Sergio Dain

Geometric inequalities for axially symmetric black holes

A geometric inequality in general relativity relates quantities that have both a physical interpretation and a geometrical definition. It is well known that the parameters that characterize the Kerr?Newman black hole satisfy several important geometric inequalities. Remarkably enough, some of these inequalities also hold for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse; they are closely related with the cosmic censorship conjecture. Axially symmetric black holes are the natural candidates to study these inequalities because the quasi-local angular momentum is well defined for them. We review recent results in this subject and we also describe the main ideas behind the proofs. Finally, a list of relevant open problems is presented.

Generalized Korn's inequality and conformal Killing vectors

Korns 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.

Trapped surfaces as boundaries for the constraint equations

Trapped surfaces are studied as inner boundary for the Einstein vacuum constraint equations. The trapped surface condition can be written as a nonlinear boundary condition for these equations. Under appropriate assumptions, we prove the existence and uniqueness of solutions in the exterior region for this boundary value problem. We also discuss the relevance of this result for the study of black-hole collisions.

Initial data for stationary spacetimes near spacelike infinity

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 result allow us to fill a gap in the proof found in the literature of the statement that all asymptotically flat, vacuum stationary space-times admit an analytic compactification at null infinity. Stationary initial data are physical important and highly non-trivial examples of a large class of data with similar regularity properties at space-like infinity, namely, initial data for which the metric and the extrinsic curvature have asymptotic expansion in terms of powers of a radial coordinate. We isolate the property of the stationary data which is responsible for this kind of expansion.

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 ofa=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 counteraligned 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 hangup effect and remnant spin. The coordinate radii of the individual apparent horizons in these cases are very small and the simulations

Black hole Area-Angular momentum inequality in non-vacuum spacetimes

We show that the area-angular-momentum inequality

Area - Angular-Momentum inequality for axisymmetric black holes

A\ensuremath{\ge}8\ensuremath{\pi}|J|

Area-charge inequality for black holes

holds for axially symmetric closed outermost stably marginally trapped surfaces. These are horizon sections (in particular, apparent horizons) contained in otherwise generic non-necessarily axisymmetric black hole spacetimes, with a non-negative cosmological constant and whose matter content satisfies the dominant energy condition.

Asymptotically Flat Initial Data with Prescribed Regularity at Infinity

We prove the local inequality A≥8π|J|, where A and J are the area and angular momentum of any axially symmetric closed stable minimal surface in an axially symmetric maximal initial data. From this theorem it is proved that the inequality is satisfied for any surface on complete asymptotically flat maximal axisymmetric data. In particular it holds for marginal or event horizons of black holes. Hence, we prove the validity of this inequality for all dynamical (not necessarily near equilibrium) axially symmetric black holes.

Horizon area―angular momentum inequality for a class of axially symmetric black holes

The inequality between area and charge A ⩾ 4πQ2 for dynamical black holes is proved. No symmetry assumption is made and charged matter fields are included. Extensions of this inequality are also proved for regions in the spacetime which are not necessarily black hole boundaries.