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Geometric relationship between the generalized Komar energy and the Arnowitt-Deser-Misner mass in dynamical spacetimes
Phys. Rev. D 114, 024079 – Published 28 July, 2026
DOI: https://doi.org/10.1103/nl77-bv95
Abstract
The standard Komar mass provides an elegant, quasilocal measure of total gravitating energy for stationary spacetimes, but it conventionally fails in dynamical scenarios that lack an exact global timelike Killing vector field. In this paper, we explore a generalized Komar energy integral bounded by a spacelike two-surface, obtained by replacing the Killing vector with the purely kinematic normal evolution vector field associated with a spacelike foliation. Through a step-by-step mathematical derivation, we show that this generalized integral reduces to the spatial boundary flux of the Eulerian four-acceleration. Furthermore, by evaluating the asymptotic vacuum constraints in generic dynamical spacetimes without restricting the metric to an isotropic or transverse-traceless spatial gauge, we establish, at leading asymptotic order, a relationship mapping this flux to the global Arnowitt-Deser-Misner (ADM) mass. By constructing a superpotential for the linearized spatial Einstein tensor, we prove that the dynamically generalized Komar energy and the ADM mass are equivalent up to a boundary flux of the temporal evolution of the conjugate momentum. We verify this relationship through nontrivial analytical test cases involving dynamical gauge foliations and physical gravitational wave radiation, revealing that the geometric momentum flux isolates longitudinal time-slicing artifacts while naturally decoupling from transverse radiative degrees of freedom.
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References (18)
- C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973).
- R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
- A. Komar, Covariant conservation laws in general relativity, Phys. Rev. 113, 934 (1959).
- R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962), pp. 227–265.
- R. Beig, Arnowitt-Deser-Misner energy and , Phys. Lett. 69A, 153 (1978).
- A. Ashtekar and A. Magnon-Ashtekar, On conserved quantities in general relativity, J. Math. Phys. (N.Y.) 20, 793 (1979).
- V. Iyer and R. M. Wald, Some properties of the Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50, 846 (1994).
- R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, R3427 (1993).
- D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, J. High Energy Phys. 10 (2020) 146.
- J. D. Brown and J. W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47, 1407 (1993).
- L. B. Szabados, Quasi-local energy-momentum and angular momentum in general relativity, Living Rev. Relativity 12, 4 (2009).
- L. Smarr and J. W. York, Jr., Kinematical conditions in the construction of spacetime, Phys. Rev. D 17, 2529 (1978).
- E. Gourgoulhon, 3 + 1 Formalism in General Relativity: Bases of Numerical Relativity, Lect. Notes Phys. Vol. 846 (Springer, Berlin, 2012).
- S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972).
- J. W. York, Jr., Kinematics and dynamics of general relativity, in Sources of Gravitational Radiation, edited by L. Smarr (Cambridge University Press, Cambridge, England, 1979), pp. 83–126.
- T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Ann. Phys. (N.Y.) 88, 286 (1974).
- L. F. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B195, 76 (1982).
- S. Deser and B. Tekin, Gravitational energy in quadratic curvature gravities, Phys. Rev. Lett. 89, 101101 (2002).
