James Isenberg

Black Hole Physics: Basic Concepts and New Developments

Preface. I. Basic Concepts. 1. Introduction: Brief History of Black Hole Physics. 2. Spherically Symmetric Black Holes. 3. Rotating Black Holes. 4. Black Hole Perturbations (with N. Andersson). 5. General Properties of Black Holes. 6. Stationary Black Holes. 7. Physical Effects in the Gravitational Field of a Black Hole. 8. Black Hole Electrodynamics. 9. Astrophysics of Black Holes. II. Further Developments. 10. Quantum Particle Creation by Black Holes. 11. Quantum Physics of Black Holes. 12. Thermodynamics of Black Holes. 13. Black Holes in Unified Theories. 14. The Interior of a Black Hole. 15. Ultimate Fate of Black and White Holes. 16. Black Holes, Wormholes, and Time Machines. Conclusions. Appendices: A: Mathematical Formulas. B: Spherically Symmetric Spacetimes. C: Rindler Frame in Minkowski Spacetime. D: Kerr-Newman Geometry. E: Newman-Penrose Formalism. F: Wave Fields in a Curved Spacetime. G: Wave Fields in the Kerr Metric. H: Quantum Fields in Kerr Spacetime. I: Quantum Oscillator. Bibliography. Index.

Constant mean curvature solutions of the Einstein constraint equations on closed manifolds

We prove in detail a theorem which completes the evaluation and parametrization of the space of constant mean curvature (CMC) solutions of the Einstein constraint equations on a closed manifold. This theorem determines which sets of CMC conformal data allow the constraint equations to be solved, and which sets of such data do not. The tools we describe and use here to prove these results have also been found to be useful for the study of non-constant mean curvature solutions of the Einstein constraints.

The Constraint Equations

Initial data for solutions of Einstein’s gravitational field equations cannot be chosen freely: the data must satisfy the four Einstein constraint equations. We first discuss the geometric origins of the Einstein constraints and the role the constraint equations play in generating solutions of the full system. We then discuss various ways of obtaining solutions of the Einstein constraint equations, and the nature of the space of solutions.

Asymptotic behavior of the gravitational field and the nature of singularities in gowdy spacetimes

Abstract In this work, we present results which support the Strong Cosmic Censorship Conjecture and some of the ideas of Belinskii, Khalatnikov, and Lifschitz. The results concern the behavior of the gravitational field in the neighborhood of singularities in the polarized Gowdy spacetimes. We rigorously show (using certain “energy” estimates) that as one approaches the singularity, the metric field of any of these vacuum solutions of Einsteins equations asymptotically approaches a solution of a truncated system of equations in which spatial curvature terms have been dropped. Hence, the gravitational fields, in a sense, spatially decouple near the singularity. Based on this asymptotic behavior, we also show that only in a very small subclass of the polarized Gowdy spacetimes is the curvature bounded near the singularity. Hence, the generic polarized Gowdy spacetime cannot be extended across a Cauchy horizon.

Symmetries of cosmological Cauchy horizons

We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions).For the special case of a null surface diffeomorphic toT3 we characterize the degenerate vacuum solutions completely. These consist of an infinite dimensional family of “plane wave” spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic toT3 and prove the generality of the previously constructed non-degenerate solutions.We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einsteins equations are essentially an artefact of symmetry.

Einstein constraints on asymptotically Euclidean manifolds

We consider the Einstein constraints on asymptotically euclidean manifolds

Non-self-dual gauge fields☆

M

Symmetries of higher dimensional black holes

of dimension

Homothetic and Conformal Symmetries of Solutions to Einstein's Equations

n \geq 3

Gluing and wormholes for the Einstein constraint equations

with sources of both scaled and unscaled types. We extend to asymptotically euclidean manifolds the constructive method of proof of existence. We also treat discontinuous scaled sources. In the last section we obtain new results in the case of non-constant mean curvature.