The Einstein tensor Gij is symmetric, divergence free, and a concomitant of the metric tensor gab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.
REFERENCES
1.
Unless otherwise specified, Latin indices run from 1 to n.
2.
A comma denotes partial differentiation.
3.
The summation convention is used throughout. The vertical bar denotes covariant differentiation.
4.
5.
H. Weyl, Space‐Time‐Matter (Dover, New York, 1922), 4th ed., pp. 315ff;
H. Vermeil, Nachr. Ges. Wiss. Göttingen, 334 (1917).
6.
If is any contravariant vector field, then we define the Riemann curvature tensor the Ricci tensor the curvature scalar R, and the Einstein tensor by and , respectively.
7.
8.
Reference 7, Theorem 4.
9.
Reference 7, Theorem 3.
10.
Reference 7, Corollary 1.
11.
For various applications of (2.6), see
D.
Lovelock
, Atti Accad. Nazi. Lincei
42
, 187
(1967
);12.
Without loss of generality, we may assume
13.
Reference 7, Theorem 5.
14.
This scalar has arisen elsewhere in an entirely different context. H. Rund, “Curvature Invariants Associated with Sets of n Fundamental Forms of Hypersurfaces of n‐Dimensional Riemannian Manifolds” [to appear in Tensor (1971)].
D. Lovelock, “Intrinsic Expressions for Curvatures of Even Order of Hypersurfaces in a Euclidean Space” [to appear in Tensor (1971)].
15.
16.
The relationship of (3.6) to Lagrangians which satisfy the Euler‐Lagrange equations identically has been investigated by R. Pavelle (private communication).
17.
This content is only available via PDF.
© 1971 The American Institute of Physics.
1971
The American Institute of Physics
You do not currently have access to this content.
