Kayll Lake

Physical acceptability of isolated, static, spherically symmetric, perfect fluid solutions of Einstein's equations

Abstract We ask the following question: Of the exact solutions to Einsteins equations extant in the literature, how many could represent the field associated with an isolated static spherically symmetric perfect fluid source? The candidate solutions were subjected to the following elementary tests: (i) isotropy of the pressure, (ii) regularity at the origin, (iii) positive definiteness of the energy density and pressure at the origin, (iv) vanishing of the pressure at some finite radius, (v) monotonic decrease of the energy density and pressure with increasing radius, and (vi) subluminal sound speed. A total of 127 candidate solutions were found. Only 16 of these passed all the tests. Of these 16, only 9 have a sound speed which monotonically decreases with radius. The analysis was facilitated by use of the computer algebra system GRTensorII.

All static spherically symmetric perfect fluid solutions of Einstein's equations

An algorithm based on the choice of a single monotone function (subject to boundary conditions) is presented which generates all regular static spherically symmetric perfect-fluid solutions of Einsteins equations. For physically relevant solutions the generating functions must be restricted by nontrivial integral-differential inequalities. Nonetheless, the algorithm is demonstrated here by the construction of an infinite number of previously unknown physically interesting exact solutions.

Junctions and thin shells in general relativity using computer algebra: I. The Darmois - Israel formalism

We present the GRjunction computer algebra program which allows the study of non-null boundary surfaces and thin shells in general relativity. Implementing the Darmois - Israel thin-shell formalism requires a careful selection of definitions and algorithms to ensure that results are generated in a straightforward way. We have used the package to correctly reproduce a wide variety of examples from the literature. In this paper GRjunction is used to perform two new calculations: joining two Kerr solutions with differing masses and angular momenta along a thin shell in the slow rotation limit, and the calculation of the stress - energy of a Curzon wormhole. The Curzon wormhole has the interesting property that shells located at radius R < 2m have regions which satisfy the weak energy condition.

Stability of transparent spherically symmetric thin shells and wormholes

The stability of transparent spherically symmetric thin shells (and wormholes) to linearized spherically symmetric perturbations about static equilibrium is examined. This work generalizes and systematizes previous studies and explores the consequences of including the cosmological constant. The approach shows how the existence (or not) of a domain wall dominates the landscape of possible equilibrium configurations.

Bending of light and the cosmological constant

The bending of light in Kottler space (the Schwarzschild vacuum with a cosmological constant) is examined. Unlike the advance of the perihelion, the cosmological constant produces no change in the bending of light. In this Brief Report we examine the conditions under which this statement holds.

Gravitational collapse of dust with a cosmological constant

The recent analysis of Markovic and Shapiro on the effect of a cosmological constant on the evolution of a spherically symmetric homogeneous dust ball is extended to include the inhomogeneous and degenerate cases. The histories are shown by way of effective potential and Penrose-Carter diagrams.

Sudden future singularities in FLRW cosmologies

The standard energy conditions of classical general relativity are applied to FLRW cosmologies containing sudden future singularities. Here we show, in a model-independent way, that although such cosmologies can satisfy the null, weak and strong energy conditions, they always fail to satisfy the dominant energy condition. They require a divergent spacelike energy flux in all but the comoving frame.

More on McVittie's Legacy: A Schwarzschild - de Sitter black and white hole embedded in an asymptotically

Recently Kaloper, Kleban and Martin reexamined the McVittie solution and argued, contrary to a very widely held belief, that the solution contains a black hole in an expanding universe. Here we corroborate their main conclusion but go on to examine, in some detail, a specific solution that asymptotes to the

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CDM cosmology

CDM cosmology. We show that part of the boundary of the solution contains the inner bifurcation two - sphere of the Schwarzschild - de Sitter spacetime and so both the black and white hole horizons together form a partial boundary of this McVittie solution. We go on to show that the null and weak energy conditions are satisfied and that the dominant energy condition is satisfied almost everywhere in the solution. The solution is understood here by way of a systematic construction of a conformal diagram based on detailed numerical integrations of the null geodesic equations. We find that the McVittie solution admits a degenerate limit in which the bifurcation two - sphere disappears. For solutions with zero cosmological constant, we find no evidence for the development of a weak null singularity. Rather, we find that in this case there is either a black hole to the future of an initial singularity or a white hole to its past.