We prove that extreme Kerr initial data set is a unique absolute minimum of the total mass in a (physically relevant) class of vacuum , maximal, asymptotically flat, axisymmetric data for Einstein equations with fixed angular momentum. These data represent non-stationary, axially symmetric, black holes. As a consequence, we obtain that any data in this… (More)
We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate.
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… (More)
We prove the existence of a family of initial data for the Einstein vacuum equation which can be interpreted as the data for two Kerr-like black holes in arbitrary location and with spin in arbitrary direction. This family of initial data has the following properties: (i) When the mass parameter of one of them is zero or when the distance between them goes… (More)
We study Cauchy initial data for asymptotically flat, stationary vacuum space-times near space-like infinity. The fall-off behavior of the intrinsic metric and the extrinsic curvature is characterized. We prove that they have an analytic expansion in powers of a radial coordinate. The coefficients of the expansion are analytic functions of the angles. This… (More)
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. The physical system we want to describe is a binary system of two black holes. It is expected that lot of such systems exist in the universe. Moreover, these systems are expected to have the… (More)
We prove that for any vacuum, maximal, asymptotically flat, ax-isymmetric initial data for Einstein equations close to extreme Kerr data, the inequality √ J ≤ 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.
The notion of center of mass for an isolated system has been previously encoded in the definition of the so called nice sections. In this article we present a generalization of the proof of existence of solutions to the linearized equation for nice sections, and formalize a local existence proof of nice sections relaxing the radiation condition. We report… (More)
Stationary, axisymmetric, vacuum, solutions of Einstein's equations are obtained as critical points of the total mass among all ax-isymmetric and (t, φ) 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… (More)
In this essay I first discuss the physical relevance of the inequality m ≥ |J| for axially symmetric (non-stationary) black holes, where m is the mass and J the angular momentum of the spacetime. Then, I present a proof of this inequality for the case of one spinning black hole. The proof involves a remarkable characterization of the extreme Kerr black hole… (More)