Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modelled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity, and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity, suggested a phenomenological model for hairy black holes, provided novel techniques to extract physics from numerical simulations, and led to new laws governing the dynamics of black holes in exact general relativity.

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Isolated and dynamical horizons and their applications

We give a derivation of the gravitational Hamiltonian starting from the Einstein - Hilbert action, keeping track of all surface terms. This derivation can be applied to any spacetime that asymptotically approaches a static background solution. The surface term that arises in the Hamiltonian can be taken as the definition of the `total energy', even for spacetimes that are not asymptotically flat. (In the asymptotically flat case, it agrees with the usual ADM energy.) We also discuss the relation between the Euclidean action and the Hamiltonian when there are horizons of infinite area (e.g. acceleration horizons) as well as the usual finite area black hole horizons. Acceleration horizons seem to be more analogous to extreme than nonextreme black holes, since we find evidence that their horizon area is not related to the total entropy.

http://cds.cern.ch/record/275032/files/9501014.pdf

The Gravitational Hamiltonian, action, entropy and surface terms

Within the spinorial version of the Cartan structure formulas with the built‐in (complex) Einstein vacuum equations some closed semi‐Einsteinian substructures are isolated and discussed. Then the idea of semigraviton is introduced, and its relationship to Penrose’s nonlinear graviton is described.

On the separation of Einsteinian substructures

A detailed description of how black holes grow in full, non-linear general relativity is presented. The starting point is the notion of dynamical horizons. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local and the energy flux is positive. Change in the horizon area is related to these fluxes. A notion of angular momentum and energy is associated with cross-sections of the horizon and balance equations, analogous to those obtained by Bondi and Sachs at null infinity, are derived. These in turn lead to generalizations of the first and second laws of black hole mechanics. The relation between dynamical horizons and their asymptotic states —the isolated horizons— is discussed briefly. The framework has potential applications to numerical, mathematical, astrophysical and quantum general relativity.

Dynamical Horizons and their Properties

We present a general procedure for transforming asymptotically flat axially symmetric solutions of the Einstein–Maxwell equations into solutions resembling Melvin’s magnetic universe. Specific applications yield metrics associated with black holes in a magnetic universe. It is hoped that these solutions will be of interest to astrophysicists studying gravitational collapse in the presence of strong magnetic fields.

Black holes in a magnetic universe

The charged C‐metric is transformed into another exact solution of the Einstein–Maxwell field equations corresponding to a massive charged particle accelerated by an electric field. When the appropriate equations of motion are satisfied, the nodal singularity associated with the C‐metric disappears.

Removal of the nodal singularity of the C‐metric

We present exact expressions for the electromagnetic fields associated with arbitrarily charged Kerr–Newman black holes in a magnetic universe. In the particular case of charge Q=2 B0 J, where B0 is the magnetic field parameter and J=ma is the angular momentum, exact expressions for the gravitational field are also presented, while for arbitrarily charged black holes metrical corrections of order B0 are evaluated.

Kerr black holes in a magnetic universe

Spin‐coefficients and null tetrad components of the Ricci tensor and the Weyl conform tensor are evaluated in terms of a single complex gravitational potential e, while null tetrad components of the electromagnetic stress energy tensor are evaluated in terms of a second complex potential φ. All the results are expressed elegantly in terms of a differential operator ð, similar to the ``thop'' of Newman and Penrose. The problem of finding physically pertinent stationary axially symmetric Einstein‐Maxwell fields is reduced to the search for a complex solution ξ0(x, y) of one nonlinear differential equation subject to simple subsidiary conditions.

Complex potential formulation of the axially symmetric gravitational field problem

We describe a procedure for generalizing static axially symmetric solutions of the vacuum Einstein field equations, and employ this procedure to append an external gravitational field to the C‐metric black hole solution. We conjecture that if the strength of the appended gravitational field is chosen appropriately, our generalized C‐metric will serve as a better model for an accelerating black hole than the original C‐metric.

Frederick J. Ernst

Papers

Black holes in a magnetic universe

Removal of the nodal singularity of the C‐metric

Kerr black holes in a magnetic universe

Complex potential formulation of the axially symmetric gravitational field problem

Generalized C-metric.