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Cited by 34Relativity and Gravitational TheoryBlack Holes and Theoretical PhysicsCosmology and Gravitation Theories

Conserved charges in general relativity

Open access
Aoki, Sinya, Onogi, Tetsuya, Yokoyama, Shuichi·2021·International Journal of Modern Physics A
DOI 10.1142/s0217751x21500986
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Abstract

We present a precise definition of a conserved quantity from an arbitrary covariantly conserved current available in a general curved space–time with Killing vectors. This definition enables us to define energy and momentum for matter by the volume integral. As a result we can compute charges of Schwarzschild and BTZ black holes by the volume integration of a delta function singularity. Employing the definition we also compute the total energy of a static compact star. It contains both the gravitational mass known as the Misner–Sharp mass in the Oppenheimer–Volkoff equation and the gravitational binding energy. We show that the gravitational binding energy has the negative contribution at maximum by 68% of the gravitational mass in the case of a constant density. We finally comment on a definition of generators associated with a vector field on a general curved manifold.

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38 of 38 references verified0 citation errors · 0 not checkable

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  • Our proposal for a covariant definition of the energy and its generalization in general relativity requires equations of motion.

    “our proposal for a covariant definition of the energy and its generalization in general relativity, which requires equations of motion[10, 11], and thus is not a charge from the 2nd theorem.”

    From Conserved non-Noether charge in general relativity: Physical definition versus Noether’s second theorem (2022) · cites this paper as [10]

  • If the metric does not contain a time coordinate x^0, the Killing vector is ξ^a=-δ^a_0, and the conserved energy is defined by (3.4).

    “If the metric does not contain a time coordinate x 0 , the Killing vector is given by ξ a = -δ a 0 in such a coordinate. Thus the conserved energy is defined by[10]”

    From Conserved non-Noether charge in general relativity: Physical definition versus Noether’s second theorem (2022) · cites this paper as [10]

  • A more distinct difference between the two definitions appears in the case of energies for a compact star [10].

    “A more distinct difference between the two definitions appears in the case of energies for a compact star[10].”

    From Conserved non-Noether charge in general relativity: Physical definition versus Noether’s second theorem (2022) · cites this paper as [10]

  • If the time-translation vector is a Killing vector, then the current formed from the energy-momentum tensor and that vector is covariantly conserved, and the energy is the conserved Noether charge.

    “then J µ := T µ ν ξ ν T is covariantly conserved as[1]”

    From Is energy conserved in general relativity? (2025) · cites this paper as [1]

  • The final metric is the regularized Schwarzschild metric as a distribution, leading to a non-zero energy-momentum tensor at the origin.

    “This metric is the 'regularized' Schwarzschild metric as a distribution, which leads to the non-zero EMT at the origin as[1]”

    From Is energy conserved in general relativity? (2025) · cites this paper as [1]

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Cites

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1 of the 39 references couldn’t be matched to a known paper.

Research works printed but not linked: 5

  1. A. Einstein, Ann. der. Phys. Ser.4, 49 (1916), pp. 769-822
  2. R. Geroch, Ann. N. Y. Acad. Sci. 224, 108 (1973);
  3. S. Aoki, T. Onogi and S. Yokoyama, [arXiv:2004.03779 [hep-th]]. · arXiv
  4. P. Townsend, [arXiv:gr-qc/9707012 [gr-qc]]. · arXiv
  5. A. S. Eddington, Nature 113, 192 (1924).

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Authors
Aoki, S., Onogi, T., & Yokoyama, S.
Affiliations
  • Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-Ku, Kyoto 606-8502, Japan
  • Kyoto University (JP)
ORCID
Venue
International Journal of Modern Physics A
Publisher
World Scientific Pub Co Pte Ltd
Published
2021-04-10
Volume
36
Issue
10
Pages
2150098
ISSN
  • 0217-751X (ISSN-L)
  • 1793-656X
DOI
10.1142/s0217751x21500986
OpenAlex
W3031379909
Cited by
34
References
39
Citations by year
2026: 4 · 2025: 5 · 2024: 5 · 2023: 8 · 2022: 8 · 2021: 3 · 2020: 1
FWCI
3.22
Type
journal article
Language
en
Open access
Locations
  • International Journal of Modern Physics A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Institutional Repositories DataBase (IRDB) (submittedVersion)
  • Kyoto University Research Information Repository (Kyoto University) (submittedVersion, cc-by)
  • arXiv (Cornell University)
Funders
Ministry of Education, Culture, Sports, Science and Technology (JP16H03978, JP18H05236, JP18K03620, JP19K03847)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
Keywords
conserved charges, Killing vector fields, Schwarzschild black hole, BTZ black hole, delta-function singularity, Misner-Sharp mass, gravitational binding energy, gravitational mass

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