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The Bianchi Identities in the Generalized Theory of Gravitation

Published online by Cambridge University Press:  20 November 2018

A. Einstein
A. Einstein*
Affiliation:
The Institute for Advanced Study Princeton, N.J.
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1. General remarks. The heuristic strength of the general principle of relativity lies in the fact that it considerably reduces the number of imaginable sets of field equations; the field equations must be covariant with respect to all continuous transformations of the four coordinates. But the problem becomes mathematically well-defined only if we have postulated the dependent variables which are to occur in the equations, and their transformation properties (field-structure). But even if we have chosen the field-structure (in such a way that there exist sufficiently strong relativistic field-equations), the principle of relativity does not determine the field-equations uniquely. The principle of “logical simplicity” must be added (which, however, cannot be formulated in a non-arbitrary way). Only then do we have a definite theory whose physical validity can be tested a posteriori.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 2 , 1950 , pp. 120 - 128
Copyright
Copyright © Canadian Mathematical Society 1950

References

1 Ann. of Math., vol. 46 (1945), no. 4; vol. 47(1946), no. 4.

2 It is a consequence of (1) that (2) and (3) are equivalent. This will be proven in sec. 5

3 The names “conjugate” and “Hermitian” can be justified as follows: an interesting possibility is to choose imaginary. Then is really the conjugate of g. Hence A is the conjugate Y of A, and the definition of “Hermitian” agrees with the usual one.

4 Thus in our theory the condition of symmetry is generalized to that of being Hermitian. are all Hermitian in .

5 Since , the two kinds of differentiation coincide when applied to g. This must be so since there is no index which could have a + or - character.