A variational principle for stationary, axisymmetric solutions of Einstein's equations

Abstract

Stationary, axisymmetric, vacuum, solutions of Einstein's equations are obtained as critical points of the total mass among all axisymmetric and $(t,\varphi )$ symmetric initial data with fixed angular momentum. In this variational principle the mass is written as a positive definite integral over a spacelike hypersurface. It is also proved that if absolute minimum exists then it is equal to the absolute minimum of the mass among all maximal, axisymmetric, vacuum, initial data with fixed angular momentum. Arguments are given to support the conjecture that this minimum exists and is the extreme Kerr initial data.

21 pages

References in corpus (4)

• Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries
• Black Holes Surrounded by Uniformly Rotating Rings
• The Constraint Equations
• Higher Dimensional Generalizations of the Kerr Black Hole

Cited by in corpus (22)

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• From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
• Geometric inequalities for axially symmetric black holes
• Angular momentum-mass inequality for axisymmetric black holes
• Killing Horizons: Negative Temperatures and Entropy Super-Additivity
• Proof of the (local) angular momemtum-mass inequality for axisymmetric black holes
• A Dain Inequality with charge
• The inequality between mass and angular momentum for axially symmetric black holes
• Extreme throat initial data set and horizon area--angular momentum inequality for axisymmetric black holes
• Geometrical inequalities bounding angular momentum and charges in General Relativity
• A uniqueness theorem for degenerate Kerr-Newman black holes
• Geometric inequalities for black holes
• Convexity of reduced energy and mass angular momentum inequalities
• Proof of the area-angular momentum-charge inequality for axisymmetric black holes
• Numerical evidences for the angular momentum-mass inequality for multiple axially symmetric black holes
• Small deformations of extreme Kerr black hole initial data
• The Yamabe invariant for axially symmetric two Kerr black holes initial data
• Initial data for rotating cosmologies
• Mass angular momentum inequality for axisymmetric vacuum data with small trace
• On a mass functional for initial data in 4+1 dimensional spacetime
• Proof of the local mass-angular momenta inequality for $U(1{)}^2$ invariant black holes
• Angular momentum-mass inequality for axisymmetric black holes