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To calculate the ADM mass, one has to specify a closed surface inside which all the mass is located. Are there any ways to calculate the mass inside a closed surface outside of which other mass still exists? Of course, I am considering the case in which the other mass is not infinitely far away.

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Yes, but …

  • it would not be called ADM mass,
  • a proper term would be “quasi-local” rather than “local” mass (energy),
  • there are many prescriptions for defining such mass (energy).

Note, while literature often uses “mass” and “energy” interchangeably, sometimes a distinction is necessary: energy is a time component of a vector (energy–momentum), while mass is the length of this vector (an invariant).

It is well known, that the energy–momentum of gravitational field cannot be defined locally as a tensor object covariant under diffeomorphisms of a manifold. This could be seen as a consequence of equivalence principle, which implies that the gravitational field cannot be detected at a point as a covariant object.

One could try to define instead a non-covariant object for the energy of a gravitational field. And so various stress-energy-momentum pseudotensors were proposed, notably by Freud, Landau & Lifshitz, Papapetrou, Bergmann & Thomson, Møller and Weinberg. The problem is, such objects are dependent on the choice of coordinates, and badly chosen coordinate system could lead to undesired results such as diverging energy of Minkowski spacetime.

A more modern approach is that gravitational energy should be quasi-local, that is associated with a closed 2-surface (and representing the energy contained inside this surface at a given moment). Such quasi-local energy is coordinate independent and independent of the slicing of constant time inside the surface but would depend on many choices: a specific formalism used for GR, what variables are considered independent, what boundary conditions are imposed, what reference spacetime is used to measure energy etc. As a result there are many proposals for such quasi-local quantities: Bartnik mass (this quantity could be considered a natural quasi-localization of the ADM mass), Hawking energy, Penrose energy–momentum, Dougan–Mason energy–momentum, Brown–York energy to name a few.

In addition there are quasi-local quantities defined for spacetimes with special properties, such as existence of timelike Killing vector field which allows one to define Komar mass, or quantities defined only for specific families of spacetimes. Also, the pseudotensors mentioned above allow quasi-local interpretation through the boundary term of the associated Hamiltonian (see e.g. here).

A comprehensive review of quasi-local quantities in general relativity is:

  • László B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in General Relativity, Living Rev. Relativity, 12, (2009), 4, doi:10.12942/lrr-2009-4.
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