TIME GEOMETRY 133 



Time Geometry. Einstein's law of gravitation controls 

 a geometrical quantity curvature in contrast to Newton's 

 law which controls a mechanical quantity force. To 

 understand the origin of this geometrisation of the world 

 in the relativity theory we must go back a little. 



The science which deals with the properties of space 

 is called geometry. Hitherto geometry has not included 

 time in its scope. But now space and time are so inter- 

 locked that there must be one science — a somewhat 

 extended geometry — embracing them both. Three- 

 dimensional space is only a section cut through four- 

 dimensional space-time, and moreover sections cut in 

 different directions form the spaces of different 

 observers. We can scarcely maintain that the study of 

 a section cut in one special direction is the proper sub- 

 ject-matter of geometry and that the study of slightly 

 different sections belongs to an altogether different 

 science. Hence the geometry of the world is now con- 

 sidered to include time as well as space. Let us follow 

 up the geometry of time. 



You will remember that although space and time are 

 mixed up there is an absolute distinction between a 

 spatial and a temporal relation of two events. Three 

 events will form a space-triangle if the three sides 

 correspond to spatial relations — if the three events are 

 absolutely elsewhere with respect to one another.* 

 Three events will form a time-triangle if the three sides 

 correspond to temporal relations — if the three events 

 are absolutely before or after one another. (It is pos- 

 sible also to have mixed triangles with two sides time-like 

 and one space-like, or vice versa.) A well-known law 

 of the space-triangle is that any two sides are together 



* This would be an instantaneous space-triangle. An enduring triangle 

 is a kind of four-dimensional prism. 



