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Energy-momentum tensors in linearized Einstein’s theory and massive gravity: The question of uniqueness

Bičák, J., Schmidt, J.

The question of the uniqueness of energy-momentum tensors in the linearized general relativity and in
the linear massive gravity is analyzed without using variational techniques. We start from a natural ansatz
for the form of the tensor (for example, that it is a linear combination of the terms quadratic in the first
derivatives), and require it to be conserved as a consequence of field equations. In the case of the linear
gravity in a general gauge we find a four-parametric system of conserved second-rank tensors which
contains a unique symmetric tensor. This turns out to be the linearized Landau-Lifshitz pseudotensor
employed often in full general relativity. We elucidate the relation of the four-parametric system to the
expression proposed recently by Butcher et al. “on physical grounds” in harmonic gauge, and we show that
the results coincide in the case of high-frequency waves in vacuum after a suitable averaging. In the massive
gravity we show how one can arrive at the expression which coincides with the “generalized linear
symmetric Landau-Lifshitz” tensor. However, there exists another uniquely given simpler symmetric tensor
which can be obtained by adding the divergence of a suitable superpotential to the canonical energymomentum
tensor following from the Fierz-Pauli action. In contrast to the symmetric tensor derived by the
Belinfante procedure which involves the second derivatives of the field variables, this expression contains
only the field and its first derivatives. It is simpler than the generalized Landau-Lifshitz tensor but both
yield the same total quantities since they differ by the divergence of a superpotential. We also discuss the
role of the gauge conditions in the proofs of the uniqueness. In the Appendix, the symbolic tensor
manipulation software CADABRA is briefly described. It is very effective in obtaining various results which
would otherwise require lengthy calculations.
journal:Phys. Rev. D
physrevd.93.024009.pdf (222.73 kB)

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