80 research outputs found
A comment on Liu and Yau's positive quasi-local mass
Liu and Yau (Phys.Rev.Lett. 90, 231102, 2003) propose a definition of quasi-local mass for any space-like, topological 2-sphere with positive Gauss curvature (and subject to a second, convexity, condition). They are able to show it is positive using a result of Shi and Tam (J.Diff.Geom. 62, 79, 2002). However, as we show here, their definition can give a strictly positive mass for a sphere in flat space
Flat foliations of spherically symmetric geometries
We examine the solution of the constraints in spherically symmetric general relativity when spacetime has a flat spatial hypersurface. We demonstrate explicitly that given one flat slice, a foliation by flat slices can be consistently evolved. We show that when the sources are finite these slices do not admit singularities and we provide an explicit bound on the maximum value assumed by the extrinsic curvature. If the dominant energy condition is satisfied, the projection of the extrinsic curvature orthogonal to the radial direction possesses a definite sign. We provide both necessary and sufficient conditions for the formation of apparent horizons in this gauge which are qualitatively identical to those established earlier for extrinsic time foliations of spacetime, Phys. Rev. D56 7658, 7666 (1997) which suggests that these conditions possess a gauge invariant validity
Bounds on 2m/R for static spherical objects
It is well known that a spherically symmetric constant density static star, modeled as a perfect fluid, possesses a bound on its mass m by its radial size R given by 2m/R \le 8/9 and that this bound continues to hold when the energy density decreases monotonically. The existence of such a bound is intriguing because it occurs well before the appearance of an apparent horizon at m = R/2. However, the assumptions made are extremely restrictive. They do not hold in a humble soap bubble and they certainly do not approximate any known topologically stable field configuration. We show that the 8/9 bound is robust by relaxing these assumptions. If the density is monotonically decreasing and the tangential stress is less than or equal to the radial stress we show that the 8/9 bound continues to hold through the entire bulk if m is replaced by the quasi-local mass. If the tangential stress exceeds the radial stress and/or the density is not monotonic we cannot recover the 8/9 bound. However, we can show that 2m/R remains strictly bounded away from unity by constructing an explicit upper bound which depends only on the ratio of the stresses and the variation of the density
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
The physical gravitational degrees of freedom
When constructing general relativity (GR), Einstein required 4D general
covariance. In contrast, we derive GR (in the compact, without boundary case)
as a theory of evolving 3-dimensional conformal Riemannian geometries obtained
by imposing two general principles: 1) time is derived from change; 2) motion
and size are relative. We write down an explicit action based on them. We
obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation but
also all the equations used in York's conformal technique for solving the
initial-value problem. This shows that the independent gravitational degrees of
freedom obtained by York do not arise from a gauge fixing but from hitherto
unrecognized fundamental symmetry principles. They can therefore be identified
as the long-sought Hamiltonian physical gravitational degrees of freedom.Comment: Replaced with published version (minor changes and added references
Existence and uniqueness of Bowen-York Trumpets
We prove the existence of initial data sets which possess an asymptotically
flat and an asymptotically cylindrical end. Such geometries are known as
trumpets in the community of numerical relativists.Comment: This corresponds to the published version in Class. Quantum Grav. 28
(2011) 24500
The Link between General Relativity and Shape Dynamics
We show that one can construct two equivalent gauge theories from a linking
theory and give a general construction principle for linking theories which we
use to construct a linking theory that proves the equivalence of General
Relativity and Shape Dynamics, a theory with fixed foliation but spatial
conformal invariance. This streamlines the rather complicated construction of
this equivalence performed previously. We use this streamlined argument to
extend the result to General Relativity with asymptotically flat boundary
conditions. The improved understanding of linking theories naturally leads to
the Lagrangian formulation of Shape Dynamics, which allows us to partially
relate the degrees of freedom.Comment: 19 pages, LaTeX, no figure
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