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?


The tensor you wrote down is identically zero, the covariant derivative of g vanishes identically. This is not just a technical problem--- the issue is that the gravitational energy is nowhere localized, while the matter energy density can be localized. This means that if gravitational energy is converted to non-gravitational source energy, it must be in a way that can't be modelled by a local equation.

An example of this (it's not so exotic) is converting gravitons to black holes. The black holes can be relatively localized, they can make a dust, but they form nonlocally from gravitational wave collapse in a region of size the Schwarzschild radius.

  • $\begingroup$ that is fine but is besides the whole point; just pick a better definition for gravitational energy density. nonlocality is not a deal-breaker, as long as you get a tensorial equation. $\endgroup$
    – lurscher
    Aug 22 '12 at 12:40
  • $\begingroup$ the bigger picture here is: all forces in the universe admit energy exchange between one form and the other, except gravity. If that doesn't look wrong or suspicious to you, well, then i guess it doesn't. $\endgroup$
    – lurscher
    Aug 22 '12 at 12:44
  • $\begingroup$ @lurscher: You can't do it locally, because you can make the grav. field vanish locally. You can and do exchange gravitational energy with other energy in GR--- the other energy is only covariantly conserved, it isn't really conserved, and this is automatic conversion of pseudo-stress-energy to other stress energy. $\endgroup$
    – Ron Maimon
    Aug 22 '12 at 16:34
  • $\begingroup$ covariantly conserved does not imply it being really conserved? $\endgroup$
    – lurscher
    Aug 22 '12 at 17:01
  • $\begingroup$ @lurscher: No, because covariant derivatives don't have an integral law. Gauss's law/Stokes theorem is for regular derivatives only, or "d"'s where the connection cancels. Only coordinate dependent regular derivatives allow you to derive the integral form of the conservation law, that the integral of the energy density is constant. The pseudo-stress tensor is what you add to the covariantly conserved matter stress-tensor to make it coordinate-conserved, so that the integral of the total energy is constant. This integral is not constant for the energy excluding the gravitational pseudo-energy. $\endgroup$
    – Ron Maimon
    Aug 23 '12 at 1:03

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