The assumption that ${T^{\nu }}_{\mu ;\nu }=0\mathrm{}$ in curved space-time is questioned. Field equations are given which are consistent with the assumption ${T^{\nu }}_{\mu ;\nu }=\lambda R_{,\mu }$, and which reduce to the Einstein equations when $\lambda =0\mathrm{}$. The equations are equivalent to the Einstein equations in empty space-time, but differ from them in the presence of matter. Applications to cosmology, stellar structure, and collapsing objects are suggested.

• Received 22 May 1972

DOI:https://doi.org/10.1103/PhysRevD.6.3357