744 research outputs found

### Shear-free Null Quasi-Spherical Spacetimes

The residual gauge freedom within the null quasi-spherical coordinate
condition is studied, for spacetimes admitting an expanding, shear-free null
foliation. The freedom consists of a boost and rotation at each coordinate
sphere, corresponding to a specification of inertial frame at each sphere.
Explicit formulae involving arbitrary functions of two variables are obtained
for the accelerated Minkowski, Schwarzschild, and Robinson-Trautman spacetimes.
These examples will be useful as test metrics in numerical relativity.Comment: 20 pages, revte

### On a Localized Riemannian Penrose Inequality

Consider a compact, orientable, three dimensional Riemannian manifold with
boundary with nonnegative scalar curvature. Suppose its boundary is the
disjoint union of two pieces: the horizon boundary and the outer boundary,
where the horizon boundary consists of the unique closed minimal surfaces in
the manifold and the outer boundary is metrically a round sphere. We obtain an
inequality relating the area of the horizon boundary to the area and the total
mean curvature of the outer boundary. Such a manifold may be thought as a
region, surrounding the outermost apparent horizons of black holes, in a
time-symmetric slice of a space-time in the context of general relativity. The
inequality we establish has close ties with the Riemannian Penrose Inequality,
proved by Huisken and Ilmanen, and by Bray.Comment: 16 page

### Initial boundary value problems for Einstein's field equations and geometric uniqueness

While there exist now formulations of initial boundary value problems for
Einstein's field equations which are well posed and preserve constraints and
gauge conditions, the question of geometric uniqueness remains unresolved. For
two different approaches we discuss how this difficulty arises under general
assumptions. So far it is not known whether it can be overcome without imposing
conditions on the geometry of the boundary. We point out a natural and
important class of initial boundary value problems which may offer
possibilities to arrive at a fully covariant formulation.Comment: 19 page

### Einstein equations in the null quasi-spherical gauge

The structure of the full Einstein equations in a coordinate gauge based on
expanding null hypersurfaces foliated by metric 2-spheres is explored. The
simple form of the resulting equations has many applications -- in the present
paper we describe the structure of timelike boundary conditions; the matching
problem across null hypersurfaces; and the propagation of gravitational shocks.Comment: 12 pages, LaTeX (revtex, amssymb), revision 18 pages, contains
expanded discussion and explanations, updated references, to appear in CQ

### Static solutions from the point of view of comparison geometry

We analyze (the harmonic map representation of) static solutions of the
Einstein Equations in dimension three from the point of view of comparison
geometry. We find simple monotonic quantities capturing sharply the influence
of the Lapse function on the focussing of geodesics. This allows, in
particular, a sharp estimation of the Laplacian of the distance function to a
given (hyper)-surface. We apply the technique to asymptotically flat solutions
with regular and connected horizons and, after a detailed analysis of the
distance function to the horizon, we recover the Penrose inequality and the
uniqueness of the Schwarzschild solution. The proof of this last result does
not require proving conformal flatness at any intermediate step.Comment: 41 page

### On the Bartnik extension problem for the static vacuum Einstein equations

We develop a framework for understanding the existence of asymptotically flat
solutions to the static vacuum Einstein equations with prescribed boundary data
consisting of the induced metric and mean curvature on a 2-sphere. A partial
existence result is obtained, giving a partial resolution of a conjecture of
Bartnik on such static vacuum extensions. The existence and uniqueness of such
extensions is closely related to Bartnik's definition of quasi-local mass.Comment: 33 pages, 1 figure. Minor revision of v2. Final version, to appear in
Class. Quantum Gravit

### Gluing Initial Data Sets for General Relativity

We establish an optimal gluing construction for general relativistic initial
data sets. The construction is optimal in two distinct ways. First, it applies
to generic initial data sets and the required (generically satisfied)
hypotheses are geometrically and physically natural. Secondly, the construction
is completely local in the sense that the initial data is left unaltered on the
complement of arbitrarily small neighborhoods of the points about which the
gluing takes place. Using this construction we establish the existence of
cosmological, maximal globally hyperbolic, vacuum space-times with no constant
mean curvature spacelike Cauchy surfaces.Comment: Final published version - PRL, 4 page

### 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.Comment: 11 pages, 10 figures. To appear in PR

### Trapped Surfaces in Vacuum Spacetimes

An earlier construction by the authors of sequences of globally regular,
asymptotically flat initial data for the Einstein vacuum equations containing
trapped surfaces for large values of the parameter is extended, from the time
symmetric case considered previously, to the case of maximal slices. The
resulting theorem shows rigorously that there exists a large class of initial
configurations for non-time symmetric pure gravitational waves satisfying the
assumptions of the Penrose singularity theorem and so must have a singularity
to the future.Comment: 14 page

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