High-order expansions of the Detweiler-Whiting singular field in Schwarzschild spacetime

The self-field of a charged particle has a singular component which diverges at the particle. We use both coordinate and covariant approaches to compute an expansion of this singular field for particles in generic geodesic orbits about a Schwarzschild black hole for scalar, electromagnetic and gravitational cases. We check that both approaches yield identical results and give, as an application, the calculation of previously unknown mode-sum regularization parameters. In the so-called mode-sum regularization approach to self-force calculations, each mode of the retarded field is finite, while their sum diverges. The sum may be rendered finite and convergent by the subtraction of appropriate regularization parameters. Higher-order parameters lead to faster convergence in the mode sum. To demonstrate the significant benefit which they yield, we use our newly derived parameters to calculate a highly accurate value of $F_r=0.000013784482575667959(3)$ for the self-force on a scalar particle in a circular orbit around a Schwarzschild black hole. Finally, as a second example application of our high-order expansions, we compute high-order expressions for use in the effective source approach to self-force calculations.

Figure 1

We expand all bitensors (which are functions of $x$ and $x^{\prime }$) about the arbitrary point $\overline{x}$ on the world line.

Figure 2

Regularization of the radial component of the self-force for the case of a scalar particle on a circular geodesic of radius $r_0=10M$ in Schwarzschild spacetime. In decreasing slope, the above lines represent the unregularized self-force, self-force regularized by subtracting from it in turn the cumulative sum of $F_{r[-1]}^l$, $F_{r[0]}^l$, $F_{r[2]}^l$, $F_{r[4]}^l$, $F_{r[6]}^l$, $F_{r[8]}^l$, $F_{r[10]}^l$, $F_{r[12]}^l$.

Figure 3

Terms in the coordinate expansion of the singular field for $\mathcal{O}({ϵ}^{-1})$ (top left) to $\mathcal{O}({ϵ}^6)$ (bottom right). Shown is the case of a circular geodesic of radius $r_0=10M$ in Schwarzschild spacetime.

Figure 4

Terms in the coordinate expansion of the singular field, $[{\Phi }^{(S)}{]}_{(n)}$, in the region of the particle for $\mathcal{O}({ϵ}^{-1})$ (top left) to $\mathcal{O}({ϵ}^6)$ (bottom right). Shown is the case of a circular geodesic of radius $r_0=10M$ in Schwarzschild spacetime.

Figure 5

Effective source for the approximate singular field of order $\mathcal{O}({ϵ}^{-1})$ (top left) to $\mathcal{O}({ϵ}^6)$ (bottom right). Shown is the case of a circular geodesic of radius $r_0=10M$ in Schwarzschild spacetime.

Figure 6

Effective source for the approximate singular field, $S^{\mathrm{eff}}{}_{(n)}$, in the region of the particle of order $\mathcal{O}({ϵ}^{-1})$ (top left) to $\mathcal{O}({ϵ}^6)$ (bottom right). Shown is the case of a circular geodesic of radius $r_0=10M$ in Schwarzschild spacetime.