# Why Shy away from non-local nature of gravitational energy/momentum?

Feel free to tear me a new one, This is something that has been bothering me a while.

The very nature of attempting to write gravitational energy/momentum (called the stress energy pseudotensor), essentially forbids it to be defined locally (via the equivalence principle because we can always shift to a frame where it is zero at a given point).

One can however, typically define integral values of said quantities which themselves are at least Lorentz-covariant quantities (Weinberg Gravitation and cosmology p167 D; and Moller, the theory of Relativity 339).

On the other side of the coin are formulations such as the ADM/Bianchi Energies and various quasi-local definitions of such quantities. All of these have in common that they have an ill defined density at a given point, yet their integral yields a physically satisfactory value, just as with the pseudotensor above.

Personally I am reminded of the uncertainty principle in which one cannot define momentum at a point and yet the integral of the momentum operator (acting on the wavefunction) over all space yields a physically significant value. Also note that the momentum operator on the wavefunction $$\bar{\psi}\hat{P}\psi$$ has units of momentum density (though it has no physical meaning). Only when we integrate it do we get the momentum. This is precisely in analogy to the above gravitational momentum/energy complex.

Considering again General Relativity, one can formulate the conservation equations in the form:

$$\partial_{\mu}\left(T^{\mu\nu}+\tau^{\mu\nu}\right)=0$$

Where $$\tau$$ is the gravitational stress-energy-pseudotensor and $$T^{\mu\nu}$$ the Stress energy tensor of the matter/fields. Integrating over a general space-time manifold one can apply Gauss's theorem, bringing out integral to some chosen three-surface $$\partial M_{\nu}$$:

$$\intop_{M}\partial_{\mu}\left(T^{\mu\nu}+\tau^{\mu\nu}\right)dM=\intop_{\partial M}\left(T^{\mu\nu}+\tau^{\mu\nu}\right)d(\partial M)_{\nu}=0$$

Choosing a spacelike hypersurface, one obtains:

$$\intop_{\partial M}\left(T^{\mu0}+\tau^{\mu0}\right)d(\partial M)=0$$

Now for some arbitrary space the right-hand-side will in general be some non-zero constant, ( this really doesn't affect what I'm attempting to point out)

$$\intop_{\partial M}\left(p^{\mu}+\tau^{\mu0}\right)d(\partial M)=0$$

For some isolated system (say a particle) on a given background, one can write the particle's momentum components in terms of it's own pseudotensor (up to a constant, not necessarily zero).

$$P^{\mu}=-\intop_{\partial M}\tau^{\mu0}d(\partial M)$$

We know however that the stress energy pseudotensor must be composed of first order derivatives of the metric tensor (in order to respect the equivalence principle), thus we can consider some operator $$\hat{P}$$ which acts on the metric perturbation h of our particle:

$$P^{\mu}=-\intop_{\partial M}\hat{P^{\mu}}hd(\partial M)$$

This also, seems similar to quantum theory.

There are an admittedly infinite number of pseudotensors, and hence operators, and the background metric will also come into play, yet the form is reminiscent of quantum mechanics, including the non-locality.

For the full (nonlinear) case, such an operator acts on the FULL metric, and could easily be written in a form which adds harmonics to the background metric (Bardeen also did this with perturbations in terms of harmonics on the background metric see his paper on Gauge invariant perturbations from the eighties). In this sense, there is a similarity to quantum field theory. One has the background metric acting similarly to the ground/vacuum state.

In fact, the Unruh radiation experienced by an accelerating observer in flat space, and the GR notion of a nonzero gravity pseudotensor observed by an accelerating observer in flat space, also fit rather well

So my question is: Why shy away from Gravitational nonlocality, it seems like it bears the closest semblence to quantum theory that one encounters in classical physics, including many of the strange phenomena one usually considers strictly the domain of quantum theory.