The Local Effects of Cosmological Variations in Physical 'Constants' and Scalar Fields I. Spherically Symmetric Spacetimes

We apply the method of matched asymptotic expansions to analyse whether cosmological variations in physical `constants' and scalar fields are detectable, locally, on the surface of local gravitationally bound systems such as planets and stars, or inside virialised systems like galaxies and clusters. We assume spherical symmetry and derive a sufficient condition for the local time variation of the scalar fields that drive varying constants to track the cosmological one. We calculate a number of specific examples in detail by matching the Schwarzschild spacetime to spherically symmetric inhomogeneous Tolman-Bondi metrics in an intermediate region by rigorously construction matched asymptotic expansions on cosmological and local astronomical scales which overlap in an intermediate domain. We conclude that, independent of the details of the scalar-field theory describing the varying `constant', the condition for cosmological variations to be measured locally is almost always satisfied in physically realistic situations. The proof of this statement provides a rigorous justification for using terrestrial experiments and solar system observations to constrain or detect any cosmological time variations in the traditional `constants' of Nature.Comment: 30 pages, 3 figures; corrected typo

Cosmic Dynamics in the Chameleon Cosmology

We study in this paper chameleon cosmology applied to Friedmann-Robertson-Walker space, which gives rise to the equation of state (EoS) parameter larger than -1 in the past and less than -1 today, satisfying current observations. We also study cosmological constraints on the model using the time evolution of the cosmological redshift of distant sources which directly probes the expansion history of the universe. Due to the evolution of the universe's expansion rate, the model independent Cosmological Redshift Drift (CRD)test is expected to experience a small, systematic drift as a function of time. The model is supported by the observational data obtained from the test.Comment: 16 pages, 9 figure

Arbitrary Dimensional Schwarzschild-FRW Black Holes

The metric of arbitrary dimensional Schwarzschild black hole in the background of Friedman-Robertson-Walker universe is presented in the cosmic coordinates system. In particular, the arbitrary dimensional Schwarzschild-de Sitter metric is rewritten in the Schwarzschild coordinates system and basing on which the even more generalized higher dimensional Schwarzschild-de Sitter metric with another extra dimensions is found. The generalized solution shows that the cosmological constant may roots in the extra dimensions of space.Comment: 10 page

Cosmological perturbations on local systems

We study the effect of cosmological expansion on orbits--galactic, planetary, or atomic--subject to an inverse-square force law. We obtain the laws of motion for gravitational or electrical interactions from general relativity--in particular, we find the gravitational field of a mass distribution in an expanding universe by applying perturbation theory to the Robertson-Walker metric. Cosmological expansion induces an ($\ddot a/a) \vec r$ force where $a(t)$ is the cosmological scale factor. In a locally Newtonian framework, we show that the $(\ddot a/a) \vec r$ term represents the effect of a continuous distribution of cosmological material in Hubble flow, and that the total force on an object, due to the cosmological material plus the matter perturbation, can be represented as the negative gradient of a gravitational potential whose source is the material actually present. We also consider the effect on local dynamics of the cosmological constant. We calculate the perihelion precession of elliptical orbits due to the cosmological constant induced force, and work out a generalized virial relation applicable to gravitationally bound clusters.Comment: 10 page

Explicit Fermi Coordinates and Tidal Dynamics in de Sitter and Goedel Spacetimes

Fermi coordinates are directly constructed in de Sitter and Goedel spacetimes and the corresponding exact coordinate transformations are given explicitly. The quasi-inertial Fermi coordinates are then employed to discuss the dynamics of a free test particle in these spacetimes and the results are compared to the corresponding generalized Jacobi equations that contain only the lowest-order tidal terms. The domain of validity of the generalized Jacobi equation is thus examined in these cases. Furthermore, the difficulty of constructing explicit Fermi coordinates in black-hole spacetimes is demonstrated.Comment: 23 pages, 3 figures; v2: expanded version (27 pages, 3 figures

Cosmological Multi-Black Hole Solutions

We present simple, analytic solutions to the Einstein-Maxwell equation, which describe an arbitrary number of charged black holes in a spacetime with positive cosmological constant $\Lambda$. In the limit $\Lambda=0$, these solutions reduce to the well known Majumdar-Papapetrou (MP) solutions. Like the MP solutions, each black hole in a $\Lambda >0$ solution has charge $Q$ equal to its mass $M$, up to a possible overall sign. Unlike the $\Lambda = 0$ limit, however, solutions with $\Lambda >0$ are highly dynamical. The black holes move with respect to one another, following natural trajectories in the background deSitter spacetime. Black holes moving apart eventually go out of causal contact. Black holes on approaching trajectories ultimately merge. To our knowledge, these solutions give the first analytic description of coalescing black holes. Likewise, the thermodynamics of the $\Lambda >0$ solutions is quite interesting. Taken individually, a $|Q|=M$ black hole is in thermal equilibrium with the background deSitter Hawking radiation. With more than one black hole, because the solutions are not static, no global equilibrium temperature can be defined. In appropriate limits, however, when the black holes are either close together or far apart, approximate equilibrium states are established.Comment: 15 pages (phyzzx), UMHEP-380 (minor referencing error corrected

Isotropy, shear, symmetry and exact solutions for relativistic fluid spheres

The symmetry method is used to derive solutions of Einstein's equations for fluid spheres using an isotropic metric and a velocity four vector that is non-comoving. Initially the Lie, classical approach is used to review and provide a connecting framework for many comoving and so shear free solutions. This provides the basis for the derivation of the classical point symmetries for the more general and mathematicaly less tractable description of Einstein's equations in the non-comoving frame. Although the range of symmetries is restrictive, existing and new symmetry solutions with non-zero shear are derived. The range is then extended using the non-classical direct symmetry approach of Clarkson and Kruskal and so additional new solutions with non-zero shear are also presented. The kinematics and pressure, energy density, mass function of these solutions are determined.Comment: To appear in Classical and Quantum Gravit

The influence of the cosmological expansion on local systems

Following renewed interest, the problem of whether the cosmological expansion affects the dynamics of local systems is reconsidered. The cosmological correction to the equations of motion in the locally inertial Fermi normal frame (the relevant frame for astronomical observations) is computed. The evolution equations for the cosmological perturbation of the two--body problem are solved in this frame. The effect on the orbit is insignificant as are the effects on the galactic and galactic--cluster scales.Comment: To appear in the Astrophysical Journal, Late

Slowly, rotating non-stationary, fluid solutions of Einstein's equations and their match to Kerr empty space-time

A general class of solutions of Einstein's equation for a slowly rotating fluid source, with supporting internal pressure, is matched using Lichnerowicz junction conditions, to the Kerr metric up to and including first order terms in angular speed parameter. It is shown that the match applies to any previously known non-rotating fluid source made to rotate slowly for which a zero pressure boundary surface exists. The method is applied to the dust source of Robertson-Walker and in outline to an interior solution due to McVittie describing gravitational collapse. The applicability of the method to additional examples is transparent. The differential angular velocity of the rotating systems is determined and the induced rotation of local inertial frame is exhibited