Quasilocal Energy Research Articles

Quasilocal energy conditions.

The classical value of the Hamiltonian for a system with timelike boundary has been interpreted as a quasilocal energy. This quasilocal energy is not positive definite. However, we derive a `quasilocal dominant energy condition' which is the natural consequence of the local dominant energy condition. We discuss some implications of this quasilocal energy condition. In particular, we find that it implies a `quasilocal weak energy condition'.

Spinors and the reference point of quasilocal energy

This paper investigates the relationship between the quasilocal energy of Brown and York and certain spinorial expressions for gravitational energy constructed from the Witten--Nester integral. A key feature of the Brown--York method for defining quasilocal energy is that it allows for the freedom to assign the reference point of the energy. When possible, it is perhaps most natural to reference the energy against flat space, i.e. assign flat space the zero value of energy. It is demonstrated that the Witten--Nester integral when evaluated on solution spinors to the Sen--Witten equation (obeying appropriate boundary conditions) is essentially the Brown--York quasilocal energy with a reference point determined by the Sen--Witten spinors. For the case of round spheres in the Schwarzschild geometry, these spinors determine the flat-space reference point. A similar viewpoint is proposed for the Schwarzschild-case quasilocal energy of Dougan and Mason.

Energy of small surfaces

The quasilocal gravitational energy expression recently proposed by Hayward using a 2+2 Hamiltonian is calculated for cross sections of a null cone near the vertex. It is shown that in vacuum, there is a dominating contribution determined by the Bel--Robinson tensor with coefficients such that the energy is negative.

Quasilocal energy for a Kerr black hole.

The quasilocal energy associated with a constant stationary time slice of the Kerr spacetime is presented. The calculations are based on a recent proposal in which quasilocal energy is derived from the Hamiltonian of spatially bounded gravitational systems. Three different classes of boundary surfaces for the Kerr slice are considered (constant radius surfaces, round spheres, and the ergosurface). Their embeddings in both the Kerr slice and flat three-dimensional space (required as a normalization of the energy) are analyzed. The energy contained within each surface is explicitly calculated in the slow rotation regime and its properties are discussed in detail. The energy is a positive, monotonically decreasing function of the boundary surface radius. It approaches the Arnowitt-Deser-Misner (ADM) mass at spatial infinity and reduces to (twice) the irreducible mass at the horizon of the Kerr black hole. The expressions possess the correct static limit and include negative contributions due to gravitational binding. The energy at the ergosurface is compared with the energies at other surfaces. Finally, the difficulties involved in an estimation of the energy in the fast rotation regime are discussed.

Two-dimensional Sen connections and quasi-local energy--momentum

The recently constructed two-dimensional Sen connection is applied in the problem of quasi-local energy--momentum in general relativity. First it is shown that, because of one of the two two-dimensional Sen--Witten identities, Penrose's quasi-local charge integral can be expressed as a Nester--Witten integral. Then, to find the spinor propagation laws appropriate to the Nester--Witten integral, all the possible first order linear differential operators that can be constructed from the irreducible chiral parts, only, of the Sen operator alone are determined and examined. It is only the holomorphy or anti-holomorphy operator that can define acceptable propagation laws. The two-dimensional Sen connection thus naturally defines a quasi-local energy--momentum, which is precisely that of Dougan and Mason. Then, provided the dominant energy condition holds and the 2-sphere is convex, we show that the following statements are equivalent: (i) the quasi-local mass (energy--momentum) associated with a 2-sphere is zero; (ii) the Cauchy development is a pp-wave geometry with pure radiation ( is flat), where is a spacelike hypersurface with ; (iii) there exists a Sen-constant spinor field (two spinor fields) on . Thus the pp-wave Cauchy developments can be characterized by the geometry of a two- rather than a three-dimensional submanifold.

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.

Canonical variables and quasilocal energy in general relativity

Recently Brown and York(1993) have devised a new method for defining quasilocal energy in general relativity. Their method employs a Hamilton-Jacobi analysis of an action functional for a spatially bounded spacetime M, and this analysis yields expressions for the quasilocal energy and momentum surface densities associated with the two-boundary B of a spacelike slice of such a spacetime. These expressions are essentially Amowitt-Deser-Misner variables, but with canonical conjugacy defined with respect to the time history 3B of the two-boundary. These boundary ADM variables match previous variables which have arisen directly in the study of black hole thermodynamics. Further, a 'microcanonical' action which features fixed-energy boundary conditions has been introduced in related work concerning the functional-integral representation of the density of quantum states for such a bounded gravitational system. This paper introduces Ashtekar-type variables on the time history 3B of the two-boundary and shows that these variables lead to elegant alternative expressions for the quasilocal surface densities. In addition, it is demonstrated here that both the boundary ADM variables and the boundary Ashtekar-type variables can be incorporated into a larger framework by appealing to the tetrad-dependent Sparling differential forms. Finally, using these results and a tetrad action principle employed by Goldberg(1988), this paper constructs two new tetrad versions of the microcanonical action.

Quasilocal energy and conserved charges derived from the gravitational action

The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on $^{3}$B, the history of the system's boundary. Energy surface density, momentum surface density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate $^{3}$B. The integral of the energy surface density over such a two-surface B is the quasilocal energy associated with a spacelike three-surface \ensuremath{\Sigma} whose orthogonal intersection with $^{3}$B is the boundary B. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary B as a surface embedded in \ensuremath{\Sigma}. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper-time translations on $^{3}$B in the timelike direction orthogonal to B. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on $^{3}$B. For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt, Deser, and Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.

Quasilocal energy and conserved charges derived from the gravitational action

Quasilocal energy and conserved charges derived from the gravitational action

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.

Quasilocal energy and the Bel-Robinson tensor

The general-relativistic field equations are examined from the point of view of a local inertial observer and a quasilocal definition of energy-momentum is thereby obtained. This definition relates to the Bel-Robinson tensor and the approach is shown to be consistent with the result obtained from the definition of energy given by Hawking.

Quasi-local acceptor levels spectrum in quantizing magnetic field Hg 1-xCd xTe materials

The paper investigates the dependence of quasi-local acceptor levels energy on the magnetic field. It is shown that under cross-section of quasi-local levels with Landau levels the energy of quasi-local levels is sharply shifted to energy decrease.

Quasi-local energy with respect to a static spacetime

This article considers the quasi-local energy in reference to a general static spacetime. We follow the approach developed by the authors in [19, 20, 7, 9] and define the quasi-local energy as a difference of surface Hamiltonians, which are derived from the Einstein-Hilbert action. The new quasi-local energy provides an effective gauge independent measurement of how far a spacetime deviates away from the reference static spacetime on a finitely extended region.