### Abstract

We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass m _{1} 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 m _{2}m _{1}. More precisely, we construct the function huuR,L(x)hμνR,Luμuν (related to Detweiler's gauge-invariant "redshift" variable), where hμνR,L( _{1}) is the regularized metric perturbation in the Lorenz gauge, uμ is the four-velocity of m _{1} in the background Schwarzschild metric of m _{2}, and [Gc ^{-}3(m _{1}+m _{2})Ω] ^{2}/ ^{3} is an invariant coordinate constructed from the orbital frequency Ω. In particular, we explore the behavior of huuR,L just outside the "light ring" at x=13 (i.e., r=3Gm _{2}/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 νm _{1}m _{2}/(m _{1}+m _{2}) ^{2}, of the main radial potential A(u,ν)=1-2u+νa(u)+O(ν2) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0<u<13 (thereby extending previous results limited to u≤15). We find that a(u) diverges like a(u)0.25(1-3u) ^{-}1 ^{/}2 at the light-ring limit, u→(13) ^{-}, 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<13 (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=16, as well as the associated O(ν) 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(ν) piece of a second EOB radial potential D̄(u)=1+νd̄(u)+O(ν2). Combining these results with our present global analytic representation of a(u), we numerically compute d̄(u) on the interval 0<u≤16.

Original language | English |
---|---|

Article number | 104041 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 86 |

Issue number | 10 |

DOIs | |

Publication status | Published - Nov 16 2012 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Cite this

*Physical Review D - Particles, Fields, Gravitation and Cosmology*,

*86*(10), [104041]. https://doi.org/10.1103/PhysRevD.86.104041

**Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring.** / Akcay, Sarp; Barack, Leor; Damour, Thibault; Sago, Norichika.

Research output: Contribution to journal › Article

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 86, no. 10, 104041. https://doi.org/10.1103/PhysRevD.86.104041

}

TY - JOUR

T1 - Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring

AU - Akcay, Sarp

AU - Barack, Leor

AU - Damour, Thibault

AU - Sago, Norichika

PY - 2012/11/16

Y1 - 2012/11/16

N2 - We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass m 1 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 m 2m 1. More precisely, we construct the function huuR,L(x)hμνR,Luμuν (related to Detweiler's gauge-invariant "redshift" variable), where hμνR,L( 1) is the regularized metric perturbation in the Lorenz gauge, uμ is the four-velocity of m 1 in the background Schwarzschild metric of m 2, and [Gc -3(m 1+m 2)Ω] 2/ 3 is an invariant coordinate constructed from the orbital frequency Ω. In particular, we explore the behavior of huuR,L just outside the "light ring" at x=13 (i.e., r=3Gm 2/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 νm 1m 2/(m 1+m 2) 2, of the main radial potential A(u,ν)=1-2u+νa(u)+O(ν2) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0-1 /2 at the light-ring limit, u→(13) -, 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

AB - We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass m 1 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 m 2m 1. More precisely, we construct the function huuR,L(x)hμνR,Luμuν (related to Detweiler's gauge-invariant "redshift" variable), where hμνR,L( 1) is the regularized metric perturbation in the Lorenz gauge, uμ is the four-velocity of m 1 in the background Schwarzschild metric of m 2, and [Gc -3(m 1+m 2)Ω] 2/ 3 is an invariant coordinate constructed from the orbital frequency Ω. In particular, we explore the behavior of huuR,L just outside the "light ring" at x=13 (i.e., r=3Gm 2/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 νm 1m 2/(m 1+m 2) 2, of the main radial potential A(u,ν)=1-2u+νa(u)+O(ν2) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0-1 /2 at the light-ring limit, u→(13) -, 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

UR - http://www.scopus.com/inward/record.url?scp=84870162975&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870162975&partnerID=8YFLogxK

U2 - 10.1103/PhysRevD.86.104041

DO - 10.1103/PhysRevD.86.104041

M3 - Article

VL - 86

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 10

M1 - 104041

ER -