Self-force on a charge outside a five-dimensional black hole

Beach, Matthew J. S.; Poisson, Eric; Nickel, Bernhard G.

doi:10.1103/physrevd.89.124014

Cited by 20 publications

(63 citation statements)

References 32 publications

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“…Another possibility would be to try to resum the purely integer-order PN series with rational coefficients that enter into the simplification or its remainder, as was done for a (likely considerably simpler) self-force series in [73].…”

Section: Discussionmentioning

confidence: 99%

Experimental mathematics meets gravitational self-force

2015

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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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Section: Discussionmentioning

confidence: 99%

Experimental mathematics meets gravitational self-force

2015

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“…We note that in Ref. [14], the authors adopt a definition for the self-force that involves a spherical averaging procedure and the smooth part of the Green's function does not contribute to the self-force after this averaging is performed. Thus, their analysis effectively corresponds to choosing the singular field for which W 0 (x, x 0 ) = 0.…”

Section: The Singular Field For Static Charges a Hadamard Green'mentioning

confidence: 99%

“…Moreover, there is as yet no known general prescription or expression for self-forces in higher dimensions. Despite the lack of formal underpinning, there has been considerable recent interest in the self-force in higher dimensions [14][15][16][17] yielding some very unexpected results in odd dimensions. In particular, Beach, Poisson and Nickel [14] calculated the selfforce on a static scalar and electric charge in a 5D black hole spacetime and found that the result depended on the radius of a sphere centered at the charge, which they interpret as the radius of the particle.…”

Section: Introductionmentioning

confidence: 99%

“…Despite the lack of formal underpinning, there has been considerable recent interest in the self-force in higher dimensions [14][15][16][17] yielding some very unexpected results in odd dimensions. In particular, Beach, Poisson and Nickel [14] calculated the selfforce on a static scalar and electric charge in a 5D black hole spacetime and found that the result depended on the radius of a sphere centered at the charge, which they interpret as the radius of the particle. Moreover, Frolov and Zelnikov [15] calculated the self-force on a uniformly accelerating charge in Minkowski spacetime (or equivalently a static charge in Rindler spacetime) and their result depends on an undetermined infra-red cutoff lengthscale which the authors postulate could be the scale over which the homogeneous gravitational field approximation is valid.…”

Section: Introductionmentioning

confidence: 99%

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Static self-forces in a five-dimensional black hole spacetime

Taylor

Flanagan

2015

Phys. Rev. D

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We obtain the electric field and scalar field for a static point charge in closed form in the 5D Schwarzschild-Tangherlini black hole spacetime. We then compute the static self-force in each of these cases by assuming that the appropriate singular field is a 4D Hadamard Green's function on the constant time Riemannian slice. It is well known that the Hadamard representation of a Green's function involves an arbitrary regular biscalar W0(x, x ), whose coincidence limit w(x) appears in the expression for the self-force. We develop an axiomatic approach to reduce this arbitrary function to a single arbitrary dimensionless coefficient. We show that in the context of this approach to regularization, the self-force does not depend on any undetermined length-scale and need not depend on the internal structure of the charge.

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“…However, using the corresponding series representation of the solution, for example, for the calculation of the self force, requires a lot of work which includes mode-by-mode subtraction of the divergences and summation of the obtained series (see e.g. [6]). In some special cases it is possible to use analytical approximations.…”

Section: Jhep04(2015)014mentioning

confidence: 99%

Bi-conformal symmetry and static Green functions in the Schwarzschild-Tangherlini spacetimes

Frolov

Zelnikov

2015

J. High Energ. Phys.

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We study a static massless minimally coupled scalar field created by a source in a static D-dimensional spacetime. We demonstrate that the corresponding equation for this field is invariant under a special transformation of the background metric. This transformation consists of the static conformal transformation of the spatial part of the metric accompanied by a properly chosen transformation of the red-shift factor. Both transformations are determined by one function Ω of the spatial coordinates. We show that in a case of higher dimensional spherically symmetric black holes one can find such a bi-conformal transformation that the symmetry of the D-dimensional metric is enhanced after its application. Namely, the metric becomes a direct sum of the metric on a unit sphere and the metric of 2D anti-de Sitter space. The method of the heat kernels is used to find the Green function in this new space, which allows one, after dimensional reduction, to obtain a static Green function in the original space of the static black hole. The general useful representation of static Green functions is obtained in the Schwarzschild-Tangherlini spacetimes of arbitrary dimension. The exact explicit expressions for the static Green functions are obtained in such metrics for D < 6. It is shown that in the four dimensional case the corresponding Green function coincides with the Copson solution.

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Self-force on a charge outside a five-dimensional black hole

Cited by 20 publications

References 32 publications

Experimental mathematics meets gravitational self-force

Experimental mathematics meets gravitational self-force

Static self-forces in a five-dimensional black hole spacetime

Bi-conformal symmetry and static Green functions in the Schwarzschild-Tangherlini spacetimes

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