Bianchi identity

Bianchi identity

[′byäŋ·kē ī′den·əd·ē]
(mathematics)
A differential identity satisfied by the Riemann curvature tensor: the antisymmetric first covariant derivative of the Riemann tensor vanishes identically.
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In this situation as well validity of the above expression for all null vectors [l.sub.a], along with two times contracted Bianchi identity and covariant conservation of matter energy momentum tensor, amounts to furnishing the ten components of Einstein's equations.
In this case besides demanding the validity of the above equation for all null vectors, it is important to use contracted Bianchi identity as well as covariant conservation of energy momentum tensor, leading to [G.sub.ab] + [LAMBDA][g.sub.ab] = [kappa][T.sub.ab].
Further it satisfies the contracted Bianchi identity, which will be sufficient for our purpose.
While in the null case, besides demanding the validity of [E.sub.ab][l.sup.a][l.sup.b] = [kappa][T.sub.ab][l.sup.a][l.sup.b] for all null vectors, one has to use the Bianchi identity associated with Lovelock theories as well as covariant conservation of matter energy momentum tensor to arrive at the Lovelock field equations [E.sub.ab] + [LAMBDA][g.sub.ab] = [kappa][T.sub.ab], which inherit the cosmological constant as well.
These assumptions could be unnecessary and generate spurious solutions, since this loss of mass is prescribed by one of the conservation equations when applying the Bianchi Identity [29, 30].
Using the conservation equations (Bianchi Identity) [T.sup.[mu].sub.v;[mu]] = 0, we obtain only three no-trivial relations:
Now, with the generalized Bianchi identity for the electromagnetic field, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], at hand, and assuming the "isochoric" condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we obtain
Now, more generally and more naturally, using the generalized Bianchi identity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we can obtain the following fundamental relation:
Note that despite the fact that the curvature tensor of the space-time [S.sub.2] is identical to that of the space-time [S.sub.1] and that both curvature tensors share common algebraic symmetries, the Bianchi identity in [S.sub.2] is not the same as the ordinary Bianchi identity in the torsion-free space-time [S.sub.1].
Hence, the second generalized Bianchi identity finally takes the somewhat more transparent form
(37) is the contracted Bianchi identity demonstrating the recovering of conservation laws also in 4D [2].