Self-force on a scalar charge in Kerr spacetime: Inclined circular orbits

Niels Warburton

Phys. Rev. D 91, 024045 – Published 30 January 2015

Abstract

Accurately modeling astrophysical extreme-mass-ratio inspirals requires calculating the gravitational self-force for orbits in Kerr spacetime. The necessary calculation techniques are typically very complex and, consequently, toy scalar-field models are often developed in order to establish a particular calculational approach. To that end, I present a calculation of the scalar-field self-force for a particle moving on a (fixed) inclined circular geodesic of a background Kerr black hole. I make the calculation in the frequency domain and demonstrate how to apply the mode-sum regularization procedure to all four components of the self-force. I present results for a number of strong-field orbits which can be used as benchmarks for emerging self-force calculation techniques in Kerr spacetime.

Received 17 August 2014

DOI:https://doi.org/10.1103/PhysRevD.91.024045

Authors & Affiliations

Niels Warburton

School of Mathematical Sciences and Complex & Adaptive Systems Laboratory, University College Dublin, Belfield, Dublin 4, Ireland

Self-force on a scalar charge in Kerr spacetime: Circular equatorial orbits

Niels Warburton and Leor Barack

Phys. Rev. D 81, 084039 (2010)

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Issue

Vol. 91, Iss. 2 — 15 January 2015

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Images

Figure 1
Projection of $Y_{\bar{l} m, θ}$ onto a basis of spherical harmonics [see Eq. (48)]. The contribution to each $l$ -mode exhibits power-law behavior (note the log-log scale). The wide bandwidth of the coupling makes practical calculations of $F_{θ}$ via this method infeasible and instead I make use of the alternative approach outlined in Sec. 3a.
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Figure 2
Convergence of $F_{θ}^{l (cons / diss)}$ components of the SSF for a sample inclined circular orbit] Convergence of $F_{θ}^{l (cons / diss)}$ components of the SSF for a sample inclined circular orbit about a black hole with $a = 0.998 M$ and with orbital parameters $(r_{0}, ι) = (3 M, 27.557 3^{°})$ , shown at $χ = 0.816814$ . The left panel depicts the contributions per $l$ -mode to the conservative component of $F_{θ}$ alongside an $l^{- 2}$ reference line. As theory predicts, the contribution of the high $l$ -modes follows closely to the reference line. The right panel shows the contributions per $l$ -mode to the dissipative component of $F_{θ}$ alongside an exponential reference line. Again, as expected from theory, the contribution from the high $l$ -modes follows closely to this line. Similar convergence behavior is observed for the conservative and dissipative pieces of the other three components ( $F_{r}, F_{t}, F_{φ}$ ) of the SSF.
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Figure 3
The radial SSF for various inclined circular orbits with radius $r_{0} = 9 M$ in motion about a black hole with spin parameter $a = 0.998 M$ . The self-force varies smoothly from the prograde equatorial orbit ( $ι = 0^{°}$ ) to the retrograde equatorial orbit ( $ι = 18 0^{°}$ ) with the largest oscillations observed for near-polar orbits. The equatorial prograde and retrograde values were computed using the code presented in Paper I.
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Figure 4
The SSF for an inclined circular orbit of radius $r_{0} = 4 M$ about a Kerr black hole with spin $a = 0.998 M$ . The angular momentum of the orbit is $L_{z} = - 0.9 M$ $(ι ≃ 102.81)$ . To make clear the relative phasing between the different components of the SSF ${\tilde{F}}_{t}, {\tilde{F}}_{r}$ and ${\tilde{F}}_{φ}$ have been shifted and rescaled as shown in the legend.
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Figure 5
The four components of the covariant SSF for a black hole with spin $a = 0.998 M$ and orbit of radius $r_{0} = 4 M$ . Reading clockwise from the top left the figures show the adimensionalized $F_{t}, F_{r}, F_{φ}$ and $F_{θ}$ . In each plot the different curves correspond to orbits with angular momentum of $L_{z} = {0.5, 0.7, 1, 1.5, 2}$ , reading from top to bottom when $χ > π$ . These correspond to orbits with inclinations of $ι \approx 81.0 3^{°}$ , 76.93°, 70.06°, 55.87°, 34.73°, respectively.
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