# variations of Einstein equations with conversion between gravitational and non-gravitational energy

I'm looking for existing papers studying a variation to Einstein equation that does not rely on the annoying matter conservation identity:

$$T_{\mu \nu; \nu} = 0$$

And instead tries to equate the divergence-free Einstein tensor with a sum of $T_{\mu \nu}$ plus some gravitational energy tensor $Y_{\mu \nu}$:

$$G_{\mu \nu} = 8 \pi G (T_{\mu \nu} + \theta Y_{\mu \nu})$$

Where the $\theta$ factor is a parameter of the ansatz.

Let me explain why this ansatz should be physically interesting: because the vanilla version of Einstein equation is based on the assumption that conversion between gravitational and non-gravitational energy does not happen never, ever. If you cherry-pick the gravitational energy tensor to be, say, the Landau Lifshitz tensor:

$$Y^{\mu \nu} = (\sqrt{-g}_{; \alpha \beta}) ( g^{\mu \nu} g^{\alpha \beta} - g^{\mu \alpha} g^{\nu \beta} )$$

(notice this is not the pseudo tensor variant; those derivatives are covariant)

this tensor is zero in the weak-field limit, is non-zero only after second order corrections in the metric, so it would match most astronomical observations that match GR in the weak field limit. Would be interesting to see what predictions does this produce in the nonlinear regime. In fact, the above argument applies just as well to any meaningful gravitational tensor that has only second order (or smaller) non-zero corrections.

Any thoughts?