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.Comment: 11 pages. Changes in presentation and typos correction

Strength asymmetry increases gait asymmetry and variability in older women.

Purpose—The aim of the research was to determine how knee extensor strength asymmetry influences gait asymmetry and variability since these gait parameters have been related to mobility and falls in older adults. Methods—Strength of the knee extensors was measured in 24 older women (65 – 80 yr). Subjects were separated into symmetrical strength (SS, n = 13) and asymmetrical strength (SA, n = 11) groups using an asymmetry cutoff of 20%. Subjects walked at a standard speed of 0.8 m s−1 and at maximal speed on an instrumented treadmill while kinetic and spatiotemporal gait variables were measured. Gait and strength asymmetry were calculated as the percent difference between legs and gait variability as the coefficient of variation over twenty sequential steps. Results—SA had greater strength asymmetry (27.4 ± 5.5%) than SS (11.7 ± 5.4%, P \u3c 0.001). Averaged across speeds, SA had greater single (7.1% vs. 2.5%) and double-limb support time asymmetry (7.0 vs. 4.3%) than SS and greater single-limb support time variability (9.7% vs. 6.6%, all P \u3c 0.05). Group × speed interactions occurred for weight acceptance force variability (P = 0.02) and weight acceptance force asymmetry (P = 0.017) with greater variability at the maximal speed in SA (5.0 ± 2.4% vs. 3.7 ± 1.2%) and greater asymmetry at the maximal speed in SA (6.4 ± 5.3% vs. 2.5 ± 2.3%). Conclusion—Gait variability and asymmetry are greater in older women with strength asymmetry and increase when they walk near their maximal capacities. The maintenance of strength symmetry, or development of symmetry through unilateral exercise, may be beneficial in reducing gait asymmetry, gait variability, and fall risk in older adults

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.Comment: 21 page

Area-charge inequality for black holes

The inequality between area and charge $A\geq 4\pi Q^2$ 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.Comment: 21 pages, 2 figure

Bounds on area and charge for marginally trapped surfaces with cosmological constant

We sharpen the known inequalities $A \Lambda \le 4\pi (1-g)$ and $A\ge 4\pi Q^2$ between the area $A$ and the electric charge $Q$ of a stable marginally outer trapped surface (MOTS) of genus g in the presence of a cosmological constant $\Lambda$. In particular, instead of requiring stability we include the principal eigenvalue $\lambda$ of the stability operator. For $\Lambda^{*} = \Lambda + \lambda > 0$ we obtain a lower and an upper bound for $\Lambda^{*} A$ in terms of $\Lambda^{*} Q^2$ as well as the upper bound $Q \le 1/(2\sqrt{\Lambda^{*}})$ for the charge, which reduces to $Q \le 1/(2\sqrt{\Lambda})$ in the stable case $\lambda \ge 0$. For $\Lambda^{*} < 0$ there remains only a lower bound on $A$. In the spherically symmetric, static, stable case one of the area inequalities is saturated iff the surface gravity vanishes. We also discuss implications of our inequalities for "jumps" and mergers of charged MOTS.Comment: minor corrections to previous version and to published versio

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 $(t,\phi)$ 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.Comment: 21 page

The interior of axisymmetric and stationary black holes: Numerical and analytical studies

We investigate the interior hyperbolic region of axisymmetric and stationary black holes surrounded by a matter distribution. First, we treat the corresponding initial value problem of the hyperbolic Einstein equations numerically in terms of a single-domain fully pseudo-spectral scheme. Thereafter, a rigorous mathematical approach is given, in which soliton methods are utilized to derive an explicit relation between the event horizon and an inner Cauchy horizon. This horizon arises as the boundary of the future domain of dependence of the event horizon. Our numerical studies provide strong evidence for the validity of the universal relation \Ap\Am = (8\pi J)^2 where \Ap and \Am are the areas of event and inner Cauchy horizon respectively, and $J$ denotes the angular momentum. With our analytical considerations we are able to prove this relation rigorously.Comment: Proceedings of the Spanish Relativity Meeting ERE 2010, 10 pages, 5 figure

Photon rockets and the Robinson-Trautman geometries

We point out the relation between the photon rocket spacetimes and the Robinson Trautman geometries. This allows a discussion of the issues related to the distinction between the gravitational and matter energy radiation that appear in these metrics in a more geometrical way, taking full advantage of their asymptotic properties at null infinity to separate the Weyl and Ricci radiations, and to clearly establish their gravitational energy content. We also give the exact solution for the generalized photon rockets.Comment: 7 pages, no figures, LaTeX2

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.Comment: 11 pages, 2 figures, RevTeX

Binary black hole initial data for numerical general relativity based on post-Newtonian data

With the goal of taking a step toward the construction of astrophysically realistic initial data for numerical simulations of black holes, we for the first time derive a family of fully general relativistic initial data based on post-2-Newtonian expansions of the 3-metric and extrinsic curvature without spin. It is expected that such initial data provide a direct connection with the early inspiral phase of the binary system. We discuss a straightforward numerical implementation, which is based on a generalized puncture method. Furthermore, we suggest a method to address some of the inherent ambiguity in mapping post-Newtonian data onto a solution of the general relativistic constraints.Comment: 13 pages, 8 figures, RevTex