In an expanding universe, what doesn't expand?

The expansion of the universe is often viewed as a uniform stretching of space that would affect compact objects, atoms and stars, as well as the separation of galaxies. One usually hears that bound systems do not take part in the general expansion, but a much more subtle question is whether bound systems expand partially. In this paper, a very definitive answer is given for a very simple system: a classical "atom" bound by electrical attraction. With a mathemical description appropriate for undergraduate physics majors, we show that this bound system either completely follows the cosmological expansion, or -- after initial transients -- completely ignores it. This "all or nothing" behavior can be understood with techniques of junior-level mechanics. Lastly, the simple description is shown to be a justifiable approximation of the relativistically correct formulation of the problem.Comment: 8 pages, 9 eps figure

Exact Cosmological Solutions of Gravitational Theories

A global picture is drawn tying together most exact cosmological solutions of gravitational theories in four or more spacetime dimensions.Comment: 11 latex article style page

Quasistationary binary inspiral. I. Einstein equations for the two Killing vector spacetime

The geometry of two infinitely long lines of mass moving in a fixed circular orbit is considered as a toy model for the inspiral of a binary system of compact objects due to gravitational radiation. The two Killing fields in the toy model are used, according to a formalism introduced by Geroch, to describe the geometry entirely in terms of a set of tensor fields on the two-manifold of Killing vector orbits. Geroch's derivation of the Einstein equations in this formalism is streamlined and generalized. The explicit Einstein equations for the toy model spacetime are derived in terms of the degrees of freedom which remain after a particular choice of gauge.Comment: 37 pages, REVTeX, one PostScript Figure included with epsfig; minor formatting changes and copyright notice added for journal publicatio

Event Horizons in Numerical Relativity I: Methods and Tests

This is the first paper in a series on event horizons in numerical relativity. In this paper we present methods for obtaining the location of an event horizon in a numerically generated spacetime. The location of an event horizon is determined based on two key ideas: (1) integrating backward in time, and (2) integrating the whole horizon surface. The accuracy and efficiency of the methods are examined with various sample spacetimes, including both analytic (Schwarzschild and Kerr) and numerically generated black holes. The numerically evolved spacetimes contain highly distorted black holes, rotating black holes, and colliding black holes. In all cases studied, our methods can find event horizons to within a very small fraction of a grid zone.Comment: 22 pages, LaTeX with RevTeX 3.0 macros, 20 uuencoded gz-compressed postscript figures. Also available at http://jean-luc.ncsa.uiuc.edu/Papers/ Submitted to Physical Review

Gravitational energy

Observers at rest in a stationary spacetime flat at infinity can measure small amounts of rest-mass+internal energies+kinetic energies+pressure energy in a small volume of fluid attached to a local inertial frame. The sum of these small amounts is the total "matter energy" for those observers. The total mass-energy minus the matter energy is the binding gravitational energy. Misner, Thorne and Wheeler evaluated the gravitational energy of a spherically symmetric static spacetime. Here we show how to calculate gravitational energy in any static and stationary spacetime for isolated sources with a set of observers at rest. The result of MTW is recovered and we find that electromagnetic and gravitational 3-covariant energy densities in conformastatic spacetimes are of opposite signs. Various examples suggest that gravitational energy is negative in spacetimes with special symmetries or when the energy-momentum tensor satisfies usual energy conditions.Comment: 12 pages. Accepted for publication in Class. Quantum Gra

Controlling Chaos through Compactification in Cosmological Models with a Collapsing Phase

We consider the effect of compactification of extra dimensions on the onset of classical chaotic "Mixmaster" behavior during cosmic contraction. Assuming a universe that is well-approximated as a four-dimensional Friedmann-Robertson--Walker model (with negligible Kaluza-Klein excitations) when the contraction phase begins, we identify compactifications that allow a smooth contraction and delay the onset of chaos until arbitrarily close the big crunch. These compactifications are defined by the de Rham cohomology (Betti numbers) and Killing vectors of the compactification manifold. We find compactifications that control chaos in vacuum Einstein gravity, as well as in string theories with N = 1 supersymmetry and M-theory. In models where chaos is controlled in this way, the universe can remain homogeneous and flat until it enters the quantum gravity regime. At this point, the classical equations leading to chaotic behavior can no longer be trusted, and quantum effects may allow a smooth approach to the big crunch and transition into a subsequent expanding phase. Our results may be useful for constructing cosmological models with contracting phases, such as the ekpyrotic/cyclic and pre-big bang models.Comment: 1 figure. v2/v3: minor typos correcte

Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities

We analyze the signal processing required for the optimal detection of a stochastic background of gravitational radiation using laser interferometric detectors. Starting with basic assumptions about the statistical properties of a stochastic gravity-wave background, we derive expressions for the optimal filter function and signal-to-noise ratio for the cross-correlation of the outputs of two gravity-wave detectors. Sensitivity levels required for detection are then calculated. Issues related to: (i) calculating the signal-to-noise ratio for arbitrarily large stochastic backgrounds, (ii) performing the data analysis in the presence of nonstationary detector noise, (iii) combining data from multiple detector pairs to increase the sensitivity of a stochastic background search, (iv) correlating the outputs of 4 or more detectors, and (v) allowing for the possibility of correlated noise in the outputs of two detectors are discussed. We briefly describe a computer simulation which mimics the generation and detection of a simulated stochastic gravity-wave signal in the presence of simulated detector noise. Numerous graphs and tables of numerical data for the five major interferometers (LIGO-WA, LIGO-LA, VIRGO, GEO-600, and TAMA-300) are also given. The treatment given in this paper should be accessible to both theorists involved in data analysis and experimentalists involved in detector design and data acquisition.Comment: 81 pages, 30 postscript figures, REVTE

Exact Hypersurface-Homogeneous Solutions in Cosmology and Astrophysics

A framework is introduced which explains the existence and similarities of most exact solutions of the Einstein equations with a wide range of sources for the class of hypersurface-homogeneous spacetimes which admit a Hamiltonian formulation. This class includes the spatially homogeneous cosmological models and the astrophysically interesting static spherically symmetric models as well as the stationary cylindrically symmetric models. The framework involves methods for finding and exploiting hidden symmetries and invariant submanifolds of the Hamiltonian formulation of the field equations. It unifies, simplifies and extends most known work on hypersurface-homogeneous exact solutions. It is shown that the same framework is also relevant to gravitational theories with a similar structure, like Brans-Dicke or higher-dimensional theories.Comment: 41 pages, REVTEX/LaTeX 2.09 file (don't use LaTeX2e !!!) Accepted for publication in Phys. Rev.

Minisuperspace Quantization of "Bubbling AdS" and Free Fermion Droplets

We quantize the space of 1/2 BPS configurations of Type IIB SUGRA found by Lin, Lunin and Maldacena (hep-th/0409174), directly in supergravity. We use the Crnkovic-Witten-Zuckerman covariant quantization method to write down the expression for the symplectic structure on this entire space of solutions. We find the symplectic form explicitly around AdS_5 x S^5 and obtain a U(1) Kac-Moody algebra, in precise agreement with the quantization of a system of N free fermions in a harmonic oscillator potential, as expected from AdS/CFT. As a cross check, we also perform the quantization around AdS_5 x S^5 by another method, using the known spectrum of physical perturbations around this background and find precise agreement with our previous calculation.Comment: 22 Pages + 2 Appendices, JHEP3; v3: explanation of factor 2 mismatch added, references reordered, published versio

Canonical Gravity, Diffeomorphisms and Objective Histories

This paper discusses the implementation of diffeomorphism invariance in purely Hamiltonian formulations of General Relativity. We observe that, if a constrained Hamiltonian formulation derives from a manifestly covariant Lagrangian, the diffeomorphism invariance of the Lagrangian results in the following properties of the constrained Hamiltonian theory: the diffeomorphisms are generated by constraints on the phase space so that a) The algebra of the generators reflects the algebra of the diffeomorphism group. b) The Poisson brackets of the basic fields with the generators reflects the space-time transformation properties of these basic fields. This suggests that in a purely Hamiltonian approach the requirement of diffeomorphism invariance should be interpreted to include b) and not just a) as one might naively suppose. Giving up b) amounts to giving up objective histories, even at the classical level. This observation has implications for Loop Quantum Gravity which are spelled out in a companion paper. We also describe an analogy between canonical gravity and Relativistic particle dynamics to illustrate our main point.Comment: Latex 16 Pages, no figures, revised in the light of referees' comments, accepted for publication in Classical and Quantum Gravit