Self-force of a scalar field for circular orbits about a Schwarzschild black hole

Detweiler, Steven; Messaritaki, E.; Whiting, B. F.

doi:10.1103/physrevd.67.104016

Cited by 100 publications

(250 citation statements)

References 25 publications

(78 reference statements)

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“…To obtain physically meaningful results, one needs to combine the MiSaTaQuWa equation of motion with the metric perturbations h µν to obtain gauge invariant quantities that can be related to physical observables. Although considerable progress has been made in the last several years to develop methods to calculate the metric perturbation and GSF at first order [238,239], the majority of the work has focused on computing the GSF on a particle that moves on a fixed worldline of the background spacetime -for example for a static particle [240], radial [241], circular [242,243] and eccentric [244,245] geodesics in Schwarzschild. Methods to compute the GSF on a particle orbiting a Kerr BH have been proposed (e.g., see [246]) and actual implementations are underway.…”

Section: Perturbation Theory and Gravitational Self Forcementioning

confidence: 99%

Sources of Gravitational Waves: Theory and Observations

Buonanno

Sathyaprakash²

2015

General Relativity and Gravitation

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Section: Perturbation Theory and Gravitational Self Forcementioning

confidence: 99%

Sources of Gravitational Waves: Theory and Observations

Buonanno

Sathyaprakash²

2015

General Relativity and Gravitation

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“…The explicit equations used in renormalization are given in [52]. The sum over ℓ modes converges quite slowly (the summand goes as ℓ −2 ), so it is customary to improve the convergence by finding higher-order regularization coefficients numerically, as in [10,52,68]. However, to obtain an accuracy of N digits in the final result of the renormalized ∆U , one has to obtaining these higher-order regularization coefficients to N digits as well, which would necessitate going to prohibitively large ℓ (e.g., ℓ max ∼ 10 3 for an accuracy of 5000 digits).…”

Section: Computing the Infinite Sum Over Renormalized ℓ Modes To Omentioning

confidence: 99%

Experimental mathematics meets gravitational self-force

2015

Self Cite

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It is now possible to compute linear in mass-ratio terms in the post-Newtonian (PN) expansion for compact binaries to very high orders using linear black hole perturbation theory applied to various invariants. For instance, a computation of the redshift invariant of a point particle in a circular orbit about a black hole in linear perturbation theory gives the linear-in-mass-ratio portion of the binding energy of a circular binary with arbitrary mass ratio. This binding energy, in turn, encodes the system's conservative dynamics. We give a method for extracting the analytic forms of these post-Newtonian coefficients from high-accuracy numerical data using experimental mathematics techniques, notably an integer relation algorithm. Such methods should be particularly important when the calculations progress to the considerably more difficult case of perturbations of the Kerr metric. As an example, we apply this method to the redshift invariant in Schwarzschild. Here we obtain analytic coefficients to 12.5PN, and higher-order terms in mixed analytic-numerical form to 21.5PN, including analytic forms for the complete 13.5PN coefficient, and all the logarithmic terms at 13PN. We have computed the individual modes to over 5000 digits, of which we use at most 1240 in the present calculation. At these high orders, an individual coefficient can have over 30 terms, including a wide variety of transcendental numbers, when written out in full. We are still able to obtain analytic forms for such coefficients from the numerical data through a careful study of the structure of the expansion. The structure we find also allows us to predict certain "leading logarithm"-type contributions to all orders. The additional terms in the expansion we obtain improve the accuracy of the PN series for the redshift observable, even in the very strongfield regime inside the innermost stable circular orbit, particularly when combined with exponential resummation.

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“…The fourth and the fifth columns show the ratio of the leading and subleading regularization parameters that we get numerically and the ones that we get from leading and subleading parts of ∂r(1/ρ) . The last column shows the value of f0 = (1 − 2M/r0) telling us that the singular part of the self-force in a radiation gauge that we use is equal to f0 ∂r(1/ρ) withÃ andB given by [9]…”

Section: Metric Pertubation and Self-forcementioning

confidence: 99%

Conservative, gravitational self-force for a particle in circular orbit around a Schwarzschild black hole in a radiation gauge

Shah

Keidl²,

Friedman

et al. 2011

Phys. Rev. D

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This is the second of two companion papers on computing the self-force in a radiation gauge; more precisely, the method uses a radiation gauge for the radiative part of the metric perturbation, together with an arbitrarily chosen gauge for the parts of the perturbation associated with changes in black-hole mass and spin and with a shift in the center of mass. In a test of the method delineated in the first paper, we compute the conservative part of the self-force for a particle in circular orbit around a Schwarzschild black hole. The gauge vector relating our radiation gauge to a Lorenz gauge is helically symmetric, implying that the quantity h αβ u α u β must have the same value for our radiation gauge as for a Lorenz gauge; and we confirm this numerically to one part in 10 14 . As outlined in the first paper, the perturbed metric is constructed from a Hertz potential that is in term obtained algebraically from the the retarded perturbed spin-2 Weyl scalar, ψ ret 0 . We use a mode-sum renormalization and find the renormalization coefficients by matching a series in L = ℓ + 1/2 to the large-L behavior of the expression for the self-force in terms of the retarded field h ret αβ ; we similarly find the leading renormalization coefficients of h αβ u α u β and the related change in the angular velocity of the particle due to its self-force. We show numerically that the singular part of the self-force has the form fα ∝ ∇αρ −1 , the part of ∇αρ −1 that is axisymmetric about a radial line through the particle. This differs only by a constant from its form for a Lorenz gauge. It is because we do not use a radiation gauge to describe the change in black-hole mass that the singular part of the self-force has no singularity along a radial line through the particle and, at least in this example, is spherically symmetric to subleading order in ρ.

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Self-force of a scalar field for circular orbits about a Schwarzschild black hole

Cited by 100 publications

References 25 publications

Sources of Gravitational Waves: Theory and Observations

Sources of Gravitational Waves: Theory and Observations

Experimental mathematics meets gravitational self-force

Conservative, gravitational self-force for a particle in circular orbit around a Schwarzschild black hole in a radiation gauge

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