A3 Spacetime curvature quantities

The (Riemann-Christoffel) curvature tensor Rabcd has the symmetries

Rabcd = R[ab][cd] = Rcdab, R[abc]d = 0, and relates to commutators of derivatives via (P&R 4.2.31)

This fixes the choice of sign convention for Rabcd. We here define the Ricci and Einstein tensors and the Ricci scalar, respectively, by

Rac=Rabcb, Eab=\Rgab - Rab, where R=Raa, and the Weyl conformal tensor Cabcd is defined by (P&R 4.8.2)

Cabcd = Rabcd -2 R[alcgbf + fRg[acgb]d, this having the same symmetries as Rabcd but, in addition, all traces vanish

In spinor terms, we find that we can write (P&R 4.6.41)

where the conformal spinor Wabcd is totally symmetric

The remaining information in Rabcd is contained in the scalar curvature R and the trace-free part of the Ricci (or Einstein) tensor, the latter being encoded in the spinor quantity Oabcd- with symmetries and Hermiticity

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