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Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring

Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring
Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring
We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass m1 as it moves along an (unstable) circular geodesic orbit between the innermost stable orbit and the light ring of a Schwarzschild black hole of mass m2?m1. More precisely, we construct the function huuR,L(x)?h??R,Lu?u? (related to Detweiler’s gauge-invariant “redshift” variable), where h??R,L(?m1) is the regularized metric perturbation in the Lorenz gauge, u? is the four-velocity of m1 in the background Schwarzschild metric of m2, and x?[Gc-3(m1+m2)?]2/3 is an invariant coordinate constructed from the orbital frequency ?. In particular, we explore the behavior of huuR,L just outside the “light ring” at x=1/3 (i.e., r=3Gm2/c2), where the circular orbit becomes null. Using the recently discovered link between huuR,L and the piece a(u), linear in the symmetric mass ratio ??m1m2/(m1+m2)2, of the main radial potential A(u,?)=1-2u+?a(u)+O(?2) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0<u<1/3 (thereby extending previous results limited to u?1/5). We find that a(u) diverges like a(u)?0.25(1-3u)-1/2 at the light-ring limit, u?(1/3)-, explain the physical origin of this divergent behavior, and discuss its consequences for the EOB formalism. We construct accurate global analytic fits for a(u), valid on the entire domain 0<u<1/3 (and possibly beyond), and give accurate numerical estimates of the values of a(u) and its first three derivatives at the innermost stable circular orbit u=1/6, as well as the associated O(?) shift in the frequency of that orbit. In previous work we used GSF data on slightly eccentric orbits to compute a certain linear combination of a(u) and its first two derivatives, involving also the O(?) piece of a second EOB radial potential D? (u)=1+?d? (u)+O(?2). Combining these results with our present global analytic representation of a(u), we numerically compute d? (u) on the interval 0<u?1/6
1550-7998
104041 -[37pp]
Akcay, Sarp
dcb16394-6b37-43a3-90ca-925458bfd668
Barack, Leor
f08e66d4-c2f7-4f2f-91b8-f2c4230d0298
Damour, Thibault
9e7fe76d-f668-4e67-a399-c806a02838d6
Sago, Norichika
c4baa9a1-e4fb-448e-8818-f7d189ed2773
Akcay, Sarp
dcb16394-6b37-43a3-90ca-925458bfd668
Barack, Leor
f08e66d4-c2f7-4f2f-91b8-f2c4230d0298
Damour, Thibault
9e7fe76d-f668-4e67-a399-c806a02838d6
Sago, Norichika
c4baa9a1-e4fb-448e-8818-f7d189ed2773

Akcay, Sarp, Barack, Leor, Damour, Thibault and Sago, Norichika (2012) Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring. Physical Review D, 86 (104041), 104041 -[37pp]. (doi:10.1103/PhysRevD.86.104041).

Record type: Article

Abstract

We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass m1 as it moves along an (unstable) circular geodesic orbit between the innermost stable orbit and the light ring of a Schwarzschild black hole of mass m2?m1. More precisely, we construct the function huuR,L(x)?h??R,Lu?u? (related to Detweiler’s gauge-invariant “redshift” variable), where h??R,L(?m1) is the regularized metric perturbation in the Lorenz gauge, u? is the four-velocity of m1 in the background Schwarzschild metric of m2, and x?[Gc-3(m1+m2)?]2/3 is an invariant coordinate constructed from the orbital frequency ?. In particular, we explore the behavior of huuR,L just outside the “light ring” at x=1/3 (i.e., r=3Gm2/c2), where the circular orbit becomes null. Using the recently discovered link between huuR,L and the piece a(u), linear in the symmetric mass ratio ??m1m2/(m1+m2)2, of the main radial potential A(u,?)=1-2u+?a(u)+O(?2) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0<u<1/3 (thereby extending previous results limited to u?1/5). We find that a(u) diverges like a(u)?0.25(1-3u)-1/2 at the light-ring limit, u?(1/3)-, explain the physical origin of this divergent behavior, and discuss its consequences for the EOB formalism. We construct accurate global analytic fits for a(u), valid on the entire domain 0<u<1/3 (and possibly beyond), and give accurate numerical estimates of the values of a(u) and its first three derivatives at the innermost stable circular orbit u=1/6, as well as the associated O(?) shift in the frequency of that orbit. In previous work we used GSF data on slightly eccentric orbits to compute a certain linear combination of a(u) and its first two derivatives, involving also the O(?) piece of a second EOB radial potential D? (u)=1+?d? (u)+O(?2). Combining these results with our present global analytic representation of a(u), we numerically compute d? (u) on the interval 0<u?1/6

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Published date: 15 November 2012
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 347766
URI: https://eprints.soton.ac.uk/id/eprint/347766
ISSN: 1550-7998
PURE UUID: 273e93c5-3a08-4975-a2b4-ad90e3be9f4e
ORCID for Leor Barack: ORCID iD orcid.org/0000-0003-4742-9413

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Date deposited: 30 Jan 2013 11:52
Last modified: 19 Nov 2019 01:46

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