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.

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.

typ: | article |
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journal: | Phys. Rev. D |

volume: | 93 |

pages: | 024009 |

year: | 2016 |