[Paper Review] The First Law of Black Hole Mechanics

Robert M. Wald|ArXiv.org|May 25, 1993

Relativity and Gravitational Theory19 citations

TL;DR

This paper presents a strengthened derivation of the first law of black hole mechanics using the Hamiltonian formulation of general relativity, proving it holds for arbitrary nonsingular, asymptotically flat perturbations of stationary, axisymmetric black holes—not just to other stationary solutions. The key result closes a gap in black hole uniqueness theorems by proving that Einstein-Maxwell black holes cannot have an ergoregion disjoint from the horizon, implying all nonrotating such black holes must be static.

ABSTRACT

A simple proof of a strengthened form of the first law of black hole mechanics is presented. The proof is based directly upon the Hamiltonian formulation of general relativity, and it shows that the the first law variational formula holds for arbitrary nonsingular, asymptotically flat perturbations of a stationary, axisymmetric black hole, not merely for perturbations to other stationary, axisymmetric black holes. As an application of this strengthened form of the first law, we prove that there cannot exist Einstein-Maxwell black holes whose ergoregion is disjoint from the horizon. This closes a gap in the black hole uniqueness theorems.

Motivation & Objective

• To derive a strengthened form of the first law of black hole mechanics applicable to arbitrary nonsingular, asymptotically flat perturbations of stationary, axisymmetric black holes.
• To close a gap in the black hole uniqueness theorems by proving that Einstein-Maxwell black holes cannot have an ergoregion disjoint from the event horizon.
• To demonstrate that any stationary, nonrotating Einstein-Maxwell black hole must be static, even without assuming global timelikeness of the time-translation Killing field.
• To show the utility of the Hamiltonian formulation in proving geometric and dynamical constraints on black hole solutions.

Proposed method

• Derive the first law using the ADM Hamiltonian formulation of Einstein-Maxwell theory, leveraging its pure constraint form and vanishing on the constraint submanifold.
• Use the variation of the ADM Hamiltonian to directly relate changes in mass, charge, and angular momentum to perturbations of initial data satisfying linearized constraints.
• Construct a specific perturbation of initial data on a maximal, asymptotically flat, and orthogonal slice intersecting the bifurcation surface, preserving charge and horizon area.
• Verify that the perturbation satisfies the linearized constraints and preserves the horizon area while yielding a negative change in ADM mass unless the extrinsic curvature vanishes.
• Use the resulting contradiction with the first law to conclude that the extrinsic curvature must vanish, implying the existence of a hypersurface-orthogonal timelike Killing field.
• Apply isometry invariance and asymptotic behavior to show the time-translation field is a global Killing field, proving staticity of nonrotating black holes.

Experimental results

Research questions

• RQ1 Can the first law of black hole mechanics be proven to hold for arbitrary nonsingular, asymptotically flat perturbations, not just to other stationary solutions?
• RQ2 Is it possible for a nonrotating Einstein-Maxwell black hole to have an ergoregion disjoint from its event horizon?
• RQ3 Does the absence of angular momentum and the preservation of horizon area under perturbations imply that the black hole must be static?
• RQ4 Can the Hamiltonian formulation of general relativity be used to derive geometric constraints on black hole solutions without assuming global symmetries a priori?
• RQ5 Why does the proof fail to generalize to Einstein-Yang-Mills theory, and what does this imply about the existence of nonstatic, nonrotating black holes in that theory?

Key findings

• The first law of black hole mechanics holds in a strengthened form for arbitrary nonsingular, asymptotically flat perturbations of stationary, axisymmetric black holes, not only for perturbations to other stationary solutions.
• Any nonrotating Einstein-Maxwell black hole with an ergoregion disjoint from the horizon must be static, resolving a gap in the black hole uniqueness theorems.
• A specific perturbation of initial data on a maximal slice preserves charge and horizon area but yields a negative change in ADM mass unless the extrinsic curvature vanishes.
• The vanishing of the extrinsic curvature implies the existence of a hypersurface-orthogonal, globally timelike Killing field, proving the spacetime is static.
• The result does not extend to Einstein-Yang-Mills theory, where nonrotating, nonstatic black holes may exist, though they are expected to be unstable.
• The proof demonstrates the power of the Hamiltonian formulation in deriving global geometric properties of black hole solutions from local constraints and variational principles.