1,637 research outputs found
Retarded coordinates based at a world line, and the motion of a small black hole in an external universe
In the first part of this article I present a system of retarded coordinates
based at an arbitrary world line of an arbitrary curved spacetime. The
retarded-time coordinate labels forward light cones that are centered on the
world line, the radial coordinate is an affine parameter on the null generators
of these light cones, and the angular coordinates are constant on each of these
generators. The spacetime metric in the retarded coordinates is displayed as an
expansion in powers of the radial coordinate and expressed in terms of the
world line's acceleration vector and the spacetime's Riemann tensor evaluated
at the world line. The formalism is illustrated in two examples, the first
involving a comoving world line of a spatially-flat cosmology, the other
featuring an observer in circular motion in the Schwarzschild spacetime. The
main application of the formalism is presented in the second part of the
article, in which I consider the motion of a small black hole in an empty
external universe. I use the retarded coordinates to construct the metric of
the small black hole perturbed by the tidal field of the external universe, and
the metric of the external universe perturbed by the presence of the black
hole. Matching these metrics produces the MiSaTaQuWa equations of motion for
the small black hole.Comment: 20 pages, revtex4, 2 figure
Radiation reaction and the self-force for a point mass in general relativity
A point particle of mass m moving on a geodesic creates a perturbation h, of
the spacetime metric g, that diverges at the particle. Simple expressions are
given for the singular m/r part of h and its quadrupole distortion caused by
the spacetime. Subtracting these from h leaves a remainder h^R that is C^1. The
self-force on the particle from its own gravitational field corrects the
worldline at O(m) to be a geodesic of g+h^R. For the case that the particle is
a small non-rotating black hole, an approximate solution to the Einstein
equations is given with error of O(m^2) as m approaches 0.Comment: 4 pages, RevTe
Rotating black hole orbit functionals in the frequency domain
In many astrophysical problems, it is important to understand the behavior of
functions that come from rotating (Kerr) black hole orbits. It can be
particularly useful to work with the frequency domain representation of those
functions, in order to bring out their harmonic dependence upon the fundamental
orbital frequencies of Kerr black holes. Although, as has recently been shown
by W. Schmidt, such a frequency domain representation must exist, the coupled
nature of a black hole orbit's and motions makes it difficult to
construct such a representation in practice. Combining Schmidt's description
with a clever choice of timelike coordinate suggested by Y. Mino, we have
developed a simple procedure that sidesteps this difficulty. One first Fourier
expands all quantities using Mino's time coordinate . In particular,
the observer's time is decomposed with . The frequency domain
description is then built from the -Fourier expansion and the
expansion of . We have found this procedure to be quite simple to implement,
and to be applicable to a wide class of functionals. We test the procedure
using a simple test function, and then apply it in a particularly interesting
case, the Weyl curvature scalar used in black hole perturbation
theory.Comment: 16 pages, 2 figures. Submitted to Phys Rev D. New version gives a
vastly improved algorithm due to Drasco for computing the Fourier transforms.
Drasco has been added as an author. Also fixed some references and
exterminated a small herd of typos; final published versio
The Transition from Inspiral to Plunge for a Compact Body in a Circular Equatorial Orbit Around a Massive, Spinning Black Hole
There are three regimes of gravitational-radiation-reaction-induced inspiral
for a compact body with mass mu, in a circular, equatorial orbit around a Kerr
black hole with mass M>>mu: (i) The "adiabatic inspiral regime", in which the
body gradually descends through a sequence of circular, geodesic orbits. (ii) A
"transition regime", near the innermost stable circular orbit (isco). (iii) The
"plunge regime", in which the body travels on a geodesic from slightly below
the isco into the hole's horizon. This paper gives an analytic treatment of the
transition regime and shows that, with some luck, gravitational waves from the
transition might be measurable by the space-based LISA mission.Comment: 8 Pages and 3 Figures; RevTeX; submitted to Physical Review
Non-precessional spin-orbit effects on gravitational waves from inspiraling compact binaries to second post-Newtonian order
We derive all second post-Newtonian (2PN), non-precessional effects of spin-
orbit coupling on the gravitational wave forms emitted by an inspiraling binary
composed of spinning, compact bodies in a quasicircular orbit. Previous post-
Newtonian calculations of spin-orbit effects (at 1.5PN order) relied on a fluid
description of the spinning bodies. We simplify the calculations by introducing
into post-Newtonian theory a delta-function description of the influence of the
spins on the bodies' energy-momentum tensor. This description was recently used
by Mino, Shibata, and Tanaka (MST) in Teukolsky-formalism analyses of particles
orbiting massive black holes, and is based on prior work by Dixon. We compute
the 2PN contributions to the wave forms by combining the MST energy-momentum
tensor with the formalism of Blanchet, Damour, and Iyer for evaluating the
binary's radiative multipoles, and with the well-known 1.5PN order equations of
motion for the binary. Our results contribute at 2PN order only to the
amplitudes of the wave forms. The secular evolution of the wave forms' phase,
the quantity most accurately measurable by LIGO, is not affected by our results
until 2.5PN order, at which point other spin-orbit effects also come into play.
We plan to evaluate the entire 2.5PN spin-orbit contribution to the secular
phase evolution in a future paper, using the techniques of this paper.Comment: 11 pages, submitted to Phys. Rev.
Black hole tidal problem in the Fermi normal coordinates
We derive a tidal potential for a self-gravitating fluid star orbiting Kerr
black hole along a timelike geodesic extending previous works by Fishbone and
Marck. In this paper, the tidal potential is calculated up to the third and
fourth-order terms in , where is the stellar radius and the
orbital separation, in the Fermi-normal coordinate system following the
framework developed by Manasse and Misner. The new formulation is applied for
determining the tidal disruption limit (Roche limit) of corotating Newtonian
stars in circular orbits moving on the equatorial plane of Kerr black holes. It
is demonstrated that the third and fourth-order terms quantitatively play an
important role in the Roche limit for close orbits with R/r \agt 0.1. It is
also indicated that the Roche limit of neutron stars orbiting a stellar-mass
black hole near the innermost stable circular orbit may depend sensitively on
the equation of state of the neutron star.Comment: Correct typo
Gravitational Waves from a Compact Star in a Circular, Inspiral Orbit, in the Equatorial Plane of a Massive, Spinning Black Hole, as Observed by LISA
Results are presented from high-precision computations of the orbital
evolution and emitted gravitational waves for a stellar-mass object spiraling
into a massive black hole in a slowly shrinking, circular, equatorial orbit.
The focus of these computations is inspiral near the innermost stable circular
orbit (isco)---more particularly, on orbits for which the angular velocity
Omega is 0.03 < Omega/Omega_{isco} < 1. The computations are based on the
Teukolsky-Sasaki-Nakamura formalism, and the results are tabulated in a set of
functions that are of order unity and represent relativistic corrections to
low-orbital-velocity formulas. These tables can form a foundation for future
design studies for the LISA space-based gravitational-wave mission. A first
survey of applications to LISA is presented: Signal to noise ratios S/N are
computed and graphed as functions of the time-evolving gravitational-wave
frequency for representative values of the hole's mass M and spin a and the
inspiraling object's mass \mu, with the distance to Earth chosen to be r_o = 1
Gpc. These S/N's show a very strong dependence on the black-hole spin, as well
as on M and \mu. A comparison with predicted event rates shows strong promise
for detecting these waves, but not beyond about 1Gpc if the inspiraling object
is a white dwarf or neutron star. This argues for a modest lowering of LISA's
noise floor. A brief discussion is given of the prospects for extracting
information from the observed wavesComment: Physical Review D, in press; 21 pages, 9 figures, 10 tables it is
present in the RevTeX fil
Innermost Stable Circular Orbit of a Spinning Particle in Kerr Spacetime
We study stability of a circular orbit of a spinning test particle in a Kerr
spacetime. We find that some of the circular orbits become unstable in the
direction perpendicular to the equatorial plane, although the orbits are still
stable in the radial direction. Then for the large spin case ($S < \sim O(1)),
the innermost stable circular orbit (ISCO) appears before the minimum of the
effective potential in the equatorial plane disappears. This changes the radius
of ISCO and then the frequency of the last circular orbit.Comment: 25 pages including 8 figure
Mapping spacetimes with LISA: inspiral of a test-body in a `quasi-Kerr' field
The future LISA detector will constitute the prime instrument for
high-precision gravitational wave observations.LISA is expected to provide
information for the properties of spacetime in the vicinity of massive black
holes which reside in galactic nuclei.Such black holes can capture stellar-mass
compact objects, which afterwards slowly inspiral,radiating gravitational
waves.The body's orbital motion and the associated waveform carry information
about the spacetime metric of the massive black hole,and it is possible to
extract this information and experimentally identify (or not!) a Kerr black
hole.In this paper we lay the foundations for a practical `spacetime-mapping'
framework. Our work is based on the assumption that the massive body is not
necessarily a Kerr black hole, and that the vacuum exterior spacetime is
stationary axisymmetric,described by a metric which deviates slightly from the
Kerr metric. We first provide a simple recipe for building such a `quasi-Kerr'
metric by adding to the Kerr metric the deviation in the value of the
quadrupole moment. We then study geodesic motion in this metric,focusing on
equatorial orbits. We proceed by computing `kludge' waveforms which we compare
with their Kerr counterparts. We find that a modest deviation from the Kerr
metric is sufficient for producing a significant mismatch between the
waveforms, provided we fix the orbital parameters. This result suggests that an
attempt to use Kerr waveform templates for studying EMRIs around a non-Kerr
object might result in serious loss of signal-to-noise ratio and total number
of detected events. The waveform comparisons also unveil a `confusion' problem,
that is the possibility of matching a true non-Kerr waveform with a Kerr
template of different orbital parameters.Comment: 19 pages, 6 figure
Radiative multipole moments of integer-spin fields in curved spacetime
Radiative multipole moments of scalar, electromagnetic, and linearized
gravitational fields in Schwarzschild spacetime are computed to third order in
v in a weak-field, slow-motion approximation, where v is a characteristic
velocity associated with the motion of the source. To zeroth order in v, a
radiative moment of order l is given by the corresponding source moment
differentiated l times with respect to retarded time. At second order in v,
additional terms appear inside the spatial integrals. These are near-zone
corrections which depend on the detailed behavior of the source. At third order
in v, the correction terms occur outside the spatial integrals, so that they do
not depend on the detailed behavior of the source. These are wave-propagation
corrections which are heuristically understood as arising from the scattering
of the radiation by the spacetime curvature surrounding the source. Our
calculations show that the wave-propagation corrections take a universal form
which is independent of multipole order and field type. We also show that in
general relativity, temporal and spatial curvatures contribute equally to the
wave-propagation corrections.Comment: 34 pages, ReVTe
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