The thermodynamic structure of Einstein tensor

We analyze the generic structure of Einstein tensor projected onto a 2-D spacelike surface S defined by unit timelike and spacelike vectors u_i and n_i respectively, which describe an accelerated observer (see text). Assuming that flow along u_i defines an approximate Killing vector X_i, we then show that near the corresponding Rindler horizon, the flux j_a=G_ab X^b along the ingoing null geodesics k_i normalised to have unit Killing energy, given by j . k, has a natural thermodynamic interpretation. Moreover, change in cross-sectional area of the k_i congruence yields the required change in area of S under virtual displacements \emph{normal} to it. The main aim of this note is to clearly demonstrate how, and why, the content of Einstein equations under such horizon deformations, originally pointed out by Padmanabhan, is essentially different from the result of Jacobson, who employed the so called Clausius relation in an attempt to derive Einstein equations from such a Clausius relation. More specifically, we show how a \emph{very specific geometric term} [reminiscent of Hawking's quasi-local expression for energy of spheres] corresponding to change in \emph{gravitational energy} arises inevitably in the first law: dE_G/d{\lambda} \alpha \int_{H} dA R_(2) (see text) -- the contribution of this purely geometric term would be missed in attempts to obtain area (and hence entropy) change by integrating the Raychaudhuri equation.Comment: added comments and references; matches final version accepted in Phys. Rev.

Geodesic Congruences in the Palatini f(R) Theory

We shall investigate the properties of a congruence of geodesics in the framework of Palatini f(R) theories. We shall evaluate the modified geodesic deviation equation and the Raychaudhuri's equation and show that f(R) Palatini theories do not necessarily lead to attractive forces. Also we shall study energy condition for f(R) Palatini gravity via a perturbative analysis of the Raychaudhuri's equation

Gauge and Averaging in Gravitational Self-force

A difficulty with previous treatments of the gravitational self-force is that an explicit formula for the force is available only in a particular gauge (Lorenz gauge), where the force in other gauges must be found through a transformation law once the Lorenz gauge force is known. For a class of gauges satisfying a ``parity condition'' ensuring that the Hamiltonian center of mass of the particle is well-defined, I show that the gravitational self-force is always given by the angle-average of the bare gravitational force. To derive this result I replace the computational strategy of previous work with a new approach, wherein the form of the force is first fixed up to a gauge-invariant piece by simple manipulations, and then that piece is determined by working in a gauge designed specifically to simplify the computation. This offers significant computational savings over the Lorenz gauge, since the Hadamard expansion is avoided entirely and the metric perturbation takes a very simple form. I also show that the rest mass of the particle does not evolve due to first-order self-force effects. Finally, I consider the ``mode sum regularization'' scheme for computing the self-force in black hole background spacetimes, and use the angle-average form of the force to show that the same mode-by-mode subtraction may be performed in all parity-regular gauges. It appears plausible that suitably modified versions of the Regge-Wheeler and radiation gauges (convenient to Schwarzschild and Kerr, respectively) are in this class

Absorption of mass and angular momentum by a black hole: Time-domain formalisms for gravitational perturbations, and the small-hole/slow-motion approximation

The first objective of this work is to obtain practical prescriptions to calculate the absorption of mass and angular momentum by a black hole when external processes produce gravitational radiation. These prescriptions are formulated in the time domain within the framework of black-hole perturbation theory. Two such prescriptions are presented. The first is based on the Teukolsky equation and it applies to general (rotating) black holes. The second is based on the Regge-Wheeler and Zerilli equations and it applies to nonrotating black holes. The second objective of this work is to apply the time-domain absorption formalisms to situations in which the black hole is either small or slowly moving. In the context of this small-hole/slow-motion approximation, the equations of black-hole perturbation theory can be solved analytically, and explicit expressions can be obtained for the absorption of mass and angular momentum. The changes in the black-hole parameters can then be understood in terms of an interaction between the tidal gravitational fields supplied by the external universe and the hole's tidally-induced mass and current quadrupole moments. For a nonrotating black hole the quadrupole moments are proportional to the rate of change of the tidal fields on the hole's world line. For a rotating black hole they are proportional to the tidal fields themselves.Comment: 36 pages, revtex4, no figures, final published versio

Second Order Gravitational Self-Force

The second-order gravitational self-force on a small body is an important problem for gravitational-wave astronomy of extreme mass-ratio inspirals. We give a first-principles derivation of a prescription for computing the first and second perturbed metric and motion of a small body moving through a vacuum background spacetime. The procedure involves solving for a "regular field" with a specified (sufficiently smooth) "effective source", and may be applied in any gauge that produces a sufficiently smooth regular field

Self-force of a point charge in the space-time of a symmetric wormhole

We consider the self-energy and the self-force for an electrically charged particle at rest in the wormhole space-time. We develop general approach and apply it to two specific profiles of the wormhole throat with singular and with smooth curvature. The self-force for these two profiles is found in manifest form; it is an attractive force. We also find an expression for the self-force in the case of arbitrary symmetric throat profile. Far from the throat the self-force is always attractive.Comment: 18 pages, 3 figures Comments: corrected pdf, enlarged pape

Dynamical Surface Gravity in Spherically Symmetric Black Hole Formation

We study dynamical surface gravity in a general spherically symmetric setting using Painlev\'{e}-Gullstrand (PG) coordinates. Our analysis includes several definitions that have been proposed in the past as well as two new definitions adapted to PG coordinates. Various properties are considered, including general covariance, value at extremality, locality and static limit. We illustrate with specific examples of "dirty" black holes that even for spacetimes possessing a global timelike Killing vector, local definitions of surface gravity can differ substantially from "non-local" ones that require an asymptotic normalization condition. Finally, we present numerical calculations of dynamical surface gravity for black hole formation via spherically symmetric scalar field collapse. Our results highlight the differences between the various definitions in a dynamical setting and provide further insight into the distinction between local and non-local definitions of surface gravity.Comment: Final version to appear in Phys. Rev. D. Slight name change, further improvements to numerics and presentation, 25 pages, 7 figure

Transition from adiabatic inspiral to plunge into a spinning black hole

A test particle of mass mu on a bound geodesic of a Kerr black hole of mass M >> mu will slowly inspiral as gravitational radiation extracts energy and angular momentum from its orbit. This inspiral can be considered adiabatic when the orbital period is much shorter than the timescale on which energy is radiated, and quasi-circular when the radial velocity is much less than the azimuthal velocity. Although the inspiral always remains adiabatic provided mu << M, the quasi-circular approximation breaks down as the particle approaches the innermost stable circular orbit (ISCO). In this paper, we relax the quasi-circular approximation and solve the radial equation of motion explicitly near the ISCO. We use the requirement that the test particle's 4-velocity remain properly normalized to calculate a new contribution to the difference between its energy and angular momentum. This difference determines how a black hole's spin changes following a test-particle merger, and can be extrapolated to help predict the mass and spin of the final black hole produced in finite-mass-ratio black-hole mergers. Our new contribution is particularly important for nearly maximally spinning black holes, as it can affect whether a merger produces a naked singularity.Comment: 9 pages, 6 figures, final version published in PRD with minor change

Light-cone coordinates based at a geodesic world line

Continuing work initiated in an earlier publication [Phys. Rev. D 69, 084007 (2004)], we construct a system of light-cone coordinates based at a geodesic world line of an arbitrary curved spacetime. The construction involves (i) an advanced-time or a retarded-time coordinate that labels past or future light cones centered on the world line, (ii) a radial coordinate that is an affine parameter on the null generators of these light cones, and (iii) angular coordinates that are constant on each generator. The spacetime metric is calculated in the light-cone coordinates, and it is expressed as an expansion in powers of the radial coordinate in terms of the irreducible components of the Riemann tensor evaluated on the world line. The formalism is illustrated in two simple applications, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer placed on the axis of symmetry of Melvin's magnetic universe.Comment: 11 pages, 1 figur

Quadrupole moments of rotating neutron stars

Numerical models of rotating neutron stars are constructed for four equations of state using the computer code RNS written by Stergioulas. For five selected values of the star's gravitational mass (in the interval between 1.0 and 1.8 solar masses) and for each equation of state, the star's angular momentum is varied from J=0 to the Keplerian limit J=J_{max}. For each neutron-star configuration we compute Q, the quadrupole moment of the mass distribution. We show that for given values of M and J, |Q| increases with the stiffness of the equation of state. For fixed mass and equation of state, the dependence on J is well reproduced with a simple quadratic fit, Q \simeq - aJ^2/M c^2, where c is the speed of light, and a is a parameter of order unity depending on the mass and the equation of state.Comment: ReVTeX, 7 pages, 5 figures, additional material, and references adde