THE SPACE-TIME MANIFOLD OF RELATIVITY. 



THE NON-EUCLIDEAN GEOMETRY OF MECHANICS 



AND ELECTROMAGNETICS. 



By Edwin B. Wilson and Gilbert N. Lewis. 



Infrodudion. 



1. The concept of space has different meanings to ditferent persons 

 according to their experience in abstract reasoning. On the one hand 

 is the common space, which for the educated person has been formu- 

 lated in the three dimensional geometry of Euclid. On the other 

 hand the mathematician has become accustomed to extend the concept 

 of space to any manifold of which the properties are completely de- 

 termined, as in Euclidean geometry, by a system of self-consistent 

 postulates. Most of these highly ingenious geometries cannot be 

 expected to be of service in the discussion of physical phenomena. 



Until recently the physicist has found the three dimensional space 

 of Euclid entirely adequate to his needs, and has therefore been in- 

 clined to attribute to it a certain reality. It is, however, inconsistent 

 with the philosophic spirit of our time to draw a sharp distinction 

 between that which is real and that which is convenient,^ and it would 

 be dogmatic to assert that no discoveries of physics might render so 

 convenient as to be almost imperative the modification or extension 

 of our present system of geometry. Indeed it seemed to Minkowski 

 that such a change was already necessitated by the facts which led 

 to the formulation of the Principle of Relativity. 



2. The possibility of associating three dimensional space and one 

 dimensional time to form a four dimensional manifold has doubtless 

 occurred to many; but as long as space and time were assumed to be 

 wholly independent, such a union seemed purely artificial. The idea 

 of abandoning once for all this assumption of independence, although 

 fore-shadowed in Lorentz's use of local time, was first clearly stated by 



1 See, for example, H. Poincard, La Science et THypothese. 



