THEORY OF RELATIVITY 
279 
rods along the circumference suffer a Lorentz contraction while 
those along the diameter do not. The ratio of circumference to 
diameter now comes out greater than tt. From this follows that 
the laws of configuration of rigid bodies does no longer conform 
to Euclidean geometry. Again if we place two synchronous 
clocks, one in the center and one in the circumference of the 
system K' in uniform rotation, the latter will go slower if ob- 
served by someone at the center. From this follows that space 
and time can not be defined with respect to K' as they were in the 
restricted theory. 
According to the principle of equivalence K' is here also to be 
considered as a frame of reference at rest with respect to which 
a gravitational field is present. (Field of centrifugal force). We 
must then come to the conclusion that a gravitational field in- 
fluences and in fact determines the metrical properties or the 
‘metrics' of the time-space continuum. If then the laws of con- 
figuration of rigid bodies are to be expressed geometrically, the 
geometry is non-Euclidean. 
A similar situation presents itself in the consideration of gen- 
eral two-dimensional surfaces as intimated above. On the plane, 
Cartesian coordinates , x^ will suffice to measure any portion 
of the same by means of rigid measuring rods ; not so on the sur- 
face of an ellipsoid for instance. Gauss introduced curvilinear 
coordinates to overcome the difficulty, which satisfying the con- 
dition of continuity are wholly arbitrary otherwise, (ffexible 
threads take the place of shord rigid rods). Later these co- 
ordinates were related to the metrical properties of the surface. 
Then Rieman extended the dimensions to any number, establish- 
ing infinitesimal geometry to n dimensions in which the general- 
ized Pythagorean theorem holds. Applying this to our space- 
time frame of reference of the general principle of relativity we 
have the arbitrary coordinates x^ , x^ , Xg , x^ numbering uniquely 
‘world points' and the invariant interval between two such points. 
ds“=g dx dx G from i — 4 g =g 
ILLV U V I^V fXV VtX 
in which the g describe with respect to the coordinates the 
UM 
metrical relations of the continuum and at the same time the 
gravitational field. 
The above expression expanded would become : 
ds2=g dx^+g, dx^-f g^ dx^-j-g dx2+2g dx dx -f-2g dx dx 
®11 1 ‘ ®22 2 ' ^33 3 ' *^44 4 ' "l2 1 2 ' ®13 1 3 
+2g„dxjdx,+2g,3dx,dx3+2g2,dx,dx4+2g34dx3dx. 
