77 research outputs found
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
is satisfied, where and 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
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
Asymptotically Flat Initial Data with Prescribed Regularity at Infinity
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.Comment: Latex 2e, 47 pages, no figure
The inequality between mass and angular momentum for axially symmetric black holes
In this essay I first discuss the physical relevance of the inequality 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 as an absolute
minimum of the total mass. Finally, I conjecture on the physical implications
of this characterization for the non linear stability problem for black holes.Comment: 8 pages, Honorable Mention in the Gravity Research Foundation Essay
Competition 200
The wave equation on the extreme Reissner-Nordstr\"om black hole
We study the scalar wave equation on the open exterior region of an extreme
Reissner-Nordstr\"om black hole and prove that, given compactly supported data
on a Cauchy surface orthogonal to the timelike Killing vector field, the
solution, together with its derivatives of arbitrary order,
a tortoise radial coordinate, is bounded by a constant that depends only on
the initial data. Our technique does not allow to study transverse derivatives
at the horizon, which is outside the coordinate patch that we use. However,
using previous results that show that second and higher transverse derivatives
at the horizon of a generic solution grow unbounded along horizon generators,
we show that any such a divergence, if present, would be milder for solutions
with compact initial data.Comment: Minor correction
Rest Frame System for Asymptotically Flat Spacetimes
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 on the differentiable
and non-self-crossing properties of this family of solutions. We also give a
proof of the global existence of nice sections.Comment: 26 pages, LaTeX2
Geometric inequalities for black holes
It is well known that the three parameters that characterize the Kerr black
hole (mass, angular momentum and horizon area) satisfy several important
inequalities. Remarkably, some of these inequalities remain valid also for
dynamical black holes. This kind of inequalities play an important role in the
characterization of the gravitational collapse. They are closed related with
the cosmic censorship conjecture. In this article recent results in this
subject are reviewed.Comment: Invited review article for General Relativity and Gravitation. Based
on my plenary lecture at GR20 and the longer review article Classical and
Quantum Gravity, 29(7):073001, 2012, arXiv:1111.3615. 27 pages. 14 figures.
Minor change
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