Generic effective source for scalar self-force calculations

A leading approach to the modeling of extreme mass ratio inspirals involves the treatment of the smaller mass as a point particle and the computation of a regularized self-force acting on that particle. In turn, this computation requires knowledge of the regularized retarded field generated by the particle. A direct calculation of this regularized field may be achieved by replacing the point particle with an effective source and solving directly a wave equation for the regularized field. This has the advantage that all quantities are finite and require no further regularization. In this work, we present a method for computing an effective source which is finite and continuous everywhere, and which is valid for a scalar point particle in arbitrary geodesic motion in an arbitrary background spacetime. We explain in detail various technical and practical considerations that underlie its use in several numerical self-force calculations. We consider as examples the cases of a particle in a circular orbit about Schwarzschild and Kerr black holes, and also the case of a particle following a generic timelike geodesic about a highly spinning Kerr black hole. We provide numerical $C$ code for computing an effective source for various orbital configurations about Schwarzschild and Kerr black holes.

Figure 1

The singular field at the point $x$ can be expressed in terms of the retarded and advanced distances to the world line, $\gamma$.

Figure 2

Relation between order of approximation to the singular field and smoothness of the corresponding effective source. Also shown is the relation to the coefficients $A_{\mu },B_{\mu },C_{\mu },D_{\mu },\cdots$ in the large-$l$ expansion of the singular field used in the mode-sum regularization method [21].

Figure 3

Coordinate series expansion approximation to the effective source close to the particle. The exact effective source (solid blue line) is well approximated in the region $\Delta r\lesssim 0.05M$ by the first term in its series expansion (dashed purple line). Including the second (dashed gold line) and third (dashed green line) terms further improves the agreement for points farther from the particle, with the curves being almost indistinguishable from the exact curve.

Figure 4

Singular field (top) and effective source (bottom) along the equatorial plane for a particle in a circular orbit around a Schwarzschild black hole. From left to right: first-, second-, third-, and fourth-order cases are shown. Note that in the third- and fourth-order cases, we used the method described in Sec. 3b to ensure periodicity in $\varphi$.

Figure 5

Close-up view of the effective source along the equatorial plane (top) and along a radial slice through the particle (bottom) for a particle in a $r=10M$ circular orbit around a Schwarzschild black hole. From left to right: first-, second-, third-, and fourth-order cases are shown.

Figure 6

Particle following a circular equatorial geodesic around a Kerr black hole with spin $a=0.99M$. Left to right: (a) fourth-order singular field, (b) fourth-order effective source, (c) close-up view of the effective source, and (d) effective source along a radial slice through the particle. In (d), we also show the Schwarzschild result as a dashed red line for comparison. Note that we used the method described in Sec. 3b to ensure periodicity in $\varphi$.

Figure 7

Particle following generic geodesic around a Kerr black hole with spin $a=0.99M$. Left to right: (a) particle’s world line (solid line) with position at which the singular field and effective source are computed indicated by a black dot, (b) fourth-order singular field, (c) fourth-order effective source, and (d) close-up view of the effective source. Note that we used the method described in Sec. 3b to ensure periodicity in $\varphi$.