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Gravitational perturbations due to a point particle moving on a static black hole background are naturally described in Regge-Wheeler gauge. The first-order field equations reduce to a single master wave equation for each radiative mode. The master function satisfying this wave equation is a linear combination of the metric perturbation amplitudes with a source term arising from the stress-energy tensor of the point particle. The original master functions were found by Regge and Wheeler (odd parity) and Zerilli (even parity). Subsequent work by Moncrief and then Cunningham, Price and Moncrief introduced new master variables which allow time domain reconstruction of the metric perturbation amplitudes. Here I explore the relationship between these different functions and develop a general procedure for deriving new higher-order master functions from ones already known. The benefit of higher-order functions is that their source terms always converge faster at large distance than their lower-order counterparts. This makes for a dramatic improvement in both the speed and accuracy of frequency domain codes when analyzing unbound motion.

Scattering of point particles by black holes: gravitational radiation

Seth Hopper, Vitor Cardoso
Paper arXiv:1706.02791 | 2017-08-06


Gravitational waves can teach us not only about sources and the environment where they were generated, but also about the gravitational interaction itself. Here we study the features of gravitational radiation produced during the scattering of a point-like mass by a black hole. Our results are exact (to numerical error) at any order in a velocity expansion, and are compared against various approximations. At large impact parameter and relatively small velocities our results agree to within percent level with various post-Newtonian and weak-field results. Further, we find good agreement with scaling predictions in the weak-field/high-energy regime. Lastly, we achieve striking agreement with zero-frequency estimates.

Spin-orbit precession along eccentric orbits for extreme mass ratio black hole binaries and its effective-one-body transcription

Chris Kavanagh, Donato Bini, Thibault Damour, Seth Hopper, Adrian C. Ottewill, Barry Wardell
Paper arXiv:1706.00459 | 2017-01-06


In this work we present an analytical gravitational self-force calculation of the spin-orbit precession along an eccentric orbit around a Schwarzschild black hole, following closely the recent prescription of Akcay, Dempsey, and Dolan. We then transcribe this quantity within the Effective-One-Body (EOB) formalism, thereby determining several new, linear-in-mass-ratio, contributions in the post-Newtonian expansion of the spin-orbit couplings entering the EOB Hamiltonian. Namely, we determine the second gyro-gravitomagnetic ratio gS∗(r,pr,pϕ) up to order pr^2/r^4 included.

Echoes of ECOs: gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale

Vitor Cardoso, Seth Hopper, Caio F. B. Macedo, Carlos Palenzuela, Paolo Pani
Paper Phys. Rev. D 94, 084031 (2016) | 2016-30-08


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.

Eccentric-orbit EMRI gravitational wave energy fluxes to 7PN order

Erik Forseth, Charles R. Evans, Seth Hopper
Paper Phys. Rev. D 93, 064058 (2016) | 2016-24-03


We present new results through 7PN order on the energy flux from eccentric extreme-mass-ratio binaries. The black hole perturbation calculations are made at very high accuracy (200 decimal places) using a Mathematica code based on the Mano-Suzuki-Takasugi (MST) analytic function expansion formalism. All published coefficients in the expansion through 3PN order are confirmed and new analytic and numeric terms are found to high order in e^2 at orders between 3.5PN and 7PN. We also show original work in finding (nearly) arbitrarily accurate expansions for hereditary terms at 1.5PN, 2.5PN, and 3PN orders. We fit to a model where at each PN order an eccentricity singular function is factored out, improving substantially the fit even as e→1.

Analytic self-force calculations in the post-Newtonian regime: eccentric orbits on a Schwarzschild background

Seth Hopper, Chris Kavanagh, Adrian C. Ottewill
Paper Phys. Rev. D 93, 044010 (2016) | 2016-04-02


We present a method for solving the first-order field equations in a post-Newtonian (PN) expansion. Our calculations generalize work of Bini and Damour and subsequently Kavanagh et al., to consider eccentric orbits on a Schwarzschild background. We derive expressions for the retarded metric perturbation at the location of the particle for all ℓ-modes. We find that, despite first appearances, the Regge-Wheeler gauge metric perturbation is C0 at the particle for all ℓ. As a first use of our solutions, we compute the gauge-invariant quantity ⟨U⟩ through 4PN while simultaneously expanding in eccentricity through e^10. By anticipating the e→1 singular behavior at each PN order, we greatly improve the accuracy of our results for large e. We use ⟨U⟩ to find 4PN contributions to the effective one body potential Q through e^10 and at linear order in the mass-ratio.

Fast spectral source integration in black hole perturbation calculations.

Seth Hopper, Erik Forseth, Thomas Osburn, and Charles Evans.
Paper Phys. Rev. D 92:044048 (2015) | 2015-15-06


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 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 e≃0.7. 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 raison d'etre) is in an arbitrary precision 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.

Lorenz gauge gravitational self-force calculations of eccentric binaries using a frequency domain procedure.

Thomas Osburn, Erik Forseth, Charles Evans, and Seth Hopper.
Paper Phys. Rev. D 89:081503 (2014) | 2014-15-12


We present an algorithm for calculating the metric perturbations and gravitational self-force for extreme-mass-ratio inspirals (EMRIs) with eccentric orbits. The massive black hole is taken to be Schwarzschild and metric perturbations are computed in Lorenz gauge. The perturbation equations are solved as coupled systems of ordinary differential equations in the frequency domain. Accurate local behavior of the metric is attained through use of the method of extended homogeneous solutions and mode-sum regularization is used to find the self-force. We focus on calculating the self-force with sufficient accuracy to ensure its error contributions to the phase in a long term orbital evolution will be δΦ≲10−2 radians. This requires the orbit-averaged force to have fractional errors ≲10−8 and the oscillatory part of the self-force to have errors ≲10−3 (a level frequently easily exceeded). Our code meets this error requirement in the oscillatory part, extending the reach to EMRIs with eccentricities of e≲0.8, if augmented by use of fluxes for the orbit-averaged force, or to eccentricities of e≲0.5 when used as a stand-alone code. Further, we demonstrate accurate calculations up to orbital separations of a≃100M, beyond that required for EMRI models and useful for comparison with post-Newtonian theory. Our principal developments include (1) use of fully constrained field equations, (2) discovery of analytic solutions for even-parity static modes, (3) finding a pre-conditioning technique for outer homogeneous solutions, (4) adaptive use of quad-precision and (5) jump conditions to handle near-static modes, and (6) a hybrid scheme for high eccentricities.

Strong-Field Scattering of Two Black Holes - Numerics Versus Analytics.

Thibault Damour, Federico Guercilena, Ian Hinder, Seth Hopper, Alessandro Nagar, Luciano Rezzolla
Paper Phys. Rev. D 89:081503 (2014) | 2014-28-02


We probe the gravitational interaction of two black holes in the strong-field regime by computing the scattering angle χ 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.


We calculate the odd-parity, radiative (ℓ≥2) 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.


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 r 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 C−1 typically) and (2) (in some cases) a delta function distribution term with a computable time-dependent amplitude.

An Exact Algebraic Evaluation of Path-Length Difference for Two-Source Interference

Seth Hopper, John Howell
Paper The Physics Teacher, 44(8):516-520 | 2006-01-11


When studying wave interference, one often wants to know the difference in path length for two waves arriving at a common point P but coming from adjacent sources. For example, in many contexts interference maxima occur where this path-length difference is an integer multiple of the wavelength. The standard approximation for the path-length difference is path-lengthdifference≈dsinθ, where d is the distance between the sources and θ is the angle between the perpendicular bisector of d and the line connecting P to the midpoint of d. A common derivation of Eq. (1) begins with the seemingly paradoxical approximation that two paths that meet at a common point can be treated as parallel. In this paper we present an alternative derivation that first finds a simple, exact expression for the path-length difference, valid even when the paths clearly are not parallel. We then show the circumstances under which Eq. (1) is a useful approximation to the exact expression and finally determine an upper limit to the error inherent in using Eq. (1) in place of the exact expression. No math is required beyond the Pythagorean theorem and simple algebra.