On the Untraced Second Bianchi Identities in General Relativity

Einstein's equations set space-time curvature traces proportional to the space-time's mechanical content expressed as a stress- energy tensor. The divergence of a stress-energy tensor is the external force density acting on the matter described by the stress-energy tensor. Einstein's equations require the total stress-energy tensor representing all the matter present. So with nothing else to push on, the divergence of the total stress-energy tensor must vanish. On the curvature side of Einstein's equations this is required by the trace of the second Bianchi identities. In this talk Einstein's equations are examined in terms of the full curvature and the untraced second Bianchi identities. It turns out, in general, that Weyl's conformal tensor (the totally traceless remains of curvature) splits into two pieces. One piece exists everywhere, inside and beyond the matter depending for its values on distant matter distributions. The other piece is strictly local, occurring only inside the matter and its value at each point depends explicitly on the matter occurring there. Thus Weyl's conformal tensor, regarded as a gravitational field has both local and non-local components. Electromagnetism and perfect fluids will be given as examples.