So normally, taking $c = 1$ and ${8\pi G = 1}$, and assuming the cosmological constant is negligible, the Einstein field equations read:
$$R_{\mu \nu} - \frac{1}2Rg_{\mu\nu} = T_{\mu \nu}.$$
However, there also exists a trace reversed form:
$$T_{\mu \nu} - \frac{1}2Tg_{\mu\nu} = R_{\mu \nu}.$$
Today while noodling around, I found that these two forms of the EFEs may be added and divided by 2 to yield what I'll refer to as the symmetric form of the EFEs:
$$R_{\mu \nu} - \frac{1}4Rg_{\mu\nu} = T_{\mu \nu} - \frac{1}4Tg_{\mu\nu} .$$
My questions:
- Is there any interpretation to be had here?
In particular, what does subtracting 1/4 of the trace times the metric do?
- Is this form the equation used in the literature anywhere?