A5 Bianchi identities

The general Bianchi identity V[oRbc]de = 0, in spinor-indexed form, becomes (P&R 4.10.7, 4.10.8)

Vg Wabcd=VfB $>CD)A-B- and Vca'Ocda-b-+§Vdb-R = 0.

When R is a constant—a situation that arises with Einstein's equations when the sources are massless—we have

VCA Qcda-b- = 0, whence VgWabcd = V^'Ocda-b-, the symmetry in BCD on the right being implied. Incorporating the Einstein equation, with massless sources, we get v^wabcd = 4nGVg' Tcdab

(see P&R 4.10.12). Note that when Tabc-d-=0, we obtain the equation (P&R 4.10.9)

Vaa'Vabcd = 0, which is the massless free-field equation in A2, for the case n = 4 (i.e. for spin 2).

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