Energy-Momentum Conservation as a Constraint to Restrict the Space of Viable Field Lagrangians: Vector Field versus Spin-Two Field
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866917521999790080 |
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| author | Nakajima, Satoshi López-Pinto, Antonio |
| author_facet | Nakajima, Satoshi López-Pinto, Antonio |
| contents | We investigate whether the requirement of total energy-momentum conservation can act as a constraint on the family of admissible Lagrangian densities for an interaction field. The aim is not to give a mere field-theoretic derivation of general relativity, nor to introduce another gravitational energy-momentum pseudotensor starting from a prescribed gravitational Lagrangian. Rather, we ask the more general question of whether a conservation principle, once a definite field energy-momentum tensor has been specified, can determine the form of the field Lagrangian itself. For a Poincaré-invariant field theory, we take the field contribution to the total energy-momentum tensor to be the symmetrized Belinfante tensor associated with the field Lagrangian. The vector-field case shows that the answer is not affirmative in general. For a symmetric rank-two tensor field, we leave the matter sector otherwise arbitrary and define its energy-momentum tensor by variation of the matter Lagrangian with respect to the field. Imposing conservation of the total energy-momentum tensor and using only the field equation then leads directly to Feynman's consistency condition. Thus, the usual consistency condition of the spin-two route to general relativity is recovered from the conservation of a definite Belinfante-type energy-momentum tensor, rather than from the simultaneous use of the matter equations of motion and the field equations. Within the class of local Lorentz-invariant Lagrangian densities that are analytic in the symmetric field, quadratic in first derivatives, and contain no non-derivative potential terms, this condition determines the Einstein Lagrangian density uniquely, up to a total divergence. For the resulting Einstein Lagrangian, the corresponding symmetrized Belinfante tensor is obtained explicitly and is shown to be related to the Papapetrou energy-momentum pseudotensor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Energy-Momentum Conservation as a Constraint to Restrict the Space of Viable Field Lagrangians: Vector Field versus Spin-Two Field Nakajima, Satoshi López-Pinto, Antonio General Relativity and Quantum Cosmology High Energy Physics - Theory We investigate whether the requirement of total energy-momentum conservation can act as a constraint on the family of admissible Lagrangian densities for an interaction field. The aim is not to give a mere field-theoretic derivation of general relativity, nor to introduce another gravitational energy-momentum pseudotensor starting from a prescribed gravitational Lagrangian. Rather, we ask the more general question of whether a conservation principle, once a definite field energy-momentum tensor has been specified, can determine the form of the field Lagrangian itself. For a Poincaré-invariant field theory, we take the field contribution to the total energy-momentum tensor to be the symmetrized Belinfante tensor associated with the field Lagrangian. The vector-field case shows that the answer is not affirmative in general. For a symmetric rank-two tensor field, we leave the matter sector otherwise arbitrary and define its energy-momentum tensor by variation of the matter Lagrangian with respect to the field. Imposing conservation of the total energy-momentum tensor and using only the field equation then leads directly to Feynman's consistency condition. Thus, the usual consistency condition of the spin-two route to general relativity is recovered from the conservation of a definite Belinfante-type energy-momentum tensor, rather than from the simultaneous use of the matter equations of motion and the field equations. Within the class of local Lorentz-invariant Lagrangian densities that are analytic in the symmetric field, quadratic in first derivatives, and contain no non-derivative potential terms, this condition determines the Einstein Lagrangian density uniquely, up to a total divergence. For the resulting Einstein Lagrangian, the corresponding symmetrized Belinfante tensor is obtained explicitly and is shown to be related to the Papapetrou energy-momentum pseudotensor. |
| title | Energy-Momentum Conservation as a Constraint to Restrict the Space of Viable Field Lagrangians: Vector Field versus Spin-Two Field |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2604.17601 |