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.

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.

Proof edit

Start with the Bianchi identity[3]


Contract both sides of the above equation with a pair of metric tensors:


The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,


The last two terms are the same (changing dummy index n to m) and can be combined into a single term which shall be moved to the right,


which is the same as


Swapping the index labels l and m on the left side yields


See also edit

Notes edit

  1. ^ Bianchi, Luigi (1902), "Sui simboli a quattro indici e sulla curvatura di Riemann", Rend. Acc. Naz. Lincei (in Italian), 11 (5): 3–7
  2. ^ Voss, A. (1880), "Zur Theorie der Transformation quadratischer Differentialausdrücke und der Krümmung höherer Mannigfaltigketien", Mathematische Annalen, 16 (2): 129–178, doi:10.1007/bf01446384, S2CID 122828265
  3. ^ Synge J.L., Schild A. (1949). Tensor Calculus. pp. 87–89–90.

References edit

  • 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