194 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1961 



coordinate numbers that denote an event. But besides these, we need 

 10 functions that tell us whether the world we describe (in a given 

 but arbitrary coordinate system) is flat or not flat, or, as we often 

 say, Euclidean or Riemannian. 



"We can now formulate Einstein's great and new idea : The 10 func- 

 tions that characterize the geometry of our four-dimensional world 

 are the same 10 functions that characterize the gravitational field. 

 A world without masses, without electrons, without an electromagnetic 

 field, is an empty world. Such an empty world is flat. But if masses 

 appear, if charged particles appear, if an electromagnetic field ap- 

 pears, then a gravitational field appears too. If the gravitational 

 field appears, then our world becomes curved. Its geometry is 

 Riemannian — that is, non-Euclidean. 



Thus the same 10 functions characterize the metric and the gravi- 

 tational field. The word "metric" indicates the connection between 

 these 10 functions and the geometry of our world. The word "grav- 

 itational" indicates that the same 10 functions describe the gravita- 

 tional phenomena of our world. The fact that we can use either or 

 both of these words indicates that the physical gravitational field 

 has its geometric counterpart. Physics — as far as the gravitational 

 field is concerned — is reflected as geometry. The geometry of our 

 world and the gravitational field are shaped, formed, by moving 

 masses, moving electric charges, and by the electromagnetic field. 

 Thus the comiection 



Physics * — > Geometry 



exists only for the gravitational field. We repeat : The gravitational 

 field is a geometric field too; the electromagnetic field is a purely 

 physical field. 



About 1920, General Relativity Theory presented a curious mixture 

 of geometry and physics. To understand Einstein's later endeavors, 

 we must understand his reason for dissatisfaction with the structure 

 of field theories as they were then known. Thus, in Maxwell's 

 equations we have : 



Given : Charges and their motion 

 Unknown : The electromagnetic field 



In Einstein's relativity theory, we have : 



Given : Masses and their motion 

 Unknown : The gravitational or metrical field 



In relativity theory, the given and unknown form a strange mixture. 

 Mass, energy has no geometrical counterpart. But the field has ! 



THE TWO SINS 

 General Relativity Theory was born because of Einstein's dissat- 

 isfaction with the classical theory of gravitation. The new theory was 



