Simple, efficient method of calculating the Detweiler-Whiting singular field to very high order

Most self-force calculations rely, in one way or another, on representations of a particle's Detweiler-Whiting singular field.We present a simple method of calculating the singular field to high order in a local expansion in powers of distance from the particle.As a demonstration, we compute the singular field to 14th order in distance, 10 orders beyond the previous state of the art, in the simple case of a scalar charge in circular orbit around a Schwarzschild black hole.We provide the result in both a 4-dimensional form and a decomposed form suitable for use in an m-mode puncture scheme.Our method should have applications in overcoming bottlenecks in current self-force calculations at both first and second order in perturbation theory. CONTENTSI. Introduction 1 II.Puncture schemes, the Detweiler-Whiting singular field, and problems of slow convergence 3 A. Formulation of a puncture scheme 3 B. Local expansion of Φ P via the Detweiler-Whiting Green's function 3 C. Accelerating convergence using high-order punctures 6 III.New algorithm for computing the singular field 7 A. Comoving coordinates 7 B. The field equation 9 C. Spectral decomposition and computation of the modes 9 IV.Results 10 A. The puncture field Φ P 10 B. m-modes in the black-hole frame 11 C.The effective source S eff 14 D. Regularity of S eff m at the particle 14 V. Conclusion 16 20 C. Mode coefficients Φ