Total energy and momentum in general relativity

Question

Discussing the energy-momentum tensor in presence of gravitational field, our professor stated that, except some particular cases, it’s not possibile to define a “global” energy and momentum in general relativity. Energy and momentum are only defined locally as densities in the energy-momentum tensor. Why?

Answer 1

A crucial problem is that one cannot unambiguously add vectors applied to different events in spacetime in the non-affine spacetime of GR. Hence you cannot add (integrate) the four momenta (densities) located on a spatial slice to construct a global quantity. The problem is even more difficult because one also would like that such a sum or integral be also constant in time (some notion of time associated to a family of spatial slices) in some situations, in order to have a meaningful notion.

Scalars can be instead integrated and this is a way out. Instead of trying to define a global four vector one can define a global component of a fourmomentum with respect to a vector field. Mathematically speaking that object is the integral over a spacelike slice of the spacetime of $${T^a}_b\xi^b n_a,$$ where $n$ is the unit normal to the slice and $\xi$ the said vector field. (That slice is a Cauchy surface if assuming that the spacetime is globally hyperbolic.) If this vector is a Killing vector and if $T$ is symmetric, then the integral of the above quantity is in fact conserved, changing the spatial slice, under mild requirements on the space infinity behaviour of $T$. If $\xi$ is a Killing timelike vector field, this procedure gives rise to a meaningful notion of conserved energy in particular.

Answer 2

Well, the situation is even worse.

1. Globally, energy (momentum) as the integral of an energy (momentum) density over a non-compact space may be divergent/non-integrable, respectively.

2. Locally, there's no well-defined gravitational stress-energy-momentum (SEM) tensor, cf. e.g. this, this, this, this Phys.SE posts and links therein. Nevertheless, check out the notion of a Landau–Lifshitz pseudotensor, which is not general covariant.