57 research outputs found
Metric perturbations from eccentric orbits on a Schwarzschild black hole: I. Odd-parity Regge-Wheeler to Lorenz gauge transformation and two new methods to circumvent the Gibbs phenomenon
We calculate the odd-parity, radiative () parts of the metric
perturbation in Lorenz gauge caused by a small compact object in eccentric
orbit about a Schwarzschild black hole. The Lorenz gauge solution is found via
gauge transformation from a corresponding one in Regge-Wheeler gauge. Like the
Regge-Wheeler gauge solution itself, the gauge generator is computed in the
frequency domain and transferred to the time domain. The wave equation for the
gauge generator has a source with a compact, moving delta-function term and a
discontinuous non-compact term. The former term allows the method of extended
homogeneous solutions to be applied (which circumvents the Gibbs phenomenon).
The latter has required the development of new means to use frequency domain
methods and yet be able to transfer to the time domain while avoiding Gibbs
problems. Two new methods are developed to achieve this: a partial annihilator
method and a method of extended particular solutions. We detail these methods
and show their application in calculating the odd-parity gauge generator and
Lorenz gauge metric perturbations. A subsequent paper will apply these methods
to the harder task of computing the even-parity parts of the gauge generator.Comment: 17 pages, 9 figures, Updated with one modified figure and minor
changes to the text. Added DOI and Journal referenc
Gravitational perturbations and metric reconstruction: Method of extended homogeneous solutions applied to eccentric orbits on a Schwarzschild black hole
We calculate the gravitational perturbations produced by a small mass in
eccentric orbit about a much more massive Schwarzschild black hole and use the
numerically computed perturbations to solve for the metric. The calculations
are initially made in the frequency domain and provide Fourier-harmonic modes
for the gauge-invariant master functions that satisfy inhomogeneous versions of
the Regge-Wheeler and Zerilli equations. These gravitational master equations
have specific singular sources containing both delta function and
derivative-of-delta function terms. We demonstrate in this paper successful
application of the method of extended homogeneous solutions, developed recently
by Barack, Ori, and Sago, to handle source terms of this type. The method
allows transformation back to the time domain, with exponential convergence of
the partial mode sums that represent the field. This rapid convergence holds
even in the region of traversed by the point mass and includes the
time-dependent location of the point mass itself. We present numerical results
of mode calculations for certain orbital parameters, including highly accurate
energy and angular momentum fluxes at infinity and at the black hole event
horizon. We then address the issue of reconstructing the metric perturbation
amplitudes from the master functions, the latter being weak solutions of a
particular form to the wave equations. The spherical harmonic amplitudes that
represent the metric in Regge-Wheeler gauge can themselves be viewed as weak
solutions. They are in general a combination of (1) two differentiable
solutions that adjoin at the instantaneous location of the point mass (a result
that has order of continuity typically) and (2) (in some cases) a
delta function distribution term with a computable time-dependent amplitude.Comment: 25 pages, 5 figures, Updated with minor change
Determination of new coefficients in the angular momentum and energy fluxes at infinity to 9PN for eccentric Schwarzschild extreme-mass-ratio inspirals using mode-by-mode fitting
We present an extension of work in an earlier paper showing high precision
comparisons between black hole perturbation theory and post-Newtonian (PN)
theory in their region of overlapping validity for bound, eccentric-orbit,
Schwarzschild extreme-mass-ratio inspirals. As before we apply a numerical
fitting scheme to extract eccentricity coefficients in the PN expansion of the
gravitational wave fluxes, which are then converted to exact analytic form
using an integer-relation algorithm. In this work, however, we fit to
individual modes to exploit simplifying factorizations that lie therein.
Since the previous paper focused solely on the energy flux, here we concentrate
initially on analyzing the angular momentum flux to infinity. A first step
involves finding convenient forms for hereditary contributions to the flux at
low-PN order, analogous to similar terms worked out previously for the energy
flux. We then apply the upgraded techniques to find new PN terms through 9PN
order and (at many PN orders) to in the power series in eccentricity.
With the new approach applied to angular momentum fluxes, we return to the
energy fluxes at infinity to extend those previous results. Like before, the
underlying method uses a \textsc{Mathematica} code based on use of the
Mano-Suzuki-Takasugi (MST) function expansion formalism to represent
gravitational perturbations and spectral source integration (SSI) to find
numerical results at arbitrarily high precision.Comment: 36 pages, 1 figur
Fast spectral source integration in black hole perturbation calculations
This paper presents a new technique for achieving spectral accuracy and fast
computational performance in a class of black hole perturbation and
gravitational self-force calculations involving extreme mass ratios and generic
orbits. Called \emph{spectral source integration} (SSI), this method should see
widespread future use in problems that entail (i) point-particle description of
the small compact object, (ii) frequency domain decomposition, and (iii) use of
the background eccentric geodesic motion. Frequency domain approaches are
widely used in both perturbation theory flux-balance calculations and in local
gravitational self-force calculations. Recent self-force calculations in Lorenz
gauge, using the frequency domain and method of extended homogeneous solutions,
have been able to accurately reach eccentricities as high as . We
show here SSI successfully applied to Lorenz gauge. In a double precision
Lorenz gauge code, SSI enhances the accuracy of results and makes a factor of
three improvement in the overall speed. The primary initial application of
SSI--for us its \emph{raison d'\^{e}tre}--is in an arbitrary precision
\emph{Mathematica} code that computes perturbations of eccentric orbits in the
Regge-Wheeler gauge to extraordinarily high accuracy (e.g., 200 decimal
places). These high accuracy eccentric orbit calculations would not be possible
without the exponential convergence of SSI. We believe the method will extend
to work for inspirals on Kerr, and will be the subject of a later publication.
SSI borrows concepts from discrete-time signal processing and is used to
calculate the mode normalization coefficients in perturbation theory via sums
over modest numbers of points around an orbit. A variant of the idea is used to
obtain spectral accuracy in solution of the geodesic orbital motion.Comment: 15 pages, 7 figure
Strong-Field Scattering of Two Black Holes: Numerics Versus Analytics
We probe the gravitational interaction of two black holes in the strong-field
regime by computing the scattering angle of hyperbolic-like, close
binary-black-hole encounters as a function of the impact parameter. The fully
general-relativistic result from numerical relativity is compared to two
analytic approximations: post-Newtonian theory and the effective-one-body
formalism. As the impact parameter decreases, so that black holes pass within a
few times their Schwarzschild radii, we find that the post-Newtonian prediction
becomes quite inaccurate, while the effective-one-body one keeps showing a good
agreement with numerical results. Because we have explored a regime which is
very different from the one considered so far with binaries in quasi-circular
orbits, our results open a new avenue to improve analytic representations of
the general-relativistic two-body Hamiltonian.Comment: 5 pages, 3 figures. Submitted to Physical Review Letter
Echoes of ECOs: gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale
Gravitational waves from binary coalescences provide one of the cleanest
signatures of the nature of compact objects. It has been recently argued that
the post-merger ringdown waveform of exotic ultracompact objects is initially
identical to that of a black-hole, and that putative corrections at the horizon
scale will appear as secondary pulses after the main burst of radiation. Here
we extend this analysis in three important directions: (i) we show that this
result applies to a large class of exotic compact objects with a photon sphere
for generic orbits in the test-particle limit; (ii) we investigate the
late-time ringdown in more detail, showing that it is universally characterized
by a modulated and distorted train of "echoes" of the modes of vibration
associated with the photon sphere; (iii) we study for the first time
equal-mass, head-on collisions of two ultracompact boson stars and compare
their gravitational-wave signal to that produced by a pair of black-holes. If
the initial objects are compact enough as to mimic a binary black-hole
collision up to the merger, the final object exceeds the maximum mass for boson
stars and collapses to a black-hole. This suggests that - in some
configurations - the coalescence of compact boson stars might be almost
indistinguishable from that of black-holes. On the other hand, generic
configurations display peculiar signatures that can be searched for in
gravitational-wave data as smoking guns of exotic compact objects.Comment: 13 pages, RevTex4. v2: typo in equation 7 corrected, references
added, to appear in PR
The gravitational field produced by extreme-mass-ratio orbits on Schwarzschild spacetime
A stellar-mass compact object orbiting a supermassive black hole will radiate energy and angular momentum in the form of gravitational waves, causing it to spiral inward. Such an extreme-mass-ratio inspiral (EMRI) is an important potential source for a direct gravity wave detection. It will require sufficiently accurate source modeling for such detections to be made and analyzed. In this thesis I present original research that has furthered the collective goal of accurate numerical EMRI simulations. I begin by giving an overview of the extensive work that has been done in this field, with an eye toward significant headway that has been made in the last decade. I then lay the groundwork for my own work by reviewing the mathematical foundations for gravity waves and black hole perturbation theory. Before attacking the subject of gravity waves on a curved background, I examine the model problem of the scalar field that is induced by an orbiting charge. This problem, while idealized, introduces many of the mathematical and numerical techniques which are necessary to solve the perturbed Einstein equations. At this point, with the foundation laid, I present new work on eccentric orbits of point masses about a Schwarzschild black hole. I show how the method of extended homogeneous solutions is generalized to find the radiative part of the first-order metric perturbation in Regge-Wheeler (RW) gauge using frequency domain techniques. Additionally, for the first time we computed the local point-singular nature of the metric perturbation in RW gauge. Due mostly to such gauge artifacts, RW gauge is not ideal of performing a local self-force calculation. Thus, I then present work on transforming the metric perturbation to Lorenz gauge. This will allow for the direct calculation of the self-force. I end this thesis by summarizing the potential and necessary areas of EMRI research in the near future.Doctor of Philosoph
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