Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula
Abstract
We present a new method of constructing a stress-energy-momentum tensor \nfor a classical field theory based on covariance considerations and Noether theory. \nThe stress-energy-momentum tensor T ^μ \n_ν that we construct is defined using \nthe (multi)momentum map associated to the spacetime diffeomorphism group. \nThe tensor T ^μ \n_ν is uniquely determined as well as gauge-covariant, and depends \nonly upon the divergence equivalence class of the Lagrangian. It satisfies \na generalized version of the classical Belinfante-Rosenfeld formula, and hence \nnaturally incorporates both the canonical stress-energy-momentum tensor and \nthe “correction terms” that are necessary to make the latter well behaved. Furthermore, \nin the presence of a metric on spacetime, our T^(μν) coincides with the \nHilbert tensor and hence is automatically symmetric.
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The multi-momentum map encodes the two Noether theorems as particular cases and generates a unique, unambiguous expression for the energy-momentum density tensor of the theory, which reduces to Noether's and Hilbert's expressions under the respective specific conditions.
“They show that a certain structure, called 'multi-momentum map, ' encodes within itself the (two) Noether's theorems as particular cases and generates a unique, unambiguous expression for the energy-momentum density tensor of the theory, which reduces to Noether's and Hilbert's expressions under the respective specific conditions.”
From A strictly local solution to the problem of energy in general relativity (2026) · cites this paper as Gotay and Marsden (1992)
The multi-momentum map of a theory reflects the reaction of the involved fields to spacetime deformations.
“The multimomentum map of a theory reflects the reaction of the involved fields to spacetime deformations (Gotay and Marsden, 1992), being physically well-motivated and mathematically bold in fundamental terms.”
From A strictly local solution to the problem of energy in general relativity (2026) · cites this paper as Gotay and Marsden, 1992
In the formulation of Gotay and Marsden (1992), the energy-momentum tensor has the basic property of null covariant divergence in a way that is very general and a priori independent of the specific theory.
“B) this tensor has the expected basic property of null covariant divergence in a way that is very general and a priori independent of the specific theory.”
From A strictly local solution to the problem of energy in general relativity (2026) · cites this paper as Gotay and Marsden (1992)
For a Lagrangian density consisting of the usual Einstein-Hilbert term plus matter terms minimally coupled to gravity (GR), the energy-momentum density tensor T αβ attributed to the full theory by the formulation of Gotay and Marsden (1992) is given by the expression with G αβ Einstein's tensor.
“if the Lagrangian density of the theory consists of the usual Einstein-Hilbert term plus matter terms minimally coupled to gravity, which means that we are only talking about GR, the energy-momentum density tensor, T αβ , attributed to the full theory by the formulation of Gotay and Marsden (1992), is”
From A strictly local solution to the problem of energy in general relativity (2026) · cites this paper as Gotay and Marsden (1992)
The formulation proposed by Gotay and Marsden (1992) supports the proposal of searching for a complete theory that would reduce to GR in some reasonable limit, offering a link between questions (FQ) and (FQQ).
“it indicates that the formulation proposed by Gotay and Marsden (1992) supports our proposal of searching for a complete theory that would reduce to GR in some reasonable limit, offering a reasonable link between both questions (FQ) and (FQQ) of the last subsection.”
From A strictly local solution to the problem of energy in general relativity (2026) · cites this paper as Gotay and Marsden (1992)
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Pitts, J. B. (2016) · Annals of Physics · doi:10.1016/j.aop.2015.12.002 · Cited by 19 · see citing works
A strictly local solution to the problem of energy in general relativity
Vieira, W. M., & Magalhaes, N. S. (2026) · Frontiers in Astronomy and Space Sciences · doi:10.3389/fspas.2025.1710235
“…consists of the usual Einstein–Hilbert term plus matter terms minimally coupled to gravity, which means that we are only talking about GR, the energy–momentum density tensor, T αβ , attributed to the full theory by the formulation of Gotay and Marsden (1992), is with G αβ being Einstein’s tensor.”
Influential
- Authors
- Gotay, M. J., & Marsden, J. E.
- Affiliations
- United States Naval Academy (US)
- Venue
- Contemporary Mathematics
- Publisher
- American Mathematical Society
- Published
- 1992
- Pages
- 367-392
- ISSN
- 0271-4132 (ISSN-L)
- 1098-3627
- OpenAlex
- W1852661351
- Cited by
- 116
- References
- 0
- Citations by year
- 2026: 1 · 2025: 2 · 2023: 3 · 2022: 5 · 2021: 4 · 2020: 33 · 2019: 5 · 2018: 8 · 2017: 1 · 2016: 7 · 2015: 7 · 2013: 1 · 2012: 3
- Type
- other
- Language
- en
- Open access
- closed
- Locations
- Contemporary mathematics - American Mathematical Society (publishedVersion)
- CaltechAUTHORS (California Institute of Technology) (submittedVersion)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Keywords
- energy-momentum tensor, Noether's theorem, moment maps, diffeomorphism groups, gauge covariance, classical field theory
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