We present a global formulation of projective theories of relativity in the framework of projective manifolds, that is, manifolds based on the pseudogroup of homogeneous transformations in R5. Apart from formulating every previously considered geometric object and physical relation in an invariant manner, some new results, such as the theorem on the semidirect product structure of the invariance group of Einstein‐Maxwell equations, and theorems on topological restrictions on the underlying five‐dimensional projective manifold, etc. have been obtained. The relationship between space‐time and the auxiliary 5‐manifold is clarified and investigated in detail. A more general geometric definition of the electromagnetic field tensor and a geometric interpretation of the charge/mass ratio is given.

Topics
Covariant formulation, Einstein Maxwell equations, Theory of relativity, Group theory
1.
H.
Weyl
,
Sitz. Ber. Preuss, Akad. Wiss.
465
(
1918
).
Google Scholar
 
2.
Th.
Kaluza
,
Sitz. Ber. Preuss, Akad. Wiss.
966
(
1921
).
Google Scholar
 
3.
O.
Klein
,
Z. Phys.
37
,
895
(
1926
).
Google Scholar
Crossref
Search ADS
 
4.
O.
Veblen
 and
B.
Hoffmann
,
Phys. Rev.
36
,
810
(
1930
).
Google Scholar
Crossref
Search ADS
 
5.
O. Veblen, Projektive Relativitätstheorie (Springer, Berlin, 1933).
6.
J. A.
Schouten
 and
D.
van Dantzig
,
Proc. Akad. Wetensch. Amsterdam
34
,
1398
(
1931
).
Google Scholar
 
7.
D.
van Dantzig
,
Math. Ann.
106
,
400
(
1932
).
Google Scholar
Crossref
Search ADS
 
8.
P. Jordan, Göttinger Nachr. 74 (1945).
9.
P. Jordan, Schwerkraft und Weltall (Vieweg, Braunschweig, 1955), 2nd. ed.
10.
G. Ludwig, Fortschritte der projektiven Relativitästheorie (Vieweg, Braunschweig, 1951).
11.
;A. Lichnerowicz, Thëories relativistes de la gravitation et de l’electromagnetisme (Masson, Paris, 1955).
12.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Wiley, New York, 1963), Vol. I.
13.
Our definition differs from the standard one given by, for example, E. Cartan Lecons sur la théorie des espaces à connexion projective (Gauthier‐Villars, Paris, 1937).
14.
Jordan, for example, assumed only that φ is a homogeneous map of degree zero. However, rigourously speaking one needs somewhat more in order to project projective vector fields from M onto V.
15.
From now on Xμ and xk will denote local coordinates in M and V respectively, and all Greek indices and all Latin indices take values 0,1,…,4 and 0,1,…,3 respectively.
16.
G. T. Evans, Ph.D. thesis (Toronto, 1973).
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© 1973 The American Institute of Physics.
1973
The American Institute of Physics
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