Newtonian and Post-Newtonian approximations of the k = 0 Friedmann Robertson Walker Cosmology

In a previous paper we derived a post-Newtonian approximation to cosmology which, in contrast to former Newtonian and post-Newtonian cosmological theories, has a well-posed initial value problem. In this paper, this new post-Newtonian theory is compared with the fully general relativistic theory, in the context of the k = 0 Friedmann Robertson Walker cosmologies. It is found that the post-Newtonian theory reproduces the results of its general relativistic counterpart, whilst the Newtonian theory does not.Comment: 11 pages, Latex, corrected typo

Integer Partitions and Exclusion Statistics

We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic point $p=0$ is a repulsive fixed point for which the limiting distribution has a Gumbel form. For all positive $p$ the distribution is shown to be Gaussian.Comment: 16 pages, 4 .eps figures include

Simple Analytic Models of Gravitational Collapse

Most general relativity textbooks devote considerable space to the simplest example of a black hole containing a singularity, the Schwarzschild geometry. However only a few discuss the dynamical process of gravitational collapse, by which black holes and singularities form. We present here two types of analytic models for this process, which we believe are the simplest available; the first involves collapsing spherical shells of light, analyzed mainly in Eddington-Finkelstein coordinates; the second involves collapsing spheres filled with a perfect fluid, analyzed mainly in Painleve-Gullstrand coordinates. Our main goal is pedagogical simplicity and algebraic completeness, but we also present some results that we believe are new, such as the collapse of a light shell in Kruskal-Szekeres coordinates.Comment: Submitted to American Journal of Physic

Some notes on the Kruskal - Szekeres completion

The Kruskal - Szekeres (KS) completion of the Schwarzschild spacetime is open to Synge's methodological criticism that the KS procedure generates "good" coordinates from "bad". This is addressed here in two ways: First I generate the KS coordinates from Israel coordinates, which are also "good", and then I generate the KS coordinates directly from a streamlined integration of the Einstein equations.Comment: One typo correcte

Post-Newtonian Cosmology

Newtonian Cosmology is commonly used in astrophysical problems, because of its obvious simplicity when compared with general relativity. However it has inherent difficulties, the most obvious of which is the non-existence of a well-posed initial value problem. In this paper we investigate how far these problems are met by using the post-Newtonian approximation in cosmology.Comment: 12 pages, Late

A Radiation Scalar for Numerical Relativity

This letter describes a scalar curvature invariant for general relativity with a certain, distinctive feature. While many such invariants exist, this one vanishes in regions of space-time which can be said unambiguously to contain no gravitational radiation. In more general regions which incontrovertibly support non-trivial radiation fields, it can be used to extract local, coordinate-independent information partially characterizing that radiation. While a clear, physical interpretation is possible only in such radiation zones, a simple algorithm can be given to extend the definition smoothly to generic regions of space-time.Comment: 4 pages, 1 EPS figur

Collisions of Einstein-Conformal Scalar Waves

A large class of solutions of the Einstein-conformal scalar equations in D=2+1 and D=3+1 is identified. They describe the collisions of asymptotic conformal scalar waves and are generated from Einstein-minimally coupled scalar spacetimes via a (generalized) Bekenstein transformation. Particular emphasis is given to the study of the global properties and the singularity structure of the obtained solutions. It is shown, that in the case of the absence of pure gravitational radiation in the initial data, the formation of the final singularity is not only generic, but is even inevitable.Comment: 17 pages, LaTe

No-Go Theorem in Spacetimes with Two Commuting Spacelike Killing Vectors

Four-dimensional Riemannian spacetimes with two commuting spacelike Killing vectors are studied in Einstein's theory of gravity, and found that no outer apparent horizons exist, provided that the dominant energy condition holds.Comment: latex, 1 figure, version published in Gen. Relativ. Grav., 37, 1919-1926 (2005