Quasilocal Energy Definitions Research Articles

The small sphere limit of quasilocal energy in higher dimensions along lightcone cuts

The problem of quasilocal energy has been extensively studied mainly in four dimensions. Here we report results regarding the quasilocal energy in spacetime dimension . After generalising three distinct quasilocal energy definitions to higher dimensions under appropriate assumptions, we evaluate their small sphere limits along lightcone cuts shrinking towards the lightcone vertex. The results in vacuum are conveniently represented in terms of the electromagnetic decompositions of the Weyl tensor. We find that the limits at presence of matter yield the stress tensor as expected, but the vacuum limits are in general not proportional to the Bel–Robinson superenergy Q in dimensions n > 4. The result defies the role of the Bel–Robinson superenergy as characterising the gravitational energy in higher dimensions, albeit the fact that it uniquely generalises. Surprisingly, the Hawking energy and the Brown–York energy exactly agree upon the small sphere limits across all dimensions. The ‘new’ vacuum limit , however, cannot be interpreted as a gravitational energy because of its non-positivity. Furthermore, we also give the small sphere limits of the Kijowski–Epp–Liu–Yau type energy in higher dimensions, and again we see in place of Q. Our work extends earlier investigations of the small sphere limits (Horowitz and Schmidt 1982 Proc. R. Soc. A 381 215–24; Bergqvist 1994 Class. Quantum Grav. 11 3013–23; Brown et al 1999 Phys. Rev. D 59 064028; Yu 2007 (arXiv:0706.1081)), and also complements (Miao et al 2017 J. Geom. Anal. 27 1323–54).

Quasilocal energy exchange and the null cone

Energy is at best defined quasilocally in general relativity. Quasilocal energy definitions depend on the conditions one imposes on the boundary Hamiltonian, i.e., how a finite region of spacetime is "isolated". Here, we propose a method to define and investigate systems in terms of their matter plus gravitational energy content. We adopt a generic construction, that involves embedding of an arbitrary dimensional world sheet into an arbitrary dimensional spacetime, to a 2 + 2 picture. In our case, the closed 2-dimensional spacelike surface $\mathbb{S}$, that is orthogonal to the 2-dimensional timelike world sheet $\mathbb{T}$ at every point, encloses the system in question. The integrability conditions of $\mathbb{T}$ and $\mathbb{S}$ correspond to three null tetrad gauge conditions once we transform our notation to the one of the null cone observables. We interpret the Raychaudhuri equation of $\mathbb{T}$ as a work-energy relation for systems that are not in equilibrium with their surroundings. We achieve this by identifying the quasilocal charge densities corresponding to rotational and nonrotational degrees of freedom, in addition to a relative work density associated with tidal fields. We define the corresponding quasilocal charges that appear in our work-energy relation and which can potentially be exchanged with the surroundings. These charges and our tetrad conditions are invariant under type-III Lorentz transformations, i.e., the boosting of the observers in the directions orthogonal to $\mathbb{S}$. We apply our construction to a radiating Vaidya spacetime, a $C$-metric and the interior of a Lanczos-van Stockum dust metric. The delicate issues related to the axially symmetric stationary spacetimes and possible extensions to the Kerr geometry are also discussed.

Quasilocal energy and thermodynamic equilibrium conditions

Equilibrium thermodynamic laws are typically applied to horizons in general relativity without stating the conditions that bring them into equilibrium. We fill this gap by applying a new thermodynamic interpretation to a generalized Raychaudhuri equation for a worldsheet orthogonal to a closed spacelike two-surface, the ‘screen’, which encompasses a system of arbitrary size in nonequilibrium with its surroundings in general. In the case of spherical symmetry this enables us to identify quasilocal thermodynamic potentials directly related to standard quasilocal energy definitions. Quasilocal thermodynamic equilibrium is defined by minimizing the mean extrinsic curvature of the screen. Moreover, without any direct reference to surface gravity, we find that the system comes into quasilocal thermodynamic equilibrium when the screen is located at a generalized apparent horizon. Examples of the Schwarzschild, Friedmann–Lemaître and Lemaître–Tolman geometries are investigated and compared. Conditions for the quasilocal thermodynamic and hydrodynamic equilibrium states to coincide are also discussed, and a quasilocal virial relation is suggested as a potential application of this approach.

Quasilocal Hamiltonians in general relativity

We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consistent definition of quasi-local energy in GR requires, at a minimum, a framework based on the (currently unknown) geometric well-posedness of the initial boundary value problem for the Einstein equations.

Angular momentum and an invariant quasilocal energy in general relativity

A key feature of the Brown-York definition of quasilocal energy is that, under local boosts of the fleet of observers measuring the energy, the quasilocal energy surface density transforms as one would expect based on the equivalence principle, namely, like E in the special relativity formula: ${E}^{2}\ensuremath{-}{p}^{2}{=m}^{2}.$ This paper provides physical motivation for the general relativistic analogue of this formula, and thereby arrives at a geometrically natural definition of an ``invariant quasilocal energy'' (IQE). In analogy with the invariant mass m, the IQE is invariant under local boosts of the fleet of observers on a given two-surface S in spacetime. A reference energy subtraction procedure is required, but in contrast with the Brown-York procedure, S is isometrically embedded in a four-dimensional reference spacetime of one's choosing. For example, it is well known that any sphere, round or not, can always be isometrically embedded into Minkowski space, even if its scalar curvature is not everywhere positive. So rather than embeddability being a concern, the problem now is that such embeddings are not unique, leading to an ambiguity in the reference IQE. However, in this codimension-two setting there are two curvatures associated with S: the curvature of its tangent bundle, and the curvature of its normal bundle. Taking advantage of this fact suggests a possible way to resolve the embedding ambiguity, which at the same time will be seen to incorporate angular momentum into the energy at the quasilocal level. The IQE is analyzed in the following cases: both the spatial and future null infinity limits of a large sphere in asymptotically flat spacetimes; a small sphere shrinking to a point along either spatial or null directions; and finally, a large sphere in asymptotically anti--de Sitter spacetimes. The last case reveals a striking similarity between the reference IQE and a certain counterterm energy recently proposed in the context of the conjectured AdS/CFT correspondence.

Conserved quasilocal quantities and general covariant theories in two dimensions.

General matterless--theories in 1+1 dimensions include dilaton gravity, Yang--Mills theory as well as non--Einsteinian gravity with dynamical torsion and higher power gravity, and even models of spherically symmetric d = 4 General Relativity. Their recent identification as special cases of 'Poisson--sigma--models' with simple general solution in an arbitrary gauge, allows a comprehensive discussion of the relation between the known absolutely conserved quantities in all those cases and Noether charges, resp. notions of quasilocal 'energy--momentum'. In contrast to Noether like quantities, quasilocal energy definitions require some sort of 'asymptotics' to allow an interpretation as a (gauge--independent) observable. Dilaton gravitation, although a little different in detail, shares this property with the other cases. We also present a simple generalization of the absolute conservation law for the case of interactions with matter of any type.

Quasilocal gravitational energy.

A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a `2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed.

Conserved quantities at spatial and null infinity: The Penrose potential.

We define a superpotential for energy-momentum and rotation momentum which is built out of the conformal tensor and a bivector. This superpotential is identified with that used by Penrose in his definition of quasilocal energy. It is applied to the definition of energy-momentum and rotation momentum at spatial and at null infinities. At spatial infinity the results are in agreement with those of Ashtekar and Hansen. At null infinity the results are unsatisfactory; they are tied to a specific Bondi frame. Thus, they are not in agreement with the results of Tamburino and Winicour, Geroch and Winicour, nor with those of Dray and Streubel. Some reasons for this failure are discussed.