Quasi-local Angular Momentum Research Articles

Non-asymptotically flat, non-AdS dilaton black holes

We show that previously known non-asymptotically flat static black hole solutions of Einstein-Maxwell-dilaton theory may be obtained as near-horizon limits of asymptotically flat black holes. Specializing to the case of the dilaton coupling constant ${\ensuremath{\alpha}}^{2}=3$, we generate from the non-asymptotically flat magnetostatic or electrostatic black holes two classes of rotating dyonic black hole solutions. The rotating dyonic black holes of the magnetic class are dimensional reductions of the five-dimensional Myers-Perry black holes relative to one of the azimuthal angles, while those of the electric class are twisted dimensional reductions of rotating dyonic Rasheed black strings. We compute the quasilocal mass and angular momentum of our rotating dyonic black holes and show that they satisfy the first law of black hole thermodynamics, as well as a generalized Smarr formula. We also discuss the construction of non-asymptotically flat multi-extreme black hole configurations.

Extremal limit for charged and rotating 2 + 1-dimensional black holes and Bertotti–Robinson geometry

We consider 2 + 1-dimensional analogues of the Bertotti–Robinson (BR) spacetimes in the sense that the coefficient at the angular part is a constant. We show that such BR-like solutions are either pure static or uncharged rotating. We trace the origin of the inconsistency between a charge and rotation, considering the BR-like spacetime as a result of the limiting transition of a non-extremal black hole to the extremal state. We also find that the quasi-local energy and angular momentum of such BR-like spacetimes calculated within the boundary l = const (l is the proper distance) are constants independent of the position of the boundary.

Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article

The present status of the quasi-local mass-energy-momentum and angular momentum constructions in general relativity is reviewed. First the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities are recalled. Then the various specific constructions and their properties (both successes and defects) are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.This review is based on the talks given at the Erwin Schrödinger Institute, Vienna, in July 1997, at the Universität Tübingen, in May 1998, and at the National Center for Theoretical Sciences in Hsinchu and at the National Central University, Chungli, Taiwan, in July 2000.

Conserved currents, superpotentials and cosmological perturbations

Conserved currents, superpotentials and cosmological perturbations

Quasi-local conservation equations in general relativity

A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza–Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner–Sharp energy. Moreover, on the event horizon of a general dynamical black hole, the quasi-local energy conservation equation coincides with the conservation equation studied by Thorne et al. We discuss the remaining quasi-local conservation equations briefly.

General analysis of self-dual solutions for the Einstein-Maxwell-Chern-Simons theory in(1+2)dimensions

The solutions of the Einstein-Maxwell-Chern-Simons theory are studied in (1+2) dimensions with the self-duality condition imposed on the Maxwell field. We give a closed form of the general solution which is determined by a single function having the physical meaning of the quasilocal angular momentum of the solution. This function completely determines the geometry of spacetime, also providing the direct computation of the conserved total mass and angular momentum of the configurations.

Quasilocal quantities for general relativity and other gravity theories

From a covariant Hamiltonian formulation, by using symplectic ideas, we obtain certain covariant boundary expressions for the quasilocal quantities of general relativity and other geometric gravity theories. The contribution from each of the independent dynamic geometric variables (the frame, metric or connection) has two possible covariant forms associated with the selected type of boundary condition. The quasilocal expressions also depend on a reference value for each dynamic variable and a displacement vector field. Integrating over a closed 2-surface with suitable choices for the vector field gives the quasilocal energy, momentum and angular momentum. For the special cases of Einstein's theory and the Poincaré gauge theory our expressions are similar to some previously known expressions and give good values for the total ADM and Bondi quantities. We apply our formalism to black hole thermodynamics obtaining the first law and an associated entropy expression for these general gravity theories. For Einstein's theory our quasilocal expressions are evaluated on static spherically symmetric solutions and compared with the findings of some other researchers. The choices needed for the formalism to associate a quasilocal expression with the boundary of a region are discussed.

Quasi-local energy - momentum and 2-surface characterization of the pp-wave spacetimes

We consider the determination of the pp-wave metric from the geometry of certain spacelike 2-surfaces. It has been shown that the vanishing of the Dougan - Mason quasi-local mass , associated with the smooth boundary of a spacelike hypersurface , is equivalent to the statement that the Cauchy development is of a pp-wave type geometry with pure radiation, provided the ingoing null normals are not diverging on and the dominant energy condition holds on . The metric on itself, however, has not been determined. Here, assuming that the matter is a zero-rest-mass field, it is shown that both the matter field and the pp-wave metric of are completely determined by the value of the zero-rest-mass field on and the two-dimensional Sen geometry of , provided a convexity condition, slightly stronger than above, holds. Thus the pp-waves can be characterized not only by the usual Cauchy data on a three-dimensional but by data on its two-dimensional boundary too. In addition, it is shown that the Ludvigsen - Vickers quasi-local angular momentum of axially symmetric pp-wave geometries has the familiar properties known for pure (matter) radiation.

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.

Difficulties with quasilocal momentum and angular momentum

The twistorial definitions of quasilocal mass, momentum and angular momentum are satisfactory in some interesting cases, but there are situations in which ambiguities are present in the definitions. It has been unclear whether these difficulties are essential. The author shows that for conformally flat spacetimes it is generically impossible to find an 'infinity twistor' which reduces the kinematic quantities to those of Minkowski, de Sitter or anti-de Sitter space. This means there is no way to extract the quasilocal momentum and angular momentum (or their (anti-) de Sitter analogues). In fact, the author shows that the kinematic twistor takes all values in an open set. This implies too that the determinant of the kinematic twistor is complex, which rules out one of the suggested definitions of quasilocal mass.

Quasilocal momentum and angular momentum in Kerr spacetime

It is shown that the Kerr spacetime possesses a vector space V and an affine space M0 allowing definitions of quasilocal momentum and angular momentum with all the desired properties. The construction of V and M0 makes use of a natural, flat, metric connection with torsion and of a generalized twistor equation. The momentum and angular momentum contained within a finite 2-surface Sigma are expressed as the integrals over Sigma of two closed 2-forms.

Twistor theory and the energy—momentum and angular momentum of the gravitational field at spatial infinity

Penrose’s ‘quasi-local mass and angular momentum’ (Penrose, Proc. R. Soc. Lond . A 381, 53 (1982)) is investigated for 2-surfaces near spatial infinity in both linearized theory on Minkowski space and full general relativity. It is shown that for space-times that are radially smooth of order one in the sense of Beig & Schmidt ( Communs math. Phys . 87, 65 (1982)), with asymptotically electric Weyl curvature, there exists a global concept of a twistor space at spatial infinity. Global conservation laws for the energy—momentum and angular momentum are obtained, and the ten conserved quantities are shown to be invariant under asymptotic coordinate transformations. The relation to other definitions is discussed briefly.

Momentum, angular momentum and their quasi-local null surface extensions

Let M be an asymptotically flat space-time which satisfies the dominant energy condition and let N be a null hypersurface of M which meets I+ in a space-like cross section S given by r= infinity , where r is an affine parameter along the generators of N. It is shown that there exists a quasi-local momentum Pa(r), defined by means of an integral over the r=constant cross sections of N, which tends to the Bondi momentum as r to infinity , and which satisfies the radial 'mass-gain condition' Pa(r2)ka>or=Pa(r1)ka when r2>or=r1, where ka is a constant future pointing null vector. This condition is then used to show that, in certain circumstances, the Bondi momentum is always future pointing or, in other words, that the Bondi mass is always positive. A quasi-local angular momentum is also defined, which tends to Bramson's (1975) angular momentum as r to infinity .

Quasi-local mass and angular momentum in general relativity

A new approach to defining energy-momentum and angular momentum in general relativity is presented which avoids some of the difficulties of previous definitions and which can be applied quasi-locally. It depends on the construction of a twistor space T α ( S ) associated with any spacelike topological 2-sphere S . Though several problems of interpretation remain to be solved, the new definition works well at I + , reproducing the Bondi-mass-momentum as four of the ten precisely determined quantities at each cut of I + . The remaining six quantities provide a definition of angular momentum which appears to be new.