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Sergio Dain

Publications and source records attributed to Sergio Dain.

At least 37 records · Page 2Linked to original sources
arXiv2010

Extreme throat initial data set and horizon area--angular momentum inequality for axisymmetric black holes

We present a formula that relates the variations of the area of extreme throat initial data with the variation of an appropriate defined mass functional. From this expression we deduce that the first variation, with fixed angular momentum, of the area is zero and the second variation is positive definite evaluated at the extreme Kerr throat initial data. This indicates that the area of the extreme Kerr throat initial data is a minimum among this class of data. And hence the area of generic throat initial data is bounded from below by the angular momentum. Also, this result strongly suggests that the inequality between area and angular momentum holds for generic asymptotically flat axially symmetric black holes. As an application, we prove this inequality in the non trivial family of spinning Bowen-York initial data.

gr-qc↗
arXiv2010

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.

gr-qc↗
arXiv2009

Small deformations of extreme Kerr black hole initial data

We prove the existence of a family of initial data for Einstein equations which represent small deformations of the extreme Kerr black hole initial data. The data in this family have the same asymptotic geometry as extreme Kerr. In particular, the deformations preserve the angular momentum and the area of the cylindrical end.

gr-qc↗
arXiv2009

On well-posedness, linear perturbations and mass conservation for axisymmetric Einstein equation

For axially symmetric solutions of Einstein equations there exists a gauge which has the remarkable property that the total mass can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides a nonlinear control of the variables along the whole evolution. In this gauge, Einstein equations reduce to a coupled hyperbolic-elliptic system which is formally singular at the axis. As a first step in analyzing this system of equations we study linear perturbations on flat background. We prove that the linear equations reduce to a very simple system of equations which provide, thought the mass formula, useful insight into the structure of the full system. However, the singular behavior of the coefficients at the axis makes the study of this linear system difficult from the analytical point of view. In order to understand the behavior of the solutions, we study the numerical evolution of them. We provide strong numerical evidence that the system is well-posed and that its solutions have the expected behavior. Finally, this linear system allows us to formulate a model problem which is physically interesting by itself, since it is connected with the linear stability of black holes solutions in axial symmetry. This model can contribute significantly to solve the nonlinear problem and at the same time it appears to be tractable.

gr-qc↗
arXiv2009

Axisymmetric evolution of Einstein equations and mass conservation

For axisymmetric evolution of isolated systems, we prove that there exists a gauge such that the total mass can be written as a positive definite integral on the spacelike hypersurfaces of the foliation and the integral is constant along the evolution. The conserved mass integral controls the square of the extrinsic curvature and the square of first derivatives of the intrinsic metric. We also discuss applications of this result for the global existence problem in axial symmetry.

gr-qc↗
arXiv2009

Numerical evidences for the angular momentum-mass inequality for multiple axially symmetric black holes

We present numerical evidences for the validity of the inequality between the total mass and the total angular momentum for multiple axially symmetric (non-stationary) black holes. We use a parabolic heat flow to solve numerically the stationary axially symmetric Einstein equations. As a by product our method, we also give numerical evidences that there are no regular solutions of Einstein equations that describe two extreme, axially symmetric black holes in equilibrium.

gr-qc↗
arXiv2008

Extra-Large Remnant Recoil Velocities and Spins from Near-Extremal-Bowen-York-Spin Black-Hole Binaries

We evolve equal-mass, equal-spin black-hole binaries with specific spins of a/mH 0.925, the highest spins simulated thus far and nearly the largest possible for Bowen-York black holes, in a set of configurations with the spins counter-aligned and pointing in the orbital plane, which maximizes the recoil velocities of the merger remnant, as well as a configuration where the two spins point in the same direction as the orbital angular momentum, which maximizes the orbital hang-up effect and remnant spin. The coordinate radii of the individual apparent horizons in these cases are very small and the simulations require very high central resolutions (h ~ M/320). We find that these highly spinning holes reach a maximum recoil velocity of ~3300 km/s (the largest simulated so far) and, for the hangup configuration, a remnant spin of a/mH 0.922. These results are consistent with our previous predictions for the maximum recoil velocity of ~4000 km/s and remnant spin; the latter reinforcing the prediction that cosmic censorship is not violated by merging highly-spinning black-hole binaries. We also numerically solve the initial data for, and evolve, a single maximal-Bowen-York-spin black hole, and confirm that the 3-metric has an O(1/r^2) singularity at the puncture, rather than the usual O(1/r^4) singularity seen for non-maximal spins.

gr-qc↗
arXiv2008

Extreme Bowen-York initial data

The Bowen-York family of spinning black hole initial data depends essentially on one, positive, free parameter. The extreme limit corresponds to making this parameter equal to zero. This choice represents a singular limit for the constraint equations. We prove that in this limit a new solution of the constraint equations is obtained. These initial data have similar properties to the extreme Kerr and Reissner-Nordstrom black hole initial data. In particular, in this limit one of the asymptotic ends changes from asymptotically flat to cylindrical. The existence proof is constructive, we actually show that a sequence of Bowen-York data converges to the extreme solution.

gr-qc↗
arXiv2008

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.

gr-qc↗
arXiv2007

The inequality between mass and angular momentum for axially symmetric black holes

In this essay I first discuss the physical relevance of the inequality $m\geq \sqrt{|J|}$ 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 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.

gr-qc↗
arXiv2007

Proof of the angular momentum-mass inequality for axisymmetric black holes

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 class satisfy the inequality $\sqrt{J} \leq m$, where $m$ and $J$ are the total mass and angular momentum of the spacetime.

gr-qc↗
arXiv2005

Angular momentum-mass inequality for axisymmetric black holes

The inequality $\sqrt{J}\leq m$ is proved for vacuum, asymptotically flat, maximal and axisymmetric data close to extreme Kerr data. The physical significance of this inequality and its relation to the standard picture of the gravitational collapse are discussed.

gr-qc↗
arXiv2005

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.

gr-qc↗
arXiv2005

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

Stationary, axisymmetric, vacuum, solutions of Einstein's equations are obtained as critical points of the total mass among all axisymmetric 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 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.

gr-qc↗
arXiv2005

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.

gr-qc↗
arXiv2004

On the existence of initial data containing isolated black holes

We present a general construction of initial data for Einstein's equations containing an arbitrary number of black holes, each of which is instantaneously in equilibrium. Each black hole is taken to be a marginally trapped surface and plays the role of the inner boundary of the Cauchy surface. The black hole is taken to be instantaneously isolated if its outgoing null rays are shear-free. Starting from the choice of a conformal metric and the freely specifiable part of the extrinsic curvature in the bulk, we give a prescription for choosing the shape of the inner boundaries and the boundary conditions that must be imposed there. We show rigorously that with these choices, the resulting non-linear elliptic system always admits solutions.

gr-qc↗
arXiv2004

A new geometric invariant on initial data for Einstein equations

For a given asymptotically flat initial data set for Einstein equations a new geometric invariant is constructed. This invariant measure the departure of the data set from the stationary regime, it vanishes if and only if the data is stationary. In vacuum, it can be interpreted as a measure of the total amount of radiation contained in the data.

gr-qc↗