1,602 research outputs found
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
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
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
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
Gravitational waveforms from a point particle orbiting a Schwarzschild black hole
We numerically solve the inhomogeneous Zerilli-Moncrief and Regge-Wheeler
equations in the time domain. We obtain the gravitational waveforms produced by
a point-particle of mass traveling around a Schwarzschild black hole of
mass M on arbitrary bound and unbound orbits. Fluxes of energy and angular
momentum at infinity and the event horizon are also calculated. Results for
circular orbits, selected cases of eccentric orbits, and parabolic orbits are
presented. The numerical results from the time-domain code indicate that, for
all three types of orbital motion, black hole absorption contributes less than
1% of the total flux, so long as the orbital radius r_p(t) satisfies r_p(t)> 5M
at all times.Comment: revtex4, 24 pages, 23 figures, 3 tables, submitted to PR
Emergence of thin shell structure during collapse in isotropic coordinates
Numerical studies of gravitational collapse in isotropic coordinates have
recently shown an interesting connection between the gravitational Lagrangian
and black hole thermodynamics. A study of the actual spacetime was not the main
focus of this work and in particular, the rich and interesting structure of the
interior has not been investigated in much detail and remains largely unknown.
We elucidate its features by performing a numerical study of the spacetime in
isotropic coordinates during gravitational collapse of a massless scalar field.
The most salient feature to emerge is the formation of a thin shell of matter
just inside the apparent horizon. The energy density and Ricci scalar peak at
the shell and there is a jump discontinuity in the extrinsic curvature across
the apparent horizon, the hallmark that a thin shell is present in its
vicinity. At late stages of the collapse, the spacetime consists of two vacuum
regions separated by the thin shell. The interior is described by an
interesting collapsing isotropic universe. It tends towards a vacuum (never
reaches a perfect vacuum) and there is a slight inhomogeneity in the interior
that plays a crucial role in the collapse process as the areal radius tends to
zero. The spacetime evolves towards a curvature (physical) singularity in the
interior, both a Weyl and Ricci singularity. In the exterior, our numerical
results match closely the analytical form of the Schwarzschild metric in
isotropic coordinates, providing a strong test of our numerical code.Comment: 24 pages, 10 figures. version to appear in Phys. Rev.
Intermediate-mass-ratio-inspirals in the Einstein Telescope. II. Parameter estimation errors
We explore the precision with which the Einstein Telescope (ET) will be able
to measure the parameters of intermediate-mass-ratio inspirals (IMRIs). We
calculate the parameter estimation errors using the Fisher Matrix formalism and
present results of a Monte Carlo simulation of these errors over choices for
the extrinsic parameters of the source. These results are obtained using two
different models for the gravitational waveform which were introduced in paper
I of this series. These two waveform models include the inspiral, merger and
ringdown phases in a consistent way. One of the models, based on the transition
scheme of Ori & Thorne [1], is valid for IMBHs of arbitrary spin, whereas the
second model, based on the Effective One Body (EOB) approach, has been
developed to cross-check our results in the non-spinning limit. In paper I of
this series, we demonstrated the excellent agreement in both phase and
amplitude between these two models for non-spinning black holes, and that their
predictions for signal-to-noise ratios (SNRs) are consistent to within ten
percent. We now use these models to estimate parameter estimation errors for
binary systems with masses 1.4+100, 10+100, 1.4+500 and 10+500 solar masses
(SMs), and various choices for the spin of the central intermediate-mass black
hole (IMBH). Assuming a detector network of three ETs, the analysis shows that
for a 10 SM compact object (CO) inspiralling into a 100 SM IMBH with spin
q=0.3, detected with an SNR of 30, we should be able to determine the CO and
IMBH masses, and the IMBH spin magnitude to fractional accuracies of 0.001,
0.0003, and 0.001, respectively. We also expect to determine the location of
the source in the sky and the luminosity distance to within 0.003 steradians,
and 10%, respectively. We also assess how the precision of parameter
determination depends on the network configuration.Comment: 21 pages, 5 figures. One reference corrected in v3 for consistency
with published version in Phys Rev
Upper limits of particle emission from high-energy collision and reaction near a maximally rotating Kerr black hole
The center-of-mass energy of two particles colliding near the horizon of a
maximally rotating black hole can be arbitrarily high if the angular momentum
of either of the incident particles is fine-tuned, which we call a critical
particle. We study particle emission from such high-energy collision and
reaction in the equatorial plane fully analytically. We show that the
unconditional upper limit of the energy of the emitted particle is given by
218.6% of that of the injected critical particle, irrespective of the details
of the reaction and this upper limit can be realized for massless particle
emission. The upper limit of the energy extraction efficiency for this emission
as a collisional Penrose process is given by 146.6%, which can be realized in
the collision of two massive particles with optimized mass ratio. Moreover, we
analyze perfectly elastic collision, Compton scattering, and pair annihilation
and show that net positive energy extraction is really possible for these three
reactions. The Compton scattering is most efficient among them and the
efficiency can reach 137.2%. On the other hand, our result is qualitatively
consistent with the earlier claim that the mass and energy of the emitted
particle are at most of order the total energy of the injected particles and
hence we can observe neither super-heavy nor super-energetic particles.Comment: 22 pages, 3 figures, typos corrected, reference updated, accepted for
publication in Physical Review D, typos correcte
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