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My question is as follows:

Why is it problematic to define energy-momentum tensor for the gravitational field?

P.S. It is well-known that in GR we get the energy-momentum tensor of "matter" by taking the variational derivative of the action $S_m$ of matter fields with respect to the metric field: $$ T_{MN}=-\cfrac{2}{\sqrt{|g|}}\cfrac{\delta S_{m}}{\delta g^{MN}} $$

Why can't we use an analogous procedure to define the energy-momentum tensor for the gravitational field $g_{MN}$? That would lead to the Einstein tensor $G_{MN}$ (at least for the Einstein-Hilbert action $S_g\sim g^{MN}Ric_{MN}$). However, for some reason the phrase "energy-momentum tensor of gravity" is persistently avoided. I understand that the problem is conceptual (for instance, then we would have the total energy-momentum of a gravitating system always being zero), but could anybody explain that in more detail?

Thanks!

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  • $\begingroup$ The energy-momentum tensor for the gravitational field doesn't exist, only a corresponding pseudo-tensor exist. Adding (e.g.) the pseudo-tensor on the right side of the EFEs, would create a non-covariant equation which would be inconsistent. The reason for this is that the energy of the gravitational field is not localizable. It can be (coordinate)-transformed away. The energy-momentum tensor of matter changes under coordinates transformations, but never gets zero under transformations. However, a pseudo-tensor does. So change appropiately the coordinates, the gravity energy has gone. $\endgroup$ May 22 '19 at 11:01
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/41662/2451 $\endgroup$
    – Qmechanic
    May 22 '19 at 14:16
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Given a complete system of equations of motion (which in this case includes Einstein's equation for $g_{MN}$), a tensor $T^{MN}$ can be called conserved if $\nabla_M T^{MN}=0$ for all histories that satisfy the equations of motion. For a conserved quantity to be useful, though, it must satisfy two additional properties:

  • Property $1$: $\nabla_M T^{MN}\neq 0$ for some histories that don't satisfy the equations of motion.

  • Property $2$: Among histories that do satisfy the equations of motion, different histories can have different values of $T^{MN}$.

The action $S$ for a gravitating system can be written $S=S_g+S_m$ where $S_g$ is the pure-gravity part and $S_m$ is the part involving "matter." As described in the OP, we can define

  • A total energy-momentum tensor $T^{MN}$ using $S$,

  • An energy-momentum tensor $T^{MN}_g$ for the gravitational field alone using $S_g$,

  • An energy-momentum tensor $T^{MN}_m$ for the "matter" alone using $S_m$.

These are related to each other by $$ T^{MN} = T^{MN}_g + T^{MN}_m, $$ and all three of them have zero divergence for histories that satisfy the equations of motion. In other words, all three of them are conserved. However, not all of them are useful:

  • The total energy-momentum tensor $T^{MN}$ is useless because it doesn't have Property $2$: it is zero for all histories that satisfy Einstein's equation. This is the point of Arnold Neumaier's answer here: https://physics.stackexchange.com/a/41663.

  • The gravity-only energy-momentum tensor $T^{MN}_g$ is useless because it doesn't have Property $1$: its divergence is identically zero. Defining it isn't a problem; it's just not useful. This is the answer to the question in the OP.

The "matter" energy-momentum tensor $T^{MN}_m$ has both Properties $1$ and $2$, which makes it useful.

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