120 GRAVITATION— THE LAW 



two-dimensional manifolds embedded in a three-dimen- 

 sional space. The absolute curvature at any point is 

 measured by a single quantity called the radius of spheri- 

 cal curvature. But space-time is a four-dimensional 

 manifold embedded in — well, as many dimensions as it 

 can find new ways to twist about in. Actually a four- 

 dimensional manifold is amazingly ingenious in discover- 

 ing new kinds of contortion, and its invention is not 

 exhausted until it has been provided with six extra 

 dimensions, making ten dimensions in all. Moreover, 

 twenty distinct measures are required at each point to 

 specify the particular sort and amount of twistiness 

 there. These measures are called coefficients of curva- 

 ture. Ten of the coefficients stand out more prominently 

 than the other ten. 



Einstein's law of gravitation asserts that the ten prin- 

 cipal coefficients of curvature are zero in empty space. 



If there were no curvature, i.e. if all the coefficients 

 were zero, there would be no gravitation. Bodies would 

 move uniformly in straight lines. If curvature were 

 unrestricted, i.e. if all the coefficients had unpredictable 

 values, gravitation would operate arbitrarily and with- 

 out law. Bodies would move just anyhow. Einstein 

 takes a condition midway between; ten of the coefficients 

 are zero and the other ten are arbitrary. That gives 

 a world containing gravitation limited by a law. The 

 coefficients are naturally separated into two groups of 

 ten, so that there is no difficulty in choosing those which 

 are to vanish. 



To the uninitiated it may seem surprising that an 

 exact law of Nature should leave some of the coefficients 

 arbitrary. But we need to leave something over to be 

 settled when we have specified the particulars of the 

 problem to which it is proposed to apply the law. A 



