We ask the following question: Of the exact solutions to Einstein's 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.2. Since the solutions must integrate from a regular origin, we check the regularity of the scalars polynomial in the Riemann tensor [6] [7] [8]. Appendix A contains the conditions for regularity.3. We require that both the pressure (p) and energy density (ρ) be positive definite at the origin. Appendix B summarizes the standard equilibrium conditions. 4. To be isolated we require that the pressure reduce to zero at some finite boundary radius r b > 0.5. We require that both the pressure and energy density be monotonically decreasing to the boundary.6. We require a subluminal sound speed (v 2 s = dp dρ < 1). Whereas at very high densities the adiabatic sound speed may not equal the actual propagation speed of the signal [9], we do not distinguish between these cases in this paper.It was found that all solutions that were studied which satisfied criteria one through three also satisfied the dominant energy condition (p/ρ < 1 for all r < r b here).

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 Einstein's 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.PACS numbers: 04.20.Cv, 04.20.Jb, 04.40.Dg Exact solutions of Einstein's field equations provide a route to the physical understanding (and discovery) of relativistic phenomena, a convenient basis from which perturbation methods can proceed and a check on numerical approximations. Here we look at static spherically symmetric perfect fluid solutions. Unfortunately, even for this simple type, very few solutions are in fact known, and of these few pass even elementary tests of physical relevance [1]. In this paper, 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 Einstein's equations. We are interested only in physically relevant solutions here and so the algorithm must be supplemented by physical considerations [2]. These additional conditions limit the generating functions allowed by way of non-trivial integral-differential inequalities. The details of how to choose physically relevant generating functions (beyond trial and error) are, at present, not known. Nonetheless, the robustness of the algorithm is demonstrated here by the construction of an infinite number of previously unknown physically interesting exact solutions.To set the notation, consider a spherically symmetric spacetime M [3]where dΩ 2 is the metric of a unit sphere (dθ 2 + sin 2 (θ)dφ 2 ) and R = R(x 1 , x 2 ) where the coordinates on the Lorentzian two space Σ are labelled as x 1 and x 2 . Consider a flow (a congruence of unit timelike vectors u α ) tangent to an open region of Σ and write n α as the normal to u α in the tangent space of Σ. Both u α and n α are uniquely determined. We suppose that (1) is generated by a fluid subject to the condition G β α u α n β = 0 where G β α is the Einstein tensor (see [4]).In the static case it follows that the flow is shear free and thatis a necessary and sufficient condition for (1) to represent a perfect fluid [5].First consider M in "curvature" coordinates,Writing out (2) [6] we obtain an expression involving Φ(r) and m(r) with derivatives to order two in Φ(r) and to order one in m(r). Viewing (2) as a differential equation in Φ(r), given m(r), we obtain a Riccati equation in the first derivative of Φ(r). However, viewing (2) as a differential equation in m(r), given Φ(r), we obtain a linear equation of first order [7]. As a consequence, we have the following algorithm for constructing all possible spherically symmetric perfect fluid solutions of Einstein's equations:Given ...

No abstract

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 ΛCDM 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.

We present the GRjunction package which allows boundary surfaces and thin-shells in general relativity to be studied with a computer algebra system. Implementing the Darmois-Israel thin shell formalism requires a careful selection of definitions and algorithms to ensure that results are generated in a straight-forward way. We have used the package to correctly reproduce a wide variety of examples from the literature. We present several of these verifications as a means of demonstrating the packages capabilities. We then use GRjunction to perform a new calculation -joining two Kerr solutions with differing masses and angular momenta along a thin shell in the slow rotation limit.