Spacetime mappings of the Brown–York quasilocal energy
In several areas of theoretical physics it is useful to know how a quasilocal energy transforms under conformal rescalings or generalized Kerr–Schild mappings. We derive the transformation properties of the Brown–York quasilocal energy in spherical symmetry and we contrast them with those of the Misner–Sharp–Hernandez energy.
The mass of a non-isolated system in General Relativity (GR) has been the subject of intense study but there is no agreement as to what the mass–energy should be. Due to the equivalence principle, the energy of the gravitational field cannot be localized and the mass–energy of a self-gravitating system includes also this energy. Unless the geometry reduces asymptotically to Minkowski (in which case the ADM energy  is appropriate), one resorts to quasilocal energy definitions. There are several quasilocal constructs in the literature, which differ from each other (see  for a recent review). Overall, quasilocal energy has been studied in the domain of formal relativity, but one ought to do better. First, the mass of a gravitating system is one of its most basic properties in astrophysics and a mass–energy definition is ultimately of no use if it cannot be employed in practical calculations (for example, in astrophysics and/or in cosmology). Second, various authors are already using, implicitly, the Hawking–Hayward quasilocal energy [3, 4] (usually in its Misner–Sharp–Hernandez form defined for spherical symmetry [5, 6]) in black hole thermodynamics [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], in which this quasilocal energy plays the role of the internal energy of the system.
Black hole thermodynamics (especially the thermodynamics of time-dependent apparent horizons) is usually studied in the context of GR and most often in spherical symmetry, where the Misner–Sharp–Hernandez mass is adopted almost universally [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] (see, however, Ref.  for an analogous study using the Brown–York mass). The Misner–Sharp–Hernandez mass is also the quasilocal construct used in spherical fluid dynamics and in black hole collapse [5, 6] and is the Noether charge associated with the covariant conservation of the Kodama energy current . But there are several other definitions of quasilocal energy  and one wonders what changes the use of another quasilocal construct, for example the Brown–York energy, would bring. When a black hole is dynamical, it is difficult to calculate its temperature unambiguously and the recent literature on the thermodynamics of dynamical black holes focuses on this quantity. If the definition of internal energy is also uncertain, the problems accumulate. Quasilocal energies have been used also in the now rather broad field of thermodynamics of spacetime [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
A full discussion of which quasilocal mass should be used, and why, requires more insight on quasilocal energies than is presently available. Here we consider a particular aspect, more related to tool-building than to core issues, which has been discussed recently in the literature. Since several analytic solutions of the Einstein field equations which describe dynamical black holes are generated by using the Schwarzschild (or another static black hole) solution as a seed and performing a conformal or a Kerr–Schild transformation [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], the transformation properties of the Misner–Sharp–Hernandez mass under these spacetime mappings were discussed . Later, relinquishing the simplifying assumption of spherical symmetry, the transformation properties of the Hawking–Hayward quasilocal energy [3, 4] (which reduces to the Misner–Sharp–Hernandez prescription [5, 6] in spherical symmetry ) were also obtained . Generalizations of the Hawking–Hayward energy to scalar-tensor gravity have also been introduced ([54, 55, 56, 57], see also [58, 59, 60, 61, 62], and [63, 64] for the case of Lovelock gravity), following earlier generalizations of the Brown–York mass to these theories [65, 66, 67]. A useful trick consists of remembering that these theories admit a representation in the Einstein conformal frame which is formally very similar to GR. If one chooses a different quasilocal energy, it becomes important to establish how this construct transforms under these spacetime mappings.
There are also other motivations for studying the transformation properties of quasilocal energies. As noted above, these quantities are defined rather formally and are not yet used in practical calculations in astrophysics and cosmology, with the exception of the recent works [68, 69, 70]. There, the Hawking–Hayward quasilocal construction was employed in a new approach to cosmological problems in which the expansion of the universe competes with the local dynamics of inhomogeneities, namely the Newtonian simulations of large scale structure formation in the early universe , the turnaround radius in the present accelerated universe , and lensing by the cosmological constant or by dark energy . To first order in the metric perturbations present in these problems, the Brown–York energy yields the same results as the Hawking–Hayward energy, provided that an appropriate gauge is chosen for the gauge-dependent Brown–York energy in the comparison . Conformal transformations were used in these works as a mere calculational tool, not for any conceptual reason. This is one more reason to establish how the Brown–York mass behaves under spacetime mappings, if it was going to replace the Hawking–Hayward/Misner–Sharp–Hernandez construct.
Since the Brown–York mass is gauge-dependent, it makes sense to derive its transformation properties under conformal and Kerr–Schild spacetime mappings only when a certain gauge is preserved by the map. This is what we do in the following sections. Since both the expression and the value of the Brown–York mass are very different in different gauges, it is meaningless to compare them in these different gauges.
2 Conformal transformations
3 Kerr–Schild transformations
Here we present examples illustrating the transformation properties of the Brown–York mass.
4.1 Conformal transformation
4.2 Kerr–Schild transformation
There are several reasons to derive the transformation properties of a quasilocal energy under conformal or (generalized) Kerr–Schild transformations. This procedure is part of the tool-building process useful in various areas of theoretical gravity (black hole thermodynamics, analytical solutions of GR describing dynamical black holes, spacetime thermodynamics, etc.). The relativity community seems to have concentrated on the Hawking–Hayward/Misner–Sharp–Hernandez quasilocal energy (see, however, Refs. [19, 75, 76]) but the Brown–York energy is also interesting in principle because it is based on the Hamilton-Jacobi formulation of GR. However, contrary to the Misner–Sharp–Hernandez mass, the Brown–York constructs suffers from a daunting gauge-dependence even in spherical symmetry. For this reason, the comparison of the “new” Brown–York energy after a spacetime mapping with the “old” one is meaningful only after restoring the gauge which is altered by the spacetime mapping. Having done this and having obtained the “new” Brown–York mass in terms of the “old” one and of the geometry, the result cannot be encapsulated in a simple formula analogous to Eqs. (17) or (35) obtained for the Misner–Sharp–Hernandez mass in . From a pragmatic point of view, the Misner–Sharp–Hernandez construct looks definitely more attractive than the Brown–York one.
We are grateful to a referee for constructing comments. This work is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (Grant no. 2016-03803 to V.F.) and by Bishop’s University.
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