A4 Massless gravitational sources

In Appendix B, we shall be particularly concerned with the Einstein field equations when the (symmetric) source tensor Tab is trace-free

Taa = 0, since this is appropriate for massless (i.e. zero rest-mass) sources, telling us that the spinor-indexed quantity Taba-b-=Ta-b-ab=Tab has the symmetry

The divergence equation VaTab=0, i.e. Vaa'Taba-b- = 0, can be re-expressed

Vg Tcda-b- = V?B Tcd)a-b'. The Einstein equations above are now (P&R 4.6.32)

Oaba-b-=4nGTab, R = 4A. When rest-mass is present, so that Tab has a trace

Taa = M, then Einstein's equations take the form

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