
,k4 ,but the observer on  the disc may  
regard his disc as a reference-body     
which is 8at rest02 on  the basis of the
 general principle of relativity he is  
justified in  doing this4 ,the force    
acting on himself1 and in fact on all   

other  bodies which are at rest relative
 to the disc1 he regards as the effect  
of a gravitational field4 ,nevertheless1
 the space-distribution of this         
gravitational field is of a kind that   
would not be possible on ,newton's      
theory of gravitation4#ah ,but since the
 observer believes in the  general      
theory of relativity1 this does not     
disturb him2 he is quite in  the right  
when he believes that a general law of  
gravitation can be  formulated--a law   
which not only explains the motion of   
the stars  correctly1 but also the field
 of force experienced by himself4       
#ah ,the field disappears at the centre 
of the disc and increases               
proportionally to the distance from the 
centre as we proceed outwards4      ,the
 observer performs experiments on his   
circular disc with clocks and           
measuring-rods4 ,in doing so1 it is his 
intention to arrive at exact            
definitions for the signification of    
time- and space-data with  reference to 

the circular disc ,k1 these definitions 
being based on  his observations4 ,what 
will be his experience in this          
enterprise8    ,to start with1 he places
 one of two identically constructed     
clocks at  the centre of the circular   
disc1 and the other on the edge of the  
disc1  so that they are at rest relative
 to it4 ,we now ask ourselves whether   
both clocks go at the same rate from the
 standpoint of the non-rotating         
,galileian reference-body ,k4 ,as judged
 from this body1 the clock at  the      
centre of the disc has no velocity1     
whereas the clock at the edge  of the   
disc is in motion relative to ,k in     
consequence of the  rotation4 ,according
 to a result obtained in ,section ,,xii1
 it follows  that the latter clock goes 
at a rate permanently slower than that  
of  the clock at the centre of the      
circular disc1 i4e4 as observed from    
,k4 ,it is obvious that the same effect 
would be noted by an observer  whom we  
will imagine sitting alongside his clock

 at the centre of the  circular disc4   
,thus on our circular disc1 or1 to make 
the case more  general1 in every        
gravitational field1 a clock will go    
more quickly or  less quickly1 according
 to the position in which the clock is  
situated  7at rest74 ,for this reason it
 is not possible to obtain a reasonable 
definition of time with the aid of      
clocks which are arranged at rest  with 
respect to the body of reference4 ,a    
similar difficulty presents  itself when
 we attempt to apply our earlier        
definition of simultaneity  in such a   
case1 but ,i do not wish to go any      
farther into this question4             
,moreover1 at this stage the definition 
of the space co-ordinates also  presents
 insurmountable difficulties4 ,if the   
observer applies his  standard          
measuring-rod 7a rod which is short as  
compared with the  radius of the disc7  
tangentially to the edge of the disc1   
then1 as  judged from the ,galileian    
system1 the length of this rod will be  

less  than #a1 since1 according to      
,section ,,xii1 moving bodies suffer a  
shortening in the direction of the      
motion4 ,on the other hand1 the         
measuring-rod will not experience a     
shortening in length1 as judged  from   
,k1 if it is applied to the disc in the 
direction of the radius4  ,if1 then1 the
 observer first measures the            
circumference of the disc  with his     
measuring-rod and then the diameter of  
the disc1 on dividing  the one by the   
other1 he will not obtain as quotient   
the familiar  number   #c4ad 4 4 41 but 
a larger number1#ai whereas of course1  
for  a disc which is at rest with       
respect to ,k1 this operation would     
yield   exactly4 ,this proves that the  
propositions of ,euclidean geometry     
cannot hold exactly on the rotating     
disc1 nor in general in a  gravitational
 field1 at least if we attribute the    
length #a to the rod  in all positions  
and in every orientation4 ,hence the    
idea of a straight  line also loses its 

meaning4 ,we are therefore not in a     
position to  define exactly the         
co-ordinates x1 y1 z relative to the    
disc by means  of the method used in    
discussing the special theory1 and as   
long as the  co-ordinates and times of  
events have not been defined1 we cannot 
assign an exact meaning to the natural  
laws in which these occur4       #ai    
,throughout this consideration we have  
to use the ,galileian   7non-rotating7  
system ,k as reference-body1 since we   
may only assume   the validity of the   
results of the special theory of        
relativity   relative to ,k 7relative to
 ,k a gravitational field prevails74    
,thus all our previous conclusions based
 on general relativity would  appear to 
be called in question4 ,in reality we   
must make a subtle  detour in order to  
be able to apply the postulate of       
general relativity  exactly4 ,i shall   
prepare the reader for this in the      
following  paragraphs4      ,,xxiv4     
,,euclidean ,,and ,,non-,,euclidean     

,,continuum    ,the surface of a marble 
table is spread out in front of me4 ,i  
can get  from any one point on this     
table to any other point by passing     
continuously from one point to a        
8neighbouring0 one1 and repeating this  
process a 7large7 number of times1 or1  
in other words1 by going from  point to 
point without executing 8jumps40 ,i am  
sure the reader will  appreciate with   
sufficient clearness what ,i mean here  
by 8neighbouring0  and by 8jumps0 7if he
 is not too pedantic74 ,we express this 
property of  the surface by describing  
the latter as a continuum4    ,let us   
now imagine that a large number of      
little rods of equal length  have been  
made1 their lengths being small compared
 with the dimensions  of the marble     
slab4 ,when ,i say they are of equal    
length1 ,i mean that  one can be laid on
 any other without the ends overlapping4
 ,we next lay  four of these little rods
 on the marble slab so that they        
constitute a  quadrilateral figure 7a   

square71 the diagonals of which are     
equally  long4 ,to ensure the equality  
of the diagonals1 we make use of a      
little  testing-rod4 ,to this square we 
add similar ones1 each of which has one 
rod in common with the first4 ,we       
proceed in like manner with each of     
these squares until finally the whole   
marble slab is laid out with  squares4  
,the arrangement is such1 that each side
 of a square belongs to  two squares and
 each corner to four squares4    ,it is 
a veritable wonder that we can carry out
 this business without  getting into the
 greatest difficulties4 ,we only need to
 think of the  following4 ,if at any    
moment three squares meet at a corner1  
then two  sides of the fourth square are
 already laid1 and1 as a consequence1   
the  arrangement of the remaining two   
sides of the square is already          
completely determined4 ,but ,i am now no
 longer able to adjust the              
quadrilateral so that its diagonals may 
be equal4 ,if they are equal of  their  

own accord1 then this is an especial    
favour of the marble slab  and of the   
little rods1 about which ,i can only be 
thankfully surprised4  ,we must         
experience many such surprises if the   
construction is to be  successful4      
,if everything has really gone smoothly1
 then ,i say that the points of  the    
marble slab constitute a ,euclidean     
continuum with respect to the  little   
rod1 which has been used as a 8distance0
 7line-interval74 ,by  choosing one     
corner of a square as 8origin0 ,i can   
characterise every  other corner of a   
square with reference to this origin by 
means of two  numbers4 ,i only need     
state how many rods ,i must pass over   
when1  starting from the origin1 ,i     
proceed towards the 8right0 and then    
8upwards10 in order to arrive at the    
corner of the square under              
consideration4 ,these two numbers are   
then the 8,cartesian co-ordinates0  of  
this corner with reference to the       
8,cartesian co-ordinate system0  which  

is determined by the arrangement of     
little rods4    ,by making use of the   
following modification of this abstract 
experiment1 we recognise that there must
 also be cases in which the  experiment 
would be unsuccessful4 ,we shall suppose
 that the rods  8expand0 by in amount   
proportional to the increase of         
temperature4 ,we  heat the central part 
of the marble slab1 but not the         
periphery1 in  which case two of our    
little rods can still be brought into   
coincidence  at every position on the   
table4 ,but our construction of squares 
must  necessarily come into disorder    
during the heating1 because the little  
rods on the central region of the table 
expand1 whereas those on the  outer part
 do not4    ,with reference to our      
little rods--defined as unit            
lengths--the marble  slab is no longer a
 ,euclidean continuum1 and we are also  
no longer in  the position of defining  
,cartesian co-ordinates directly with   
their  aid1 since the above construction

 can no longer be carried out4 ,but     
since there are other things which are  
not influenced in a similar  manner to  
the little rods 7or perhaps not at all7 
by the temperature of  the table1 it is 
possible quite naturally to maintain the
 point of view  that the marble slab is 
a 8,euclidean continuum40 ,this can be  
done in a  satisfactory manner by making
 a more subtle stipulation about the    
measurement or the comparison of        
lengths4    ,but if rods of every kind  
7i4e4 of every material7 were to behave 
in  the same way as regards the         
influence of temperature when they are  
on  the variably heated marble slab1 and
 if we had no other means of  detecting 
the effect of temperature than the      
geometrical behaviour of  our rods in   
experiments analogous to the one        
described above1 then our  best plan    
would be to assign the distance one to  
two points on the  slab1 provided that  
the ends of one of our rods could be    
made to  coincide with these two points2

 for how else should we define the      
distance without our proceeding being in
 the highest measure grossly  arbitrary8
 ,the method of ,cartesian coordinates  
must then be discarded1  and replaced by
 another which does not assume the      
validity of ,euclidean  geometry for    
rigid bodies4#bj ,the reader will notice
 that the  situation depicted here      
corresponds to the one brought about by 
the  general postulate of relativity    
7,section ,,xxiii74       #bj           
,mathematicians have been confronted    
with our problem in the   following     
form4 ,if we are given a surface 7e4g4  
an ellipsoid7 in   ,euclidean           
three-dimensional space1 then there     
exists for this surface   a             
two-dimensional geometry1 just as much  
as for a plane surface4 ,gauss          
undertook the task of treating this     
two-dimensional geometry from   first   
principles1 without making use of the   
fact that the surface   belongs to a    
,euclidean continuum of three           

dimensions4 ,if we imagine              
constructions to be made with rigid rods
 in the surface 7similar to   that above
 with the marble slab71 we should find  
that different laws   hold for these    
from those resulting on the basis of    
,euclidean plane   geometry4 ,the       
surface is not a ,euclidean continuum   
with respect to the   rods1 and we      
cannot define ,cartesian co-ordinates in
 the surface4   ,gauss indicated the    
principles according to which we can    
treat the   geometrical relationships in
 the surface1 and thus pointed out the  
way   to the method of ,riemann of      
treating multi-dimensional1             
non-,euclidean   continuum4 ,thus it is 
that mathematicians long ago solved the 
formal   problems to which we are led by
 the general postulate of relativity4   
,,xxv4    ,,gaussian ,,co-,,ordinates   
image#jcc      ,according to ,gauss1    
this combined analytical and geometrical
 mode of  handling the problem can be   
arrived at in the following way4 ,we    

imagine  a system of arbitrary curves   
7see ,fig4 #d7 drawn on the surface of  
the  table4 ,these we designate as      
u-curves1 and we indicate each of them  
by means of a number4 ,the ,curves u    
#a1 u  #b and u  #c are drawn  in the   
diagram4 ,between the curves u  #a and u
  #b we must imagine  an infinitely     
large number to be drawn1 all of which  
correspond to real  numbers lying       
between #a and #b4 ,we have then a      
system of u-curves1 and  this           
8infinitely dense0 system covers the    
whole surface of the table4  ,these     
u-curves must not intersect each other1 
and through each point  of the surface  
one and only one curve must pass4 ,thus 
a perfectly  definite value of u belongs
 to every point on the surface of the   
marble slab4 ,in like manner we imagine 
a system of v-curves drawn on  the      
surface4 ,these satisfy the same        
conditions as the u-curves1 they  are   
provided with numbers in a corresponding
 manner1 and they may  likewise be of   

arbitrary shape4 ,it follows that a     
value of u and a  value of v belong to  
every point on the surface of the table4
 ,we call  these two numbers the        
co-ordinates of the surface of the table
  7,gaussian co-ordinates74 ,for        
example1 the point ,p in the diagram has
  the ,gaussian co-ordinates u  #c1 v   
#a4 ,two neighbouring points ,p  and ,p 
on the surface then correspond to the   
co-ordinates    ,p3 u1 v    ,p3 u  du1 v
  dv1    where du and dv signify very   
small numbers4 ,in a similar manner we  
may indicate the distance               
7line-interval7 between ,p and ,p1 as   
measured with a little rod1 by means of 
the very small number ds4  ,then        
according to ,gauss we have    ds#b     
g#aa;du#b  #b;g#ab;du dv  g#bb;dv#b1    
where g#aa1 g#ab1 g#bb1 are magnitudes  
which depend in a perfectly  definite   
way on u and v4 ,the magnitudes g#aa1   
g#ab and g#bb1  determine the behaviour 
of the rods relative to the u-curves and
  v-curves1 and thus also relative to   

the surface of the table4 ,for the  case
 in which the points of the surface     
considered form a ,euclidean  continuum 
with reference to the measuring-rods1   
but only in this case1  it is possible  
to draw the u-curves and v-curves and to
 attach  numbers to them1 in such a     
manner1 that we simply have3    ds#b    
du#b  dv#b    ,under these conditions1  
the u-curves and v-curves are straight  
lines in the sense of ,euclidean        
geometry1 and they are perpendicular to 
each other4 ,here the ,gaussian         
coordinates are simply ,cartesian ones4 
,it  is clear that ,gauss co-ordinates  
are nothing more than an association  of
 two sets of numbers with the points of 
the surface considered1 of  such a      
nature that numerical values differing  
very slightly from each  other are      
associated with neighbouring points 8in 
space40    ,so far1 these considerations
 hold for a continuum of two dimensions4
  ,but the ,gaussian method can be      
applied also to a continuum of three1   

four or more dimensions4 ,if1 for       
instance1 a continuum of four           
dimensions be supposed available1 we may
 represent it in the following  way4    
,with every point of the continuum1 we  
associate arbitrarily four  numbers1    
x#a1 x#b1 x#c1 x#d1 which are known as  
8co-ordinates40  ,adjacent points       
correspond to adjacent values of the    
coordinates4 ,if a  distance ds is      
associated with the adjacent points ,p  
and ,p1 this  distance being measurable 
and well defined from a physical point  
of  view1 then the following formula    
holds3    ds#b  g#aa;dx#ab              
#b;g#ab;dx#a;dx#b 4 4 4 4  g#dd;dx#db1  
where the magnitudes g#aa1 etc41 have   
values which vary with the  position in 
the continuum4 ,only when the continuum 
is a ,euclidean one  is it possible to  
associate the co-ordinates x#a 4 4 x#d4 
with the  points of the continuum so    
that we have simply    ds#b  dx#ab      
dx#bb  dx#cb  dx#db4    ,in this case   
relations hold in the four-dimensional  

continuum which are  analogous to those 
holding in our three-dimensional        
measurements4    ,however1 the ,gauss   
treatment for ds#b which we have given  
above is not  always possible4 ,it is   
only possible when sufficiently small   
regions of  the continuum under         
consideration may be regarded as        
,euclidean  continua4 ,for example1 this
 obviously holds in the case of the     
marble  slab of the table and local     
variation of temperature4 ,the          
temperature  is practically constant for
 a small part of the slab1 and thus the 
geometrical behaviour of the rods is    
almost as it ought to be  according to  
the rules of ,euclidean geometry4 ,hence
 the imperfections  of the construction 
of squares in the previous section do   
not show  themselves clearly until this 
construction is extended over a         
considerable portion of the surface of  
the table4    ,we can sum this up as    
follows3 ,gauss invented a method for   
the  mathematical treatment of continua 

in general1 in which  8size-relations0  
78distances0 between neighbouring       
points7 are defined4  ,to every point of
 a continuum are assigned as many       
numbers 7,gaussian  coordinates7 as the 
continuum has dimensions4 ,this is done 
in such a  way1 that only one meaning   
can be attached to the assignment1 and  
that  numbers 7,gaussian coordinates7   
which differ by an indefinitely small   
amount are assigned to adjacent points4 
,the ,gaussian coordinate system  is a  
logical generalisation of the ,cartesian
 co-ordinate system4 ,it is  also       
applicable to non-,euclidean continua1  
but only when1 with respect  to the     
defined 8size0 or 8distance10 small     
parts of the continuum under            
consideration behave more nearly like a 
,euclidean system1 the smaller  the part
 of the continuum under our notice4     
,,xxvi4    ,,the ,,space-,,time         
,,continuum ,,of ,,the ,,special        
,,theory ,,of ,,relativity ,,considered 
,,as ,a ,,euclidean ,,continuum    ,we  

are now in a position to formulate more 
exactly the idea of  ,minkowski1 which  
was only vaguely indicated in ,section  
,,xvii4 ,in  accordance with the special
 theory of relativity1 certain          
co-ordinate  systems are given          
preference for the description of the   
four-dimensional1 space-time continuum4 
,we called these 8,galileian            
co-ordinate systems40 ,for these        
systems1 the four co-ordinates x1 y1  z1
 t1 which determine an event or--in     
other words--a point of the             
four-dimensional continuum1 are defined 
physically in a simple manner1  as set  
forth in detail in the first part of    
this book4 ,for the  transition from one
 ,galileian system to another1 which is 
moving  uniformly with reference to the 
first1 the equations of the ,lorentz    
transformation are valid4 ,these last   
form the basis for the derivation  of   
deductions from the special theory of   
relativity1 and in themselves  they are 
nothing more than the expression of the 

universal validity of  the law of       
transmission of light for all ,galileian
 systems of  reference4    ,minkowski   
found that the ,lorentz transformations 
satisfy the following  simple           
conditions4 ,let us consider two        
neighbouring events1 the  relative      
position of which in the                
four-dimensional continuum is given     
with respect to a ,galileian            
reference-body ,k by the space          
co-ordinate  differences dx1 dy1 dz and 
the time-difference dt4 ,with reference 
to a second ,galileian system we shall  
suppose that the corresponding          
differences for these two events are dx1
 dy1 dz1 dt4 ,then these  magnitudes    
always fulfill the condition4#ba        
#ba ,cf4 ,appendixes ,i and ,,ii4 ,the  
relations which are derived there   for 
the co-ordinates themselves are valid   
also for co-ordinate   differences1 and 
thus also for co-ordinate differentials 
7indefinitely small differences74       
dx#b  dy#b  dz#b - c#b;dt#b  dx#b  dy#b 

dz#b -  c#b;dt#b4      ,the validity of 
the ,lorentz transformation follows from
 this condition4  ,we can express this  
as follows3 ,the magnitude    ds#b  dx#b
  dy#b  dz#b - c#b dt#b1      which     
belongs to two adjacent points of the   
four-dimensional space-time  continuum1 
has the same value for all selected     
7,galileian7  reference-bodies4 ,if we  
replace x1 y1 z1    image#jcd      by   
x#a1 x#b1 x#c1 x#d1 we also obtain the  
result that    ds#b  dx#ab  dx#bb  dx#cb
  dx#db4      is independent of the     
choice of the body of reference4 ,we    
call the  magnitude ds the 8distance0   
apart of the two events or              
four-dimensional points4    ,thus1 if we
 choose as time-variable the imaginary  
variable    image#jce      instead of   
the real quantity t1 we can regard the  
space-time  contintium--accordance with 
the special theory of relativity--as a  
8,euclidean0 four-dimensional continuum1
 a result which follows from the        
considerations of the preceding section4

      ,,xxvii4    ,,the ,,space-,,time  
,,continuum ,,of ,,the ,,general        
,,theory ,,of ,,relativity ,,is ,,not ,a
  ,,euclidean ,,continuum    ,in the    
first part of this book we were able to 
make use of space-time  co-ordinates    
which allowed of a simple and direct    
physical  interpretation1 and which1    
according to ,section ,,xxvi1 can be    
regarded  as four-dimensional ,cartesian
 co-ordinates4 ,this was possible on the
  basis of the law of the constancy of  
the velocity of light4 ,but  according  
to ,section ,,xxi the general theory of 
relativity cannot retain  this law4 ,on 
the contrary1 we arrived at the result  
that according to  this latter theory   
the velocity of light must always depend
 on the  co-ordinates when a            
gravitational field is present4 ,in     
connection with  a specific illustration
 in ,section ,,xxiii1 we found that the 
presence of  a gravitational field      
invalidates the definition of the       
coordinates and  the time1 which led us 

to our objective in the special theory  
of  relativity4    ,in view of the      
resu,its of these considerations we are 
led to the  conviction that1 according  
to the general principle of relativity1 
the  space-time continuum cannot be     
regarded as a ,euclidean one1 but that  
here we have the general case1          
corresponding to the marble slab with   
local variations of temperature1 and    
with which we made acquaintance as  an  
example of a two-dimensional continuum4 
,just as it was there  impossible to    
construct a ,cartesian co-ordinate      
system from equal rods1  so here it is  
impossible to build up a system         
7reference-body7 from  rigid bodies and 
clocks1 which shall be of such a nature 
that  measuring-rods and clocks1        
arranged rigidly with respect to one    
another1 shall indicate position and    
time directly4 ,such was the  essence of
 the difficulty with which we were      
confronted in ,section  ,,xxiii4    ,but
 the considerations of ,sections ,,xxv  

and ,,xxvi show us the way to  surmount 
this difficulty4 ,we refer the          
four-dimensional space-time  continuum  
in an arbitrary manner to ,gauss        
co-ordinates4 ,we assign to  every point
 of the continuum 7event7 four numbers1 
x#a1 x#b1 x#c1  x#d 7co-ordinates71     
which have not the least direct physical
  significance1 but only serve the      
purpose of numbering the points of the  
continuum in a definite but arbitrary   
manner4 ,this arrangement does not  even
 need to be of such a kind that we must 
regard x#a1 x#b1 x#c1 as  8space0       
co-ordinates and x#d1 as a 8time0       
co-ordinate4    ,the reader may think   
that such a description of the world    
would be  quite inadequate4 ,what does  
it mean to assign to an event the       
particular co-ordinates x#a1 x#b1 x#c1  
x#d1 if in themselves these             
co-ordinates have no significance8 ,more
 careful consideration shows1  however1 
that this anxiety is unfounded4 ,let us 
consider1 for instance1  a material     

point with any kind of motion4 ,if this 
point had only a  momentary existence   
without duration1 then it would to      
described in  space-time by a single    
system of values x#a1 x#b1 x#c1 x#d4    
,thus  its permanent existence must be  
characterised by an infinitely large    
number of such systems of values1 the   
co-ordinate values of which are  so     
close together as to give continuity2   
corresponding to the material  point1 we
 thus have a 7uni-dimensional7 line in  
the four-dimensional  continuum4 ,in the
 same way1 any such lines in our        
continuum correspond  to many points in 
motion4 ,the only statements having     
regard to these  points which can claim 
a physical existence are in reality the 
statements about their encounters4 ,in  
our mathematical treatment1 such  an    
encounter is expressed in the fact that 
the two lines which  represent the      
motions of the points in question have a
 particular  system of co-ordinate      
values1 x#a1 x#b1 x#c1 x#d1 in common4  

,after  mature consideration the reader 
will doubtless admit that in reality    
such encounters constitute the only     
actual evidence of a time-space  nature 
with which we meet in physical          
statements4    ,when we were describing 
the motion of a material point relative 
to a  body of reference1 we stated      
nothing more than the encounters of this
  point with particular points of the   
reference-body4 ,we can also  determine 
the corresponding values of the time by 
the observation of  encounters of the   
body with clocks1 in conjunction with   
the observation  of the encounter of the
 hands of clocks with particular points 
on the  dials4 ,it is just the same in  
the case of space-measurements by means 
of measuring-rods1 as a little          
consideration will show4    ,the        
following statements hold generally3    
,every physical description  resolves   
itself into a number of statements1 each
 of which refers to  the space-time     
coincidence of two events ,a and ,b4 ,in

 terms of  ,gaussian co-ordinates1 every
 such statement is expressed by the     
agreement of their four co-ordinates    
x#a1 x#b1 x#c1 x#d4 ,thus in  reality1  
the description of the time-space       
continuum by means of ,gauss            
co-ordinates completely replaces the    
description with the aid of a body  of  
reference1 without suffering from the   
defects of the latter mode of           
description2 it is not tied down to the 
,euclidean character of the  continuum  
which has to be represented4            
,,xxviii4    ,,exact ,,formulation ,,of 
,,the ,,general ,,principle ,,of        
,,relativity    ,we are now in a        
position to replace the provisional     
formulation of the  general principle of
 relativity given in ,section ,,xviii by
 an exact  formulation4 ,the form there 
used1 8,all bodies of reference ,k1 ,k1 
etc41 are equivalent for the description
 of natural phenomena  7formulation of  
the general laws of nature71 whatever   
may be their  state of motion10 cannot  

be maintained1 because the use of rigid 
reference-bodies1 in the sense of the   
method followed in the special  theory  
of relativity1 is in general not        
possible in space-time  description4    
,the ,gauss co-ordinate system has to   
take the place of the  body of          
reference4 ,the following statement     
corresponds to the  fundamental idea of 
the general principle of relativity3    
8,all ,gaussian  co-ordinate systems are
 essentially equivalent for the         
formulation of  the general laws of     
nature40    ,we can state this general  
principle of relativity in still another
  form1 which renders it yet more       
clearly intelligible than it is when in 
the form of the natural extension of the
 special principle of  relativity4      
,according to the special theory of     
relativity1 the  equations which express
 the general laws of nature pass over   
into  equations of the same form when1  
by making use of the ,lorentz           
transformation1 we replace the          

space-time variables x1 y1 z1 t1 of a   
7,galileian7 reference-body ,k by the   
space-time variables x1 y1 z1  t1 of a  
new reference-body ,k4 ,according to the
 general theory of  relativity1 on the  
other hand1 by application of arbitrary 
substitutions of the ,gauss variables   
x#a1 x#b1 x#c1 x#d1 the  equations must 
pass over into equations of the same    
form2 for every  transformation 7not    
only the ,lorentz transformation7       
corresponds to the  transition of one   
,gauss co-ordinate system into another4 
,if we desire to adhere to our          
8old-time0 three-dimensional view of    
things1 then we can characterise the    
development which is being  undergone by
 the fundamental idea of the general    
theory of relativity  as follows3 ,the  
special theory of relativity has        
reference to ,galileian  domains1 i4e4  
to those in which no gravitational field
 exists4 ,in  this connection a         
,galileian reference-body serves as body
 of reference1  i4e4 a rigid body the   

state of motion of which is so chosen   
that the  ,galileian law of the uniform 
rectilinear motion of 8isolated0        
material  points holds relatively to it4
    ,certain considerations suggest that
 we should refer the same ,galileian    
domains to non-,galileian               
reference-bodies also4 ,a gravitational 
field  of a special kind is then present
 with respect to these bodies 7cf4      
,sections ,,xx and ,,xxiii74    ,in     
gravitational fields there are no such  
things as rigid bodies with  ,euclidean 
properties2 thus the fictitious rigid   
body of reference is of  no avail in the
 general theory of relativity4 ,the     
motion of clocks is  also influenced by 
gravitational fields1 and in such a way 
that a  physical definition of time     
which is made directly with the aid of  
clocks has by no means the same degree  
of plausibility as in the  special      
theory of relativity4    ,for this      
reason non-rigid reference-bodies are   
used1 which are as a  whole not only    

moving in any way whatsoever1 but which 
also suffer  alterations in form ad lib4
 during their motion4 ,clocks1 for which
  the law of motion is of any kind1     
however irregular1 serve for the        
definition of time4 ,we have to imagine 
each of these clocks fixed at a  point  
on the non-rigid reference-body4 ,these 
clocks satisfy only the  one condition1 
that the 8readings0 which are observed  
simultaneously on  adjacent clocks 7in  
space7 differ from each other by an     
indefinitely  small amount4 ,this       
non-rigid reference-body1 which might   
appropriately  be termed a              
8reference-mollusc01 is in the main     
equivalent to a  ,gaussian              
four-dimensional co-ordinate system     
chosen arbitrarily4 ,that  which gives  
the 8mollusc0 a certain                 
comprehensibility as compared with  the 
,gauss co-ordinate system is the 7really
 unjustified7 formal  retention of the  
separate existence of the    space      
co-ordinates as opposed to the time     

co-ordinate4 ,every point on  the       
mollusc is treated as a space-point1 and
 every material point which  is at rest 
relatively to it as at rest1 so long as 
the mollusc is  considered as           
reference-body4 ,the general principle  
of relativity  requires that all these  
molluscs can be used as reference-bodies
 with  equal right and equal success in 
the formulation of the general laws of  
nature2 the laws themselves must be     
quite independent of the choice of      
mollusc4    ,the great power possessed  
by the general principle of relativity  
lies  in the comprehensive limitation   
which is imposed on the laws of nature  
in consequence of what we have seen     
above4      ,,xxix4    ,,the ,,solution 
,,of ,,the ,,problem ,,of ,,gravitation 
,,on ,,the ,,basis ,,of ,,the ,,general 
,,principle ,,of ,,relativity    ,if the
 reader has followed all our previous   
considerations1 he will  have no further
 difficulty in understanding the methods
 leading to the  solution of the problem

 of gravitation4    ,we start off on a  
consideration of a ,galileian domain1   
i4e4 a domain  in which there is no     
gravitational field relative to the     
,galileian  reference-body ,k4 ,the     
behaviour of measuring-rods and clocks  
with  reference to ,k is known from the 
special theory of relativity1  likewise 
the behaviour of 8isolated0 material    
points2 the latter move  uniformly and  
in straight lines4    ,now let us refer 
this domain to a random ,gauss          
coordinate system or to  a 8mollusc0 as 
reference-body ,k4 ,then with respect to
 ,k there is  a gravitational field ,g  
7of a particular kind74 ,we learn the   
behaviour of measuring-rods and clocks  
and also of freely-moving  material     
points with reference to ,k simply by   
mathematical  transformation4 ,we       
interpret this behaviour as the         
behaviour of  measuring-rods1 clocks and
 material points under the influence of 
the  gravitational field ,g4 ,hereupon  
we introduce a hypothesis3 that the     

influence of the gravitational field on 
measuring-rods1 clocks and              
freely-moving material points continues 
to take place according to the  same    
laws1 even in the case where the        
prevailing gravitational field is  not  
derivable from the ,galileian special   
case1 simply by means of a              
transformation of co-ordinates4    ,the 
next step is to investigate the         
space-time behaviour of the             
gravitational field ,g1 which was       
derived from the ,galileian special     
case simply by transformation of the    
coordinates4 ,this behaviour is         
formulated in a law1 which is always    
valid1 no matter how the  reference-body
 7mollusc7 used in the description may  
be chosen4    ,this law is not yet the  
general law of the gravitational field1 
since  the gravitational field under    
consideration is of a special kind4 ,in 
order to find out the general           
law-of-field of gravitation we still    
require to obtain a generalisation of   

the law as found above4 ,this can  be   
obtained without caprice1 however1 by   
taking into consideration the  following
 demands3    7a7 ,the required          
generalisation must likewise satisfy the
 general  postulate of relativity4      
7b7 ,if there is any matter in the      
domain under consideration1 only  its   
inertial mass1 and thus according to    
,section ,,xv only its energy is  of    
importance for its effect in exciting a 
field4      7c7 ,gravitational field and
 matter together must satisfy the law of
  the conservation of energy 7and of    
impulse74      ,finally1 the general    
principle of relativity permits us to   
determine  the influence of the         
gravitational field on the course of all
 those  processes which take place      
according to known laws when a          
gravitational  field is absent i4e4     
which have already been fitted into the 
frame of  the special theory of         
relativity4 ,in this connection we      
proceed in  principle according to the  

method which has already been explained 
for  measuring-rods1 clocks and freely  
moving material points4    ,the theory  
of gravitation derived in this way from 
the general  postulate of relativity    
excels not only in its beauty2 nor in   
removing  the defect attaching to       
classical mechanics which was brought to
 light  in ,section ,,xxi2 nor in       
interpreting the empirical law of the   
equality  of inertial and gravitational 
mass2 but it has also already explained 
a  result of observation in astronomy1  
against which classical mechanics  is   
powerless4    ,if we confine the        
application of the theory to the case   
where the  gravitational fields can be  
regarded as being weak1 and in which all
  masses move with respect to the       
coordinate system with velocities which 
are small compared with the velocity of 
light1 we then obtain as a  first       
approximation the ,newtonian theory4    
,thus the latter theory is  obtained    
here without any particular assumption1 

whereas ,newton had to  introduce the   
hypothesis that the force of attraction 
between mutually  attracting material   
points is inversely proportional to the 
square of  the distance between them4   
,if we increase the accuracy of the     
calculation1 deviations from the theory 
of ,newton make their  appearance1      
practically all of which must           
nevertheless escape the test  of        
observation owing to their smallness4   
,we must draw attention here to one of  
these deviations4 ,according to         
,newton's theory1 a planet moves round  
the sun in an ellipse1 which  would     
permanently maintain its position with  
respect to the fixed  stars1 if we could
 disregard the motion of the fixed stars
 themselves  and the action of the other
 planets under consideration4 ,thus1 if 
we  correct the observed motion of the  
planets for these two influences1  and  
if ,newton's theory be strictly correct1
 we ought to obtain for the  orbit of   
the planet an ellipse1 which is fixed   

with reference to the  fixed stars4     
,this deduction1 which can be tested    
with great accuracy1  has been confirmed
 for all the planets save one1 with the 
precision  that is capable of being     
obtained by the delicacy of observation 
attainable at the present time4 ,the    
sole exception is ,mercury1 the  planet 
which lies nearest the sun4 ,since the  
time of ,leverrier1 it has  been known  
that the ellipse corresponding to the   
orbit of ,mercury1  after it has been   
corrected for the influences mentioned  
above1 is not  stationary with respect  
to the fixed stars1 but that it rotates 
exceedingly slowly in the plane of the  
orbit and in the sense of the  orbital  
motion4 ,the value obtained for this    
rotary movement of the  orbital ellipse 
was #dc seconds of arc per century1 an  
amount ensured to  be correct to within 
a few seconds of arc4 ,this effect can  
be explained  by means of classical     
mechanics only on the assumption of     
hypotheses  which have little           

probability1 and which were devised     
solely for this  purponse4    ,on the   
basis of the general theory of          
relativity1 it is found that the        
ellipse of every planet round the sun   
must necessarily rotate in the  manner  
indicated above2 that for all the       
planets1 with the exception of          
,mercury1 this rotation is too small to 
be detected with the delicacy of        
observation possible at the present     
time2 but that in the case of  ,mercury 
it must amount to #dc seconds of arc per
 century1 a result which  is strictly in
 agreement with observation4    ,apart  
from this one1 it has hitherto been     
possible to make only two  deductions   
from the theory which admit of being    
tested by observation1  to wit1 the     
curvature of light rays by the          
gravitational field of the  sun1#bb and 
a displacement of the spectral lines of 
light reaching us  from large stars1 as 
compared with the corresponding lines   
for light  produced in an analogous     

manner terrestrially 7i4e4 by the same  
kind  of atom74#bc ,these two deductions
 from the theory have both been         
confirmed4       #bb ,first observed by 
,eddington and others in #aiai4 7,cf4   
,appendix   ,,iii74       #bc           
,established by ,adams in #aibd4 7,cf4  
p4 #acb7      ,,part ,,iii3             
,,considerations ,,on ,,the ,,universe  
,,as ,a ,,whole      ,,xxx4             
,,cosmological ,,difficulties ,,of      
,,newton',s ,,theory    ,part from the  
difficulty discussed in ,section ,,xxi1 
there is a second  fundamental          
difficulty attending classical celestial
 mechanics1 which1  to the best of my   
knowledge1 was first discussed in detail
 by the  astronomer ,seeliger4 ,if we   
ponder over the question as to how the  
universe1 considered as a whole1 is to  
be regarded1 the first answer  that     
suggests itself to us is surely this3   
,as regards space 7and time7  the       
universe is infinite4 ,there are stars  
everywhere1 so that the  density of     

matter1 although very variable in       
detail1 is nevertheless on  the average 
everywhere the same4 ,in other words3   
,however far we might  travel through   
space1 we should find everywhere an     
attenuated swarm of  fixed stars of     
approrimately the same kind and density4
    ,this view is not in harmony with   
the theory of ,newton4 ,the latter      
theory rather requires that the universe
 should have a kind of centre  in which 
the density of the stars is a maximum1  
and that as we proceed  outwards from   
this centre the group-density of the    
stars should  diminish1 until finally1  
at great distances1 it is succeeded by  
an  infinite region of emptiness4 ,the  
stellar universe ought to be a finite   
island in the infinite ocean of         
space4#bd       #bd ,proof--,according  
to the theory of ,newton1 the number of 
8lines   of force0 which come from      
infinity and terminate in a mass m is   
proportional to the mass m4 ,if1 on the 
average1 the mass density #j   is       

constant throughout the universe1 then a
 sphere of volume ,v will   enclose the 
average mass #j,v4 ,thus the number of  
lines of force   passing through the    
surface ,f of the sphere into its       
interior is   proportional to #j,v4 ,for
 unit area of the surface of the sphere 
the   number of lines of force which    
enters the sphere is thus proportional  
to #j,v_/,f or to #j,r4 ,hence the      
intensity of the field at the   surface 
would ultimately become infinite with   
increasing radius ,r of   the sphere1   
which is impossible4      ,this         
conception is in itself not very        
satisfactory4 ,it is still less         
satisfactory because it leads to the    
result that the light emitted by  the   
stars and also individual stars of the  
stellar system are  perpetually passing 
out into infinite space1 never to       
return1 and  without ever again coming  
into interaction with other objects of  
nature4 ,such a finite material universe
 would be destined to become  gradually 

but systematically impoverished4    ,in 
order to escape this dilemma1 ,seeliger 
suggested a modification of  ,newton's  
law1 in which he assumes that for great 
distances the force of  attraction      
between two masses diminishes more      
rapidly than would result  from the     
inverse square law4 ,in this way it is  
possible for the mean  density of matter
 to be constant everywhere1 even to     
infinity1 without  infinitely large     
gravitational fields being produced4 ,we
 thus free  ourselves from the          
distasteful conception that the material
 universe  ought to possess something of
 the nature of a centre4 ,of course we  
purchase our emancipation from the      
fundamental difficulties mentioned1  at 
the cost of a modification and          
complication of ,newton's law which  has
 neither empirical nor theoretical      
foundation4 ,we can imagine  innumerable
 laws which would serve the same        
purpose1 without our being  able to     
state a reason why one of them is to be 

preferred to the  others2 for any one of
 these laws would be founded just as    
little on  more general theoretical     
principles as is the law of ,newton4    
,,xxxi4    ,,the ,,possibility ,,of ,a  
8,,finite0 ,,and ,,yet 8,,unbounded0    
,,universe    ,but speculations on the  
structure of the universe also move in  
quite  another direction4 ,the          
development of non-,euclidean geometry  
led to the  recognition of the fact1    
that we can cast doubt on the           
infiniteness  of our space without      
coming into conflict with the laws of   
thought or  with experience 7,riemann1  
,helmholtz74 ,these questions have      
already been  treated in detail and with
 unsurpassable lucidity by ,helmholtz   
and  ,poincar1 whereas ,i can only touch
 on them briefly here4    ,in the first 
place1 we imagine an existence in two   
dimensional space4  ,flat beings with   
flat implements1 and in particular flat 
rigid  measuring-rods1 are free to move 
in a plane4 ,for them nothing exists    

outside of this plane3 that which they  
observe to happen to themselves  and to 
their flat 8things0 is the all-inclusive
 reality of their plane4  ,in           
particular1 the constructions of plane  
,euclidean geometry can be  carried out 
by means of the rods e4g4 the lattice   
construction1  considered in ,section   
,,xxiv4 ,in contrast to ours1 the       
universe of these  beings is            
two-dimensional2 but1 like ours1 it     
extends to infinity4 ,in  their universe
 there is room for an infinite number of
 identical  squares made up of rods1    
i4e4 its volume 7surface7 is infinite4  
,if  these beings say their universe is 
8plane10 there is sense in the          
statement1 because they mean that they  
can perform the constructions of  plane 
,euclidean geometry with their rods4 ,in
 this connection the  individual rods   
always represent the same distance1     
independently of  their position4       
,let us consider now a second           
two-dimensional existence1 but this time

  on a spherical surface instead of on a
 plane4 ,the flat beings with  their    
measuring-rods and other objects fit    
exactly on this surface and  they are   
unable to leave it4 ,their whole        
universe of observation  extends        
exclusively over the surface of the     
sphere4 ,are these beings  able to      
regard the geometry of their universe as
 being plane geometry  and their rods   
withal as the realisation of 8distance08
 ,they cannot do  this4 ,for if they    
attempt to realise a straight line1 they
 will obtain  a curve1 which we         
8three-dimensional beings0 designate as 
a great  circle1 i4e4 a self-contained  
line of definite finite length1 which   
can be measured up by means of a        
measuring-rod4 ,similarly1 this         
universe has a finite area that can be  
compared with the area1 of a  square    
constructed with rods4 ,the great charm 
resulting from this  consideration lies 
in the recognition of the fact that the 
universe of  these beings is finite and 

yet has no limits4    ,but the          
spherical-surface beings do not need to 
go on a world-tour in  order to perceive
 that they are not living in a          
,euclidean universe4  ,they can convince
 themselves of this on every part of    
their 8world10  provided they do not use
 too small a piece of it4 ,starting from
 a  point1 they draw 8straight lines0   
7arcs of circles as judged in three     
dimensional space7 of equal length in   
all directions4 ,they will call  the    
line joining the free ends of these     
lines a 8circle40 ,for a plane  surface1
 the ratio of the circumference of a    
circle to its diameter1  both lengths   
being measured with the same rod1 is1   
according to  ,euclidean geometry of the
 plane1 equal to a constant value 1     
which is  independent of the diameter of
 the circle4 ,on their spherical surface
  our flat beings would find for this   
ratio the value    image#jcf      i4e4 a
 smaller value than 1 the difference    
being the more  considerable1 the       

greater is the radius of the circle in  
comparison  with the radius ,r of the   
8world-sphere40 ,by means of this       
relation  the spherical beings can      
determine the radius of their universe  
78world071 even when only a relatively  
small part of their worldsphere  is     
available for their measurements4 ,but  
if this part is very small  indeed1 they
 will no longer be able to demonstrate  
that they are on a  spherical 8world0   
and not on a ,euclidean plane1 for a    
small part of a  spherical surface      
differs only slightly from a piece of a 
plane of the  same size4    ,thus if the
 spherical surface beings are living on 
a planet of which  the solar system     
occupies only a negligibly small part of
 the spherical  universe1 they have no  
means of determining whether they are   
living in  a finite or in an infinite   
universe1 because the 8piece of         
universe0 to  which they have access is 
in both cases practically plane1 or     
,euclidean4 ,it follows directly from   

this discussion1 that for our           
sphere-beings the circumference of a    
circle first increases with the  radius 
until the 8circumference of the         
universe0 is reached1 and that  it      
thenceforward gradually decreases to    
zero for still further  increasing      
values of the radius4 ,during this      
process the area of the  circle         
continues to increase more and more1    
until finally it becomes  equal to the  
total area of the whole 8world-sphere40 
,perhaps the reader will wonder why we  
have placed our 8beings0 on a  sphere   
rather than on another closed surface4  
,but this choice has its  justification 
in the fact that1 of all closed         
surfaces1 the sphere is  unique in      
possessing the property that all points 
on it are equivalent4  ,i admit that the
 ratio of the circumference c of a      
circle to its  radius r depends on r1   
but for a given value of r it is the    
same  for all points of the             
8worldsphere02 in other words1 the      

8world-sphere0  is a 8surface of        
constant curvature40    ,to this        
two-dimensional sphere-universe there is
 a three-dimensional  analogy1 namely1  
the three-dimensional spherical space   
which was  discovered by ,riemann4 its  
points are likewise all equivalent4 ,it 
possesses a finite volume1 which is     
determined by its 8radius0  7#bb,r#c74  
,is it possible to imagine a spherical  
space8 ,to imagine a  space means       
nothing else than that we imagine an    
epitome of our 8space0  experience1 i4e4
 of experience that we can have in the  
movement of  8rigid0 bodies4 ,in this   
sense we can imagine a spherical space4 
,suppose we draw lines or stretch       
strings in all directions from a  point1
 and mark off from each of these the    
distance r with a  measuring-rod4 ,all  
the free end-points of these lengths lie
 on a  spherical surface4 ,we can       
specially measure up the area 7,f7 of   
this  surface by means of a square made 
up of measuring-rods4 ,if the universe  

is ,euclidean1 then ,f  #dr#b2 if it is 
spherical1 then ,f is always  less than 
#dr#b4 ,with increasing values of r1 ,f 
increases from zero  up to a maximum    
value which is determined by the        
8world-radius10 but  for still further  
increasing values of r1 the area        
gradually  diminishes to zero4 ,at      
first1 the straight lines which radiate 
from the  starting point diverge farther
 and farther from one another1 but later
  they approach each other1 and finally 
they run together again at a            
8counter-point0 to the starting point4  
,under such conditions they have        
traversed the whole spherical space4 ,it
 is easily seen that the                
three-dimensional spherical space is    
quite analogous to the  two-dimensional 
spherical surface4 ,it is finite 7i4e4  
of finite  volume71 and has no bounds4  
,it may be mentioned that there is yet  
another kind of curved space3           
8elliptical space40 ,it can be regarded 
as a curved space in which the  two     

8counter-points0 are identical          
7indistinguishable from each other74    
,an elliptical universe can thus be     
considered to some extent as a  curved  
universe possessing central symmetry4   
,it follows from what has been said1    
that closed spaces without limits  are  
conceivable4 ,from amongst these1 the   
spherical space 7and the  elliptical7   
excels in its simplicity1 since all     
points on it are  equivalent4 ,as a     
result of this discussion1 a most       
interesting question  arises for        
astronomers and physicists1 and that is 
whether the universe  in which we live  
is infinite1 or whether it is finite in 
the manner of  the spherical universe4  
,our experience is far from being       
sufficient to  enable us to answer this 
question4 ,but the general theory of    
relativity  permits of our answering it 
with a moderate degree of certainty1 and
 in  this connection the difficulty     
mentioned in ,section ,,xxx finds its   
solution4      ,,xxxii4    ,,the        

,,structure ,,of ,,space ,,according    
,,to ,,the ,,general ,,theory ,,of      
,,relativity    ,according to the       
general theory of relativity1 the       
geometrical  properties of space are not
 independent1 but they are determined by
  matter4 ,thus we can draw conclusions 
about the geometrical structure of  the 
universe only if we base our            
considerations on the state of the      
matter as being something that is known4
 ,we know from experience that1  for a  
suitably chosen co-ordinate system1 the 
velocities of the stars  are small as   
compared with the velocity of           
transmission of light4 ,we  can thus as 
a rough approximation arrive at a       
conclusion as to the  nature of the     
universe as a whole1 if we treat the    
matter as being at  rest4    ,we already
 know from our previous discussion that 
the behaviour of  measuring-rods and    
clocks is influenced by gravitational   
fields1 i4e4  by the distribution of    
matter4 ,this in itself is sufficient to

 exclude  the possibility of the exact  
validity of ,euclidean geometry in our  
universe4 ,but it is conceivable that   
our universe differs only slightly  from
 a ,euclidean one1 and this notion seems
 all the more probable1  since          
calculations show that the metrics of   
surrounding space is  influenced only to
 an exceedingly small extent by masses  
even of the  magnitude of our sun4 ,we  
might imagine that1 as regards geometry1
 our  universe behaves analogously to a 
surface which is irregularly curved  in 
its individual parts1 but which nowhere 
departs appreciably from a  plane3      
something like the rippled surface of a 
lake4 ,such a universe  might fittingly 
be called a quasi-,euclidean universe4  
,as regards its  space it would be      
infinite4 ,but calculation shows that in
 a  quasi-,euclidean universe the       
average density of matter would         
necessarily be nil4 ,thus such a        
universe could not be inhabited by      
matter everywhere2 it would present to  

us that unsatisfactory picture  which we
 portrayed in ,section ,,xxx4    ,if we 
are to have in the universe an average  
density of matter which  differs from   
zero1 however small may be that         
difference1 then the  universe cannot be
 quasi-,euclidean4 ,on the contrary1 the
 results of  calculation indicate that  
if matter be distributed uniformly1 the 
universe would necessarily be spherical 
7or elliptical74 ,since in  reality the 
detailed distribution of matter is not  
uniform1 the real  universe will deviate
 in individual parts from the spherical1
 i4e4  the universe will be             
quasi-spherical4 ,but it will be        
necessarily  finite4 ,in fact1 the      
theory supplies us with a simple        
connection#be  between the space-expanse
 of the universe and the average density
 of  matter in it4       #be ,for the   
radius ,r of the universe we obtain the 
equation      image#jcg      ,the use of
 the ,c4,g4,s4 system in this equation  
gives #b_/k  #a4jh x #ajbg2   is the    

average density of the matter and k is a
 constant connected  with the ,newtonian
 constant of gravitation4               
,,appendices      ,,appendix ,i         
,,simple ,,derivation ,,of ,,the        
,,lorentz ,,transformation              
7,,supplementary ,,to ,,section ,,xi7   
,for the relative orientation of the    
co-ordinate systems indicated in  ,fig4 
#b1 the x-axes of both systems          
permanently coincide4 ,in the  present  
case we can divide the problem into     
parts by considering first  only events 
which are localised on the x-axis4 ,any 
such event is  represented with respect 
to the co-ordinate system ,k by the     
abscissa  x and the time t1 and with    
respect to the system ,k by the         
abscissa x and the time t4 ,we require  
to find x and t when  x and t are given4
    ,a light-signal1 which is proceeding
 along the positive axis of x1 is       
transmitted according to the equation   
x  ct      or    x - ct  #j 4 4 4 4 4   
7#a74      ,since the same light-signal 

has to be transmitted relative to ,k    
with  the velocity c1 the propagation   
relative to the system ,k will be       
represented by the analogous formula    
x - ct  #j 4 4 4 4 4 7#b7      ,those   
space-time points 7events7 which satisfy
 7#a7 must also satisfy  7#b74          
,obviously this will be the case when   
the relation    7x - ct7  7x - ct7 4 4 4
 7#c74      is fulfilled in general1    
where  indicates a constant2 for1       
according  to 7#c71 the disappearance of
 7x - ct7 involves the disappearance of 
7x - ct74    ,if we apply quite similar 
considerations to light rays which are  
being  transmitted along the negative   
x-axis1 we obtain the condition    7x   
ct7  7x  ct7 4 4 4 7#d74      ,by adding
 7or subtracting7 equations 7#c7 and    
7#d71 and introducing for  convenience  
the constants a and b in place of the   
constants  and   where    image#jch     
and    image#jci      we obtain the     
equations    image#jdj      ,we should  
thus have the solution of our problem1  

if the constants a  and b were known4   
,these result from the following        
discussion4    ,for the origin of ,k we 
have permanently x  #j1 and hence       
according to the first of the equations 
7#e7    image#jda      ,if we call v the
 velocity with which the origin of ,k is
 moving  relative to ,k1 we then have   
image#jdb      ,the same value v can be 
obtained from equations 7#e71 if we     
calculate  the velocity of another point
 of ,k relative to ,k1 or the velocity  
7directed towards the negative x-axis7  
of a point of ,k with respect  to ,k4   
,in short1 we can designate v as the    
relative velocity of the  two systems4  
,furthermore1 the principle of          
relativity teaches us that1 as judged   
from ,k1 the length of a unit           
measuring-rod which is at rest with     
reference to ,k must be exactly the same
 as the length1 as judged  from ,k1 of a
 unit measuring-rod which is at rest    
relative to ,k4 ,in  order to see how   
the points of the x-axis appear as      

viewed from ,k1  we only require to take
 a 8snapshot0 of ,k from ,k2 this means 
that  we have to insert a particular    
value of t 7time of ,k71 e4g4 t   #j4   
,for this value of t we then obtain from
 the first of the equations  7#e7    x  
ax      ,two points of the x-axis which 
are separated by the distance x   #a    
when measured in the ,k system are thus 
separated in our  instantaneous         
photograph by the distance    image#jdc 
,but if the snapshot be taken from ,k7t 
#j71 and if we eliminate  t from the    
equations 7#e71 taking into account the 
expression 7#f71 we  obtain    image#jdd
      ,from this we conclude that two   
points on the x-axis separated by the   
distance #a 7relative to ,k7 will be    
represented on our snapshot by the      
distance    image#jde      ,but from    
what has been said1 the two snapshots   
must be identical2 hence  x in 7#g7 must
 be equal to x in 7#g;a71 so that we    
obtain    image#jdf      ,the equations 
7#f7 and 7#g;b7 determine the constants 

a and b4 ,by  inserting the values of   
these constants in 7#e71 we obtain the  
first and  the fourth of the equations  
given in ,section ,,xi4    image#jdg    
,thus we have obtained the ,lorentz     
transformation for events on the        
x-axis4 ,it satisfies the condition     
x#b - c#bt#b  x#b - c#bt#b 4 4 4 4 4 4  
7#h;a74      ,the extension of this     
result1 to include events which take    
place  outside the x-axis1 is obtained  
by retaining equations 7#h7 and         
supplementing them by the relations     
image#jdh      ,in this way we satisfy  
the postulate of the constancy of the   
velocity  of light in vacuo for rays of 
light of arbitrary direction1 both for  
the system ,k and for the system ,k4    
,this may be shown in the  following    
manner4    ,we suppose a light-signal   
sent out from the origin of ,k at the   
time  t  #j4 ,it will be propagated     
according to the equation    image#jdi  
or1 if we square this equation1         
according to the equation    x#b  y#b   

z#b - c#bt#b  #j 4 4 4 4 4 7#aj74       
,it is required by the law of           
propagation of light1 in conjunction    
with  the postulate of relativity1 that 
the transmission of the signal in       
question should take place--as judged   
from ,k--in accordance with the         
corresponding formula    r  ct      or1 
x#b  y#b  z#b - c#bt#b  #j 4 4 4 4 4 4  
7#aj;a74      ,in order that equation   
7#aj;a7 may be a consequence of equation
 7#aj71  we must have    x#b  y#b  z#b -
 c#bt#b   7x#b  y#b  z#b - c#bt#b7      
7#aa74      ,since equation 7#h;a7 must 
hold for points on the x-axis1 we thus  
have   #a4 ,it is easily seen that the  
,lorentz transformation really          
satisfies equation 7#aa7 for   #a2 for  
7#aa7 is a consequence of 7#h;a7  and   
7#i71 and hence also of 7#h7 and 7#i74  
,we have thus derived the  ,lorentz     
transformation4    ,the ,lorentz        
transformation represented by 7#h7 and  
7#i7 still requires to  be generalised4 
,obviously it is immaterial whether the 

axes of ,k be  chosen so that they are  
spatially parallel to those of ,k4 ,it  
is also  not essential that the velocity
 of translation of ,k with respect to   
,k should be in the direction of the    
x-axis4 ,a simple consideration  shows  
that we are able to construct the       
,lorentz transformation in this  general
 sense from two kinds of                
transformations1 viz4 from ,lorentz     
transformations in the special sense and
 from purely spatial  transformations4  
which corresponds to the replacement of 
the  rectangular co-ordinate system by a
 new system with its axes pointing  in  
other directions4    ,mathematically1 we
 can characterise the generalised       
,lorentz  transformation thus3    ,it   
expresses x1 y1 x1 t1 in terms of linear
 homogeneous functions  of x1 y1 x1 t1  
of such a kind that the relation    x#b 
y#b  z#b - c#bt#b  x#b  y#b  z#b -      
c#bt#b  7#aa;a74      is satisficd      
identically4 ,that is to say3 ,if we    
substitute their  expressions in x1 y1  

x1 t1 in place of x1 y1 x1 t1 on the    
left-hand side1 then the left-hand side 
of 7#aa;a7 agrees with the  right-hand  
side4      ,,appendix ,,ii              
,,minkowski',s ,,four-,,dimensional     
,,space 78,,world07    7,,supplementary 
,,to ,,section ,,xvii7    ,we can       
characterise the ,lorentz transformation
 still more simply if we  introduce the 
imaginary    image#jca      in place of 
t1 as time-variable4 ,if1 in accordance 
with this1 we  insert    image#jej      
and similarly for the accented system   
,k1 then the condition which is         
identically satisfied by the            
transformation can be expressed thus3   
x#ab  x#bb  x#cb  x#db  x#ab  x#bb  x#cb
  x#db 7#ab74      ,that is1 by the     
afore-mentioned choice of 8coordinates10
 7#aa;a7 see  the end of ,appendix ,,ii 
is transformed into this equation4      
,we see from 7#ab7 that the imaginary   
time co-ordinate x#d1 enters into  the  
condition of transformation in exactly  
the same way as the space  co-ordinates 

x#a1 x#b1 x#c4 ,it is due to this fact  
that1 according  to the theory of       
relativity1 the 8time0 x#d1 enters into 
natural laws  in the same form as the   
space co ordinates x#a1 x#b1 x#c4    ,a 
four-dimensional continuum described by 
the 8co-ordinates0 x#a1  x#b1 x#c1 x#d1 
was called 8world0 by ,minkowski1 who   
also termed a  point-event a            
8world-point40 ,from a 8happening0 in   
three-dimensional  space1 physics       
becomes1 as it were1 an 8existence0 in  
the  four-dimensional 8world40    ,this 
four-dimensional 8world0 bears a close  
similarity to the  three-dimensional    
8space0 of 7,euclidean7 analytical      
geometry4 ,if we  introduce into the    
latter a new ,cartesian co-ordinate     
system 7x#a1  x#b1 x#c7 with the same   
origin1 then x#a1 x#b1 x#c1 are  linear 
homogeneous functions of x#a1 x#b1 x#c  
which identically  satisfy the equation 
x#ab  x#bb  x#cb  x#ab  x#bb  x#cb      
,the analogy with 7#ab7 is a complete   
one4 ,we can regard ,minkowski's        

8world0 in a formal manner as a         
four-dimensional ,euclidean space 7with 
an imaginary time coordinate72 the      
,lorentz transformation corresponds  to 
a 8rotation0 of the co-ordinate system  
in the four-dimensional  8world40       
,,appendix ,,iii    ,,the ,,experimental
 ,,confirmation ,,of ,,the ,,general    
,,theory ,,of ,,relativity    ,from a   
systematic theoretical point of view1 we
 may imagine the process  of evolution  
of an empirical science to be a         
continuous process of  induction4       
,theories are evolved and are expressed 
in short compass as  statements of a    
large number of individual observations 
in the form of  empirical laws1 from    
which the general laws can be           
ascertained by  comparison4 ,regarded in
 this way1 the development of a science 
bears  some resemblance to the          
compilation of a classified catalogue4  
,it is1  as it were1 a purely empirical 
enterprise4    ,but this point of view  
by no means embraces the whole of the   

actual  process2 for it slurs over the  
important part played by intuition and  
deductive thought in the development of 
an exact science4 ,as soon as a  science
 has emerged from its initial stages1   
theoretical advances are  no longer     
achieved merely by a process of         
arrangement4 ,guided by  empirical data1
 the investigator rather develops a     
system of thought  which1 in general1 is
 built up logically from a small number 
of  fundamental assumptions1 the        
so-called axioms4 ,we call such a system
 of  thought a theory4 ,the theory finds
 the justification for its  existence in
 the fact that it correlates a large    
number of single  observations1 and it  
is just here that the 8truth0 of the    
theory lies4    ,corresponding to the   
same complex of empirical data1 there   
may be  several theories1 which differ  
from one another to a considerable      
extent4 ,but as regards the deductions  
from the theories which are  capable of 
being tested1 the agreement between the 

theories may be so  complete that it    
becomes difficult to find any deductions
 in which the  two theories differ from 
each other4 ,as an example1 a case of   
general  interest is available in the   
province of biology1 in the ,darwinian  
theory of the development of species by 
selection in the struggle for           
existence1 and in the theory of         
development which is based on the       
hypothesis of the hereditary            
transmission of acquired characters4    
,we have another instance of            
far-reaching agreement between the      
deductions from two theories in         
,newtonian mechanics on the one hand1   
and the general theory of relativity on 
the other4 ,this agreement goes  so far1
 that up to the present we have been    
able to find only a few  deductions from
 the general theory of relativity which 
are capable of  investigation1 and to   
which the physics of pre-relativity days
 does not  also lead1 and this despite  
the profound difference in the          

fundamental  assumptions of the two     
theories4 ,in what follows1 we shall    
again  consider these important         
deductions1 and we shall also discuss   
the  empirical evidence appertaining to 
them which has hitherto been  obtained4 
7a7 ,motion of the ,perihelion of       
,mercury    ,according to ,newtonian    
mechanics and ,newton's law of          
gravitation1 a  planet which is         
revolving round the sun would describe  
an ellipse round  the latter1 or1 more  
correctly1 round the common centre of   
gravity of  the sun and the planet4 ,in 
such a system1 the sun1 or the common   
centre  of gravity1 lies in one of the  
foci of the orbital ellipse in such a   
manner that1 in the course of a         
planet-year1 the distance sun-planet    
grows from a minimum to a maximum1 and  
then decreases again to a  minimum4 ,if 
instead of ,newton's law we insert a    
somewhat different law  of attraction   
into the calculation1 we find that1     
according to this new  law1 the motion  

would still take place in such a manner 
that the  distance sun-planet exhibits  
periodic variations2 but in this case   
the  angle described by the line joining
 sun and planet during such a period    
7from perihelion--closest proximity to  
the sun--to perihelion7 would  differ   
from #cfj4 ,the line of the orbit would 
not then be a closed one  but in the    
course of time it would fill up an      
annular part of the  orbital plane1 viz4
 between the circle of least and the    
circle of  greatest distance of the     
planet from the sun4    ,according also 
to the general theory of relativity1    
which differs of  course from the theory
 of ,newton1 a small variation from the 
,newton-,kepler motion of a planet in   
its orbit should take place1 and in     
such away1 that the angle described by  
the radius sun-planet between  one      
perhelion and the next should exceed    
that corresponding to one  complete     
revolution by an amount given by        
image#jea      7,n4,b4--,one complete   

revolution corresponds to the angle #b  
in the  absolute angular measure        
customary in physics1 and the above     
expression  given the amount by which   
the radius sun-planet exceeds this angle
  during the interval between one       
perihelion and the next47 ,in this      
expression a represents the major       
semi-axis of the ellipse1 e its         
eccentricity1 c the velocity of light1  
and ,t the period of  revolution of the 
planet4 ,our result may also be stated  
as follows3  ,according to the general  
theory of relativity1 the major axis of 
the  ellipse rotates round the sun in   
the same sense as the orbital motion  of
 the planet4 ,theory requires that this 
rotation should amount to #dc  seconds  
of arc per century for the planet       
,mercury1 but for the other  ,planets of
 our solar system its magnitude should  
be so small that it  would necessarily  
escape detection4#bf       #bf          
,especially since the next planet ,venus
 has an orbit that is   almost an exact 

circle1 which makes it more difficult to
 locate the   perihelion with precision4
      ,in point of fact1 astronomers    
have found that the theory of ,newton   
does  not suffice to calculate the      
observed motion of ,mercury with an     
exactness corresponding to that of the  
delicacy of observation  attainable at  
the present time4 ,after taking account 
of all the  disturbing influences       
exerted on ,mercury by the remaining    
planets1 it  was found 7,leverrier3     
#ahei2 and ,newcomb3 #ahie7 that an     
unexplained  perihelial movement of the 
orbit of ,mercury remained over1 the    
amount  of which does not differ        
sensibly from the above mentioned #dc   
seconds  of arc per century4 ,the       
uncertainty of the empirical result     
amounts to  a few seconds only4    7b7  
,deflection of ,light by a              
,gravitational ,field    image#jeb      
,in ,section ,,xxii it has been already 
mentioned that according to the  general
 theory of relativity1 a ray of light   

will experience a  curvature of its path
 when passing through a gravitational   
field1 this  curvature being similar to 
that experienced by the path of a body  
which  is projected through a           
gravitational field4 ,as a result of    
this theory1  we should expect that a   
ray of light which is passing close to a
  heavenly body would be deviated       
towards the latter4 ,for a ray of light 
which passes the sun at a distance of   
sun-radii from its centre1 the  angle of
 deflection 77 should amount to         
image#jec      ,it may be added that1   
according to the theory1 half of this   
deflection  is produced by the          
,newtonian field of attraction of the   
sun1 and the  other half by the         
geometrical modification 78curvature07  
of space  caused by the sun4    ,this   
result admits of an experimental test by
 means of the photographic  registration
 of stars during a total eclipse of the 
sun4 ,the only  reason why we must wait 
for a total eclipse is because at every 

other  time the atmosphere is so        
strongly illuminated by the light from  
the  sun that the stars situated near   
the sun's disc are invisible4 ,the      
predicted effect can be seen clearly    
from the accompanying diagram4 ,if  the 
sun 7,s7 were not present1 a star which 
is practically infinitely  distant would
 be seen in the direction ,d#a1 as      
observed front the  earth4 ,but as a    
consequence of the deflection of light  
from the star by  the sun1 the star will
 be seen in the direction ,d#b1 i4e4 at 
a  somewhat greater distance from the   
centre of the sun than corresponds  to  
its real position4    ,in practice1 the 
question is tested in the following way4
 ,the stars in  the neighbourhood of the
 sun are photographed during a solar    
eclipse4    ,in addition1 a second      
photograph of the same stars is taken   
when the  sun is situated at another    
position in the sky1 i4e4 a few months  
earlier or later4 ,as compared with the 
standard photograph1 the  positions of  

the stars on the eclipse-photograph     
ought to appear  displaced radially     
outwards 7away from the centre of the   
sun7 by an  amount corresponding to the 
angle a4    ,we are indebted to the     
,british ,royal ,society and to the     
,royal  ,astronomical ,society for the  
investigation of this important         
deduction4  ,undaunted by the first     
world war and by difficulties of both a 
material and a psychological nature     
aroused by the war1 these societies     
equipped two expeditions--to ,sobral    
7,brazil71 and to the island of         
,principe 7,west ,africa7--and sent     
several of ,britain's most celebrated   
astronomers 7,eddington1 ,cottingham1   
,crommelin1 ,davidson71 in order to     
obtain photographs of the solar eclipse 
of #bith ,may1 #aiai4 ,the relative     
discrepancies to be expected between the
 stellar photographs obtained  during   
the eclipse and the comparison          
photographs amounted to a few           
hundredths of a millimetre only4 ,thus  

great accuracy was necessary in  making 
the adjustments required for the taking 
of the photographs1 and  in their       
subsequent measurement4    ,the results 
of the measurements confirmed the theory
 in a thoroughly  satisfactory manner4  
,the rectangular components of the      
observed and of  the calculated         
deviations of the stars 7in seconds of  
arc7 are set  forth in the following    
table of results3    image#jed      7c7 
,displacement of ,spectral ,lines       
,towards the ,red    ,in ,section       
,,xxiii it has been shown that in a     
system ,k which is in  rotation with    
regard to a ,galileian system ,k1 clocks
 of identical  construction1 and which  
are considered at rest with respect to  
the  rotating reference-body1 go at     
rates which are dependent on the        
positions of the clocks4 ,we shall now  
examine this dependence  quantitatively4
 ,a clock1 which is situated at a       
distance r from the  centre of the disc1
 has a velocity relative to ,k which is 

given by    v  r1      where  represents
 the angular velocity of rotation of the
 disc ,k  with respect to ,k4 ,if v#j1  
represents the number of ticks of the   
clock per unit time 78rate0 of the      
clock7 relative to ,k when the  clock is
 at rest1 then the 8rate0 of the clock  
7v7 when it is moving  relative to ,k   
with a velocity v1 but at rest with     
respect to the  disc1 will1 in          
accordance with ,section ,,xii1 be given
 by    image#jee      or with sufficient
 accuracy by    image#jef      ,this    
expression may also be stated in the    
following form3    image#jeg      ,if we
 represent the difference of potential  
of the centrifugal force  between the   
position of the clock and the centre of 
the disc by 1  i4e4 the work1 considered
 negatively1 which must be performed on 
the  unit of mass against the           
centrifugal force in order to transport 
it  from the position of the clock on   
the rotating disc to the centre of  the 
disc1 then we have    image#jeh         

,from this it follows that    image#jei 
,in the first place1 we see from this   
expression that two clocks of  identical
 construction will go at different rates
 when situated at  different distances  
from the centre of the disc4 ,this      
result is also  valid from the          
standpoint of an observer who is        
rotating with the disc4    ,now1 as     
judged from the disc1 the latter is in a
 gravitational field of  potential 1    
hence the result we have obtained will  
hold quite  generally for gravitational 
fields4 ,furthermore1 we can regard an  
atom  which is emitting spectral lines  
as a clock1 so that the following       
statement will hold3    ,an atom absorbs
 or emits light of a frequency which is 
dependent on  the potential of the      
gravitational field in which it is      
situated4    ,the frequency of an atom  
situated on the surface of a heavenly   
body  will be somewhat less than the    
frequency of an atom of the same element
  which is situated in free space 7or on

 the surface of a smaller  celestial    
body74    ,now   - ,k 7,m_/r71 where ,k 
is ,newton's constant of gravitation1   
and  ,m is the mass of the heavenly     
body4 ,thus a displacement towards the  
red ought to take place for spectral    
lines produced at the surface of  stars 
as compared with the spectral lines of  
the same element produced  at the       
surface of the earth1 the amount of this
 displacement being    image#jfj        
,for the sun1 the displacement towards  
the red predicted by theory  amounts to 
about two millionths of the wave-length4
 ,a trustworthy  calculation is not     
possible in the case of the stars1      
because in  general neither the mass ,m 
nor the radius r are known4    ,it is an
 open question whether or not this      
effect exists1 and at the  present time 
7#aibj7 astronomers are working with    
great zeal towards the  solution4 ,owing
 to the smallness of the effect in the  
case of the sun1  it is difficult to    
form an opinion as to its existence4    

,whereas ,grebe  and ,bachem 7,bonn71 as
 a result of their own measurements and 
those of  ,evershed and ,schwarzschild  
on the cyanogen bands1 have placed the  
existence of the effect almost beyond   
doubt1 while other investigators1       
particularly ,st4 ,john1 have been led  
to the opposite opinion in  consequence 
of their measurements4    ,mean         
displacements of lines towards the less 
refrangible end of the  spectrum are    
certainly revealed by statistical       
investigations of the  fixed stars2 but 
up to the present the examination of the
 available  data does not allow of any  
definite decision being arrived at1 as  
to  whether or not these displacements  
are to be referred in reality to the    
effect of gravitation4 ,the results of  
observation have been collected         
together1 and discussed in detail from  
the standpoint of the question  which   
has been engaging our attention here1 in
 a paper by ,e4 ,freundlich  entitled   
8,zur ,prfung der allgemeinen           

,relativitts-,theorie0 7,die            
,naturwissenschaften1 #aiai1 ,no4 #ce1  
p4 #ebj3 ,julius ,springer1 ,berlin74   
,at all events1 a definite decision will
 be reached during the next few  years4 
,if the displacement of spectral lines  
towards the red by the  gravitational   
potential does not exist1 then the      
general theory of  relativity will be   
untenable4 ,on the other hand1 if the   
cause of the  displacement of spectral  
lines be definitely traced to the       
gravitational potential1 then the study 
of this displacement will  furnish us   
with important information as to the    
mass of the heavenly  bodies4#bg        
#bg ,the displacement of spectral lines 
towards the red end of the   spectrum   
was definitely established by ,adams in 
#aibd1 by observations   on the dense   
companion of ,sirius1 for which the     
effect is about thirty   times greater  
than for the ,sun4 ,r4,w4,l4--translator
      ,,appendix ,,iv    ,,the          
,,structure ,,of ,,space ,,according    

,,to ,,the ,,general ,,theory ,,of      
,,relativity    7,,supplementary ,,to   
,,section ,,xxxii7    ,since the        
publication of the first edition of this
 little book1 our  knowledge about the  
structure of space in the large         
78cosmological  problem07 has had an    
important development1 which ought to be
 mentioned  even in a popular           
presentation of the subject4    ,my     
original considerations on the subject  
were based on two hypotheses3    7#a7   
,there exists an average density of     
matter in the whole of space  which is  
everywhere the same and different from  
zero4    7#b7 ,the magnitude 78radius07 
of space is independent of time4        
,both these hypotheses proved to be     
consistent1 according to the general    
theory of relativity1 but only after a  
hypothetical term was added to  the     
field equations1 a term which was not   
required by the theory as  such nor did 
it seem natural from a theoretical point
 of view  78cosmological term of the    

field equations074    ,hypothesis 7#b7  
appeared unavoidable to me at the time1 
since ,i thought  that one would get    
into bottomless speculations if one     
departed from  it4    ,however1 already 
in the 'twenties1 the ,russian          
mathematician ,friedman  showed that a  
different hypothesis was natural from a 
purely  theoretical point of view4 ,he  
realized that it was possible to        
preserve  hypothesis 7#a7 without       
introducing the less natural            
cosmological term  into the field       
equations of gravitation1 if one was    
ready to drop  hypothesis 7#b74 ,namely1
 the original field equations admit a   
solution  in which the 8world radius0   
depends on time 7expanding space74 ,in  
that  sense one can say1 according to   
,friedman1 that the theory demands an   
expansion of space4    ,a few years     
later ,hubble showed1 by a special      
investigation of the  extra-galactic    
nebulae 78milky ways071 that the        
spectral lines emitted  showed a red    

shift which increased regularly with the
 distance of the  nebulae4 ,this can be 
interpreted in regard to our present    
knowledge  only in the sense of         
,doppler's principle1 as an expansive   
motion of the  system of stars in the   
large--as required1 according to        
,friedman1 by the  field equations of   
gravitation4 ,hubble's discovery can1   
therefore1 be  considered to some extent
 as a confirmation of the theory4       
,there does arise1 however1 a strange   
difficulty4 ,the interpretation of  the 
galactic line-shift discovered by       
,hubble as an expansion 7which can      
hardly be doubted from a theoretical    
point of view71 leads to an origin  of  
this expansion which lies 8only0 about  
#aji years ago1 while physical          
astronomy makes it appear likely that   
the development of individual  stars and
 systems of stars takes considerably    
longer4 ,it is in no way  known how this
 incongruity is to be overcome4    ,i   
further want to remark that the theory  

of expanding space1 together  with the  
empirical data of astronomy1 permit no  
decision to be reached  about the finite
 or infinite character of               
7three-dimensional7 space1  while the   
original 8static0 hypothesis of space   
yielded the closure  7finiteness7 of    
space4    ,k  co-ordinate system    x1 y
  two-dimensional co-ordinates    x1 y1 
z  three-dimensional co-ordinates    x1 
y1 z1 t  four-dimensional co-ordinates  
t  time    ,i  distance    v  velocity  
,f  force    ,g  gravitational field