Linear perturbations for the vacuum axisymmetric Einstein equations

In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. This property is expected to play an important role in the global evolution. In this gauge the equations reduce to a coupled hyperbolic-elliptic system which is formally singular at the axis. Due to the rather peculiar properties of the system, the local in time existence has proved to resist analysis by standard methods. To analyze the principal part of the equations, which may represent the main source of the difficulties, we study linear perturbation around the flat Minkowski solution in this gauge. In this article we solve this linearized system explicitly in terms of integral transformations in a remarkable simple form. This representation is well suited to obtain useful estimates to apply in the non-linear case.Comment: 13 pages. We suppressed the statements about decay at infinity. The proofs of these statements were incomplete. The complete proofs will require extensive technical analysis. We will studied this in a subsequent work. We also have rewritten the introduction and slighted changed the titl

Angular momentum-mass inequality for axisymmetric black holes

In these notes we describe recent results concerning the inequality $m\geq \sqrt{|J|}$ for axially symmetric black holes.Comment: 7 pages, 1 figur

The Yamabe invariant for axially symmetric two Kerr black holes initial data

An explicit 3-dimensional Riemannian metric is constructed which can be interpreted as the (conformal) sum of two Kerr black holes with aligned angular momentum. When the separation distance between them is large we prove that this metric has positive Ricci scalar and hence positive Yamabe invariant. This metric can be used to construct axially symmetric initial data for two Kerr black holes with large angular momentum.Comment: 14 pages, 2 figure

Initial data for black hole collisions

I describe the construction of initial data for the Einstein vacuum equations that can represent a collision of two black holes. I stress in the main physical ideas.Comment: 5 pages, 2 figures. To appear in the Proceedings of the Spanish Relativity Meeting Gravitation and Cosmology ERE - 2002; isbn: 978844752738

Proof of the (local) angular momemtum-mass inequality for axisymmetric black holes

We prove that for any vacuum, maximal, asymptotically flat, axisymmetric initial data for Einstein equations close to extreme Kerr data, the inequality $\sqrt{J} \leq m$ is satisfied, where $m$ and $J$ are the total mass and angular momentum of the data. The proof consists in showing that extreme Kerr is a local minimum of the mass.Comment: 14 page

On black holes as inner boundaries for the constraint equations

General aspects of the boundary value problem for the constraint equations and their application to black holes are discussed.Comment: 8 pages. Seventh Hungarian Relativity Workshop, Sarospatak, Hungary, 10-15 August, 2003. To appear in the proceedings; isbn: 978963058187

Generalized Korn's inequality and conformal Killing vectors

Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity.Comment: 8 page