Trace-reversed stress-energy tensor (${\overline{T}}_{\mu \nu }=T_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }T$)

In four spacetime dimensions, the trace reversal of a stress-energy tensor has the displayed form, where $T=g^{\mu \nu }{T}_{\mu \nu }$. Its trace is $-T$, so applying trace reversal twice returns the original tensor. Taking the trace of the Einstein field equations without a cosmological constant gives $R=-\kappa T$; substitution yields $R_{\mu \nu }=\kappa {\overline{T}}_{\mu \nu }$. In a general dimension $d>2$, solving for the Ricci tensor instead uses $T_{\mu \nu }-g_{\mu \nu }T/(d-2)$.