Contracted Bianchi identities

In general relativity and tensor calculus, the contracted Bianchi identities are:[1]

where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.

A proof can be found in the entry Proofs involving covariant derivatives.

These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880.[2] In the Einstein field equations, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter stress-energy tensor.

See also

Notes

References

  • Lovelock, David; Hanno Rund (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
  • Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2.
  • J.R. Tyldesley (1975), An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, ISBN 0-582-44355-5
  • D.C. Kay (1988), Tensor Calculus, Schaum’s Outlines, McGraw Hill (USA), ISBN 0-07-033484-6
  • T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, ISBN 978-1107-602601
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