General relativity includes a dynamical spacetime, so it is difficult to see how to identify the conserved energy and momentum. Noether's theorem allows these quantities to be determined from a Lagrangian with translation invariance, but general covariance makes translation invariance into something of a gauge symmetry.
The energy and momentum derived within general relativity by Noether's
presecriptions do not make a real tensor for this reason.
Einstein argued that this is true for fundamental reasons, because
the gravitational field could be made to vanish by a choice of
coordinates. He maintained that the non-covariant energy momentum
pseudotensor was in fact the best description of the energy momentum
distribution in a gravitational field. This approach has been echoed by Lev Landau and Evgeny Lifshitz, and others, and has become standard.
The use of non-covariant objects like pseudotensors was heavily criticized in 1917 by Erwin Schrödinger and others.
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