arXiv subjects
Publications and source records attributed to Niall Ó Murchadha.
We give a detailed description of the constant mean curvature foliations in the Schwarzschild solution; show that the lapse collapses exponentially, and compute the exponent.
We give a derivation of general relativity and the gauge principle that is novel in presupposing neither spacetime nor the relativity principle. We consider a class of actions defined on superspace with two key properties. The first is 3-coordinate invariance. This is the only postulated symmetry, and it leads to the standard momentum constraint. The second property is that the Lagrangian is constructed from a `local' square root of an expression quadratic in the velocities, `local' because it is taken before integration over 3-space. It gives rise to quadratic constraints that do not correspond to any symmetry and are not, in general, propagated by the Euler-Lagrange equations. Therefore these actions are internally inconsistent. Only one action of this form is well behaved: the Baierlein-Sharp-Wheeler reparametrisation-invariant action for GR. From this viewpoint, spacetime symmetry is emergent. It appears as a `hidden' symmetry in the (underdetermined) solutions of the evolution equations, without being manifestly coded into the action itself. In addition, propagation of the constraints acts as a striking selection mechanism beyond pure gravity. If a scalar field is included in the configuration space, it must have the same characteristic speed as gravity. Thus Einstein causality emerges. Finally, self-consistency requires that any 3-vector field must satisfy Einstein causality, the equivalence principle and, in addition, the Gauss constraint. Therefore we recover the standard (massless) Maxwell equations.
This article investigates the interaction of a spherically symmetric massless scalar field with a strong gravitational field. It focuses on the propagation of waves in regions outside any horizons. The two factors acting on the waves can be identified as a redshift and a backscattering. The influence of backscattering on the intensity of the outgoing radiation is studied and rigorous quantitative upper bounds obtained. These show that the total flux may be decreased if the sources are placed in a region adjoining an apparent horizon. Backscattering can be neglected in the case $2m_0 /R<< 1$, that is when the emitter is located at a distance from a black hole much larger than the Schwarzschild radius. This backscattering may have noticeable astrophysical consequences.
We examine the constraints of spherically symmetric general relativity with one asymptotically flat region, exploiting both the traditional metric variables and variables constructed from the optical scalars. With respect to the latter variables, there exist two linear combinations of the Hamiltonian and momentum constraints which are related by time reversal. We introduce a one-parameter family of linear extrinsic time foliations of spacetime. The values of the parameter yielding globally valid gauges correspond to the vanishing of a timelike vector in the superspace of spherically symmetric geometries. We define a quasi-local mass on spheres of fixed proper radius which we prove is positive when the constraints are satisfied. Underpinning the proof are various local bounds on the configuration variables. We prove that a reasonable definition of the gravitational binding energy is always negative. Finally, we provide a tentative characterization of the configuration space of the theory in terms of closed bounded trajectories on the parameter space of the optical scalars.
We continue our investigation of the configuration space of general relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We show that MS configurations satisfy both the positive quasi-local mass (QLM) theorem and its converse. We derive an analytical expression for the spatial metric in the neighborhood of a generic singularity. The corresponding curvature singularity shows up in the traceless component of the Ricci tensor. We show that if the energy density of matter is monotonically decreasing, the geometry cannot be singular. A supermetric on the configuration space which distinguishes between singular geometries and non-singular ones is constructed explicitly. Global necessary and sufficient criteria for the formation of trapped surfaces and singularities are framed in terms of inequalities which relate appropriate measures of the material energy content on a given support to a measure of its volume. The strength of these inequalities is gauged by exploiting the exactly solvable piece-wise constant density star as a template.
We continue our examination of the constraints in spherically symmetric general relativity begun in I (gr-qc/9411009) and II (gr-qc/9411010). We extend to general configurations with $J\ne 0$ the analysis of II which treated a moment of time symmetry. We exploit the one parameter family of foliations introduced in I which are linear and homogeneous in the extrinsic curvature to characterize apparent horizons and spatial singularities in the initial data. In particular, we demonstrate that these characterizations do not depend sensitively on the foliation.
We investigate the formation of trapped surfaces in asymptotically flat spherical spacetimes, using constant mean curvature slicing.
We investigate the formation of trapped surfaces in cosmological spacetimes, using constant mean curvature slicing. Quantitative criteria for the formation of trapped surfaces demonstrate that cosmological regions enclosed by trapped surfaces may have matter density exceeding significantly the background matter density of the flat and homogeneous cosmological model.
We demonstrate that the Penrose inequality is valid for spherically symmetric geometries even when the horizon is immersed in matter. The matter field need not be at rest. The only restriction is that the source satisfies the weak energy condition outside the horizon. No restrictions are placed on the matter inside the horizon. The proof of the Penrose inequality gives a new necessary condition for the formation of trapped surfaces. This formulation can also be adapted to give a sufficient condition. We show that a modification of the Penrose inequality proposed by Gibbons for charged black holes can be broken in early stages of gravitational collapse. This investigation is based exclusively on the initial data formulation of General Relativity.
The assumption that a solution to the Einstein equations is static (or stationary) very strongly constrains the asymptotic behaviour of the metric. It is shown that one need only impose very weak differentiability and decay conditions {\it a priori} on the metric for the field equations to force the metric to be analytic near infinity and to have the standard Schwarzschildian falloff.
Consider spherically symmetric initial data for a cosmology which, in the large, approximates an open $k = -1 ,Λ= 0$ Friedmann-Lema{\^ı}tre universe. Further assume that the data is chosen so that the trace of the extrinsic curvature is a constant and that the matter field is at rest at this instant of time. One expects that no trapped surfaces appear in the data if no significant clump of excess matter is to be found. This letter confirms this belief by displaying a necessary condition for the existence of trapped surfaces.This necessary condition, simply stated, says that a relatively large amount of excess matter must be concentrated in a small volume for trapped surfaces to appear.
Brill waves are the simplest (non-trivial) solutions to the vacuum constraints of general relativity. They are also rich enough in structure to allow us believe that they capture, at least in part, the generic properties of solutions of the Einstein equations. As such, they deserve the closest attention. This article illustrates this point by showing how Brill waves can be used to investigate the structure of conformal superspace. This article is written as a contribution to the Dieter Brill Festschrift.
The Cosmological Principle states that the universe is both homogeneous and isotropic. This, alone, is not enough to specify the global geometry of the spacetime. If we were able to measure both the Hubble constant and the energy density we could determine whether the universe is open or closed. Unfortunately, while some agreement exists on the value of the Hubble constant, the question of the energy density seems quite intractable. This Letter describes a possible way of avoiding this difficulty and shows that if one could measure the rate at which light-rays emerging from a surface expand, one might well be able to deduce whether the universe is closed.
This article gives necessary and sufficient conditions for the formation of trapped surfaces in spherically symmetric initial data defined on a closed manifold. Such trapped surfaces surround a region in which there occurs an enhancement of matter over the average. The conditions are posed directly in terms of physical variables and show that what one needs is a relatively large amount of excess matter confined to a small volume. The expansion of the universe and an outward flow of matter oppose the formation of trapped surfaces; an inward flow of matter helps. The model can be regarded as a Friedmann-Lema\^ıtre-Walker cosmology with localized spherical inhomogeneities. We show that the total excess mass cannot be too large.
Gravitational instantons, solutions to the euclidean Einstein equations, with topology $R^3 XS^1$ arise naturally in any discussion of finite temperature quantum gravity. This Letter shows that all such instantons (irrespective of their interior behaviour) must have the same asymptotic structure as the Schwarzschild instanton. Using this, it can be shown that if the Ricci tensor of the manifold is non-negative it must be flat. One special case is when the Ricci tensor vanishes; hence one can conclude that there is no nontrivial vacuum gravitational instanton. This result has uses both in quantum and classical gravity. It places a significant restriction on the instabilities of hot flat space. It also can be used to show that any static vacuum Lorentzian Kaluza-Klein solution is flat.