Following on my previous question, I am interested in understanding a quantity that arises in the derivation of the Einstein equations described there, which appears to be of the nature of a Hamiltonian.

Here's how I see the Hamiltonian's proper definition. Given for example a one-derivative Lagrangian $\mathcal{L}(q, \dot{q})$, we can form its variation with respect to some derivation $\delta q$ of the generalized position:

$$ \delta \mathcal{L}(q, \dot{q}) = \frac{\partial\mathcal{L}}{\partial q} \delta q + \frac{\partial\mathcal{L}}{\partial \dot{q}} \delta \dot{q}, $$

assume that the dot and the $\delta$ commute, and integrate by parts to get

$$ \delta \mathcal{L} = \left( \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} \right) \delta q + \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \delta q \right). $$

If $\delta$ happens to be the time derivative $d/dt$, then we just reorganize to get

$$ \frac{d}{dt} \left( \mathcal{L} - \frac{\partial \mathcal{L}}{\partial \dot{q}} \dot{q} \right) = \left( \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} \right) \dot{q} $$

The parenthesized expression on the left is minus the Hamiltonian as it is typically defined. This shows that for a nontrivial ($\dot q \neq 0$) motion, the equations of motion are satisfied if and only if the Hamiltonian is conserved, i.e. conservation of energy.

Now the same thing in GR. According to the derivation here, we have

$$ \delta (\sqrt{-g} R) = \sqrt{-g} G_{ab} \, \delta g^{ab} + \partial_d \sqrt{-g}(g^{ac} \delta\Gamma^d_{\mathstrut ac} - g^{cd} \delta\Gamma^b_{bc}) $$

Taking $\delta = \partial/\partial x^e$, and using the identity $\sqrt{-g} \nabla_e A^e = \partial_e \sqrt{-g} A^e$ referenced in the article, we get

$$ \nabla_d \bigl(R\, \delta^d_e - (g^{ac} \partial_e\Gamma^d_{\mathstrut ac} - g^{cd} \partial_e\Gamma^b_{bc})\bigr) = G_{ab} \, \partial_d g^{ab} $$

From this equation I conclude that the odd quantity inside the derivative on the left should play the role of a Hamiltonian. Based on the fact that the Hamiltonian is the energy, and also that this derivation is just the standard derivation of momentum as the conserved Noether current for spatial translations (as you can see, if the vacuum Einstein equation $G_{ab} = 0$ holds, then the "Hamiltonian" is indeed locally conserved), I feel like the left-hand side should be the divergence of the energy-momentum tensor. Rewriting it with some index raisings made explicit:

$$ \nabla_d g^{df}\bigl(R\,g_{ef} - (g^{ac} g_{df} \partial_e\Gamma^d_{\mathstrut ac} - \partial_e\Gamma^b_{bf})\bigr) = G_{ab} \, \partial_d g^{ab}, $$

I guess my question is:

Is the tensor $T_{ef} = g^{ac} g_{df} \partial_e\Gamma^d_{\mathstrut ac} - \partial_e\Gamma^b_{bf} - R\,g_{ef}$ in any reasonable sense the "energy-momentum of gravity", and likewise is it the Hamiltonian in any reasonable sense? Also, it would be nice to know if it has a more...expressive...form.

Following on my previous question, I am interested in understanding a quantity that arises in the derivation of the Einstein equations described there, which appears to be of the nature of a Hamiltonian.

Here's how I see the Hamiltonian's proper definition. Given for example a one-derivative Lagrangian $\mathcal{L}(q, \dot{q})$, we can form its variation with respect to some derivation $\delta q$ of the generalized position:

$$ \delta \mathcal{L}(q, \dot{q}) = \frac{\partial\mathcal{L}}{\partial q} \delta q + \frac{\partial\mathcal{L}}{\partial \dot{q}} \delta \dot{q}, $$

assume that the dot and the $\delta$ commute, and integrate by parts to get

$$ \delta \mathcal{L} = \left( \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} \right) \delta q + \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \delta q \right). $$

If $\delta$ happens to be the time derivative $d/dt$, then we just reorganize to get

$$ \frac{d}{dt} \left( \mathcal{L} - \frac{\partial \mathcal{L}}{\partial \dot{q}} \dot{q} \right) = \left( \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} \right) \dot{q} $$

The parenthesized expression on the left is minus the Hamiltonian as it is typically defined (sans the substitution of generalized momenta for generalized velocities, so energy). This shows that for a nontrivial ($\dot q \neq 0$) motion, the equations of motion are satisfied if and only if the Hamiltonian is conserved, i.e. conservation of energy.

Now the same thing in GR. According to the derivation here, we have

$$ \delta (\sqrt{-g} R) = \sqrt{-g} G_{ab} \, \delta g^{ab} + \partial_d \sqrt{-g}(g^{ac} \delta\Gamma^d_{\mathstrut ac} - g^{cd} \delta\Gamma^b_{bc}) $$

Taking $\delta = \partial/\partial x^e$, and using the identity $\sqrt{-g} \nabla_e A^e = \partial_e \sqrt{-g} A^e$ referenced in the article, we get

$$ \nabla_d \bigl(R\, \delta^d_e - (g^{ac} \partial_e\Gamma^d_{\mathstrut ac} - g^{cd} \partial_e\Gamma^b_{bc})\bigr) = G_{ab} \, \partial_d g^{ab} $$

From this equation I conclude that the odd quantity inside the derivative on the left should play the role of a Hamiltonian. Based on the fact that the Hamiltonian is the energy, and also that this derivation is just the standard derivation of momentum as the conserved Noether current for spatial translations (as you can see, if the vacuum Einstein equation $G_{ab} = 0$ holds, then the "Hamiltonian" is indeed locally conserved), I feel like the left-hand side should be the divergence of the energy-momentum tensor. Rewriting it with some index raisings made explicit:

$$ \nabla_d g^{df}\bigl(R\,g_{ef} - (g^{ac} g_{df} \partial_e\Gamma^d_{\mathstrut ac} - \partial_e\Gamma^b_{bf})\bigr) = G_{ab} \, \partial_d g^{ab}, $$

I guess my question is:

Is the tensor $T_{ef} = g^{ac} g_{df} \partial_e\Gamma^d_{\mathstrut ac} - \partial_e\Gamma^b_{bf} - R\,g_{ef}$ in any reasonable sense the "energy-momentum of gravity", and likewise is it the Hamiltonian in any reasonable sense? Also, it would be nice to know if it has a more...expressive...form.