[Paper Review] Regularization parameters for the self-force of a scalar particle in a general orbit about a Schwarzschild black hole

Dong-Hoon Kim|arXiv (Cornell University)|Jan 25, 2010

Astrophysical Phenomena and Observations1 references3 citations

TL;DR

This paper derives analytical regularization parameters for computing the scalar self-force on a particle in generic orbit around a Schwarzschild black hole using a Dirac-like splitting of the retarded field into singular and regular parts. By employing a multipole decomposition of the singular field and expressing the regularization parameters in terms of complete elliptic integrals, the method enables accurate mode-sum self-force calculations, with results matching prior work by Barack and Ori.

ABSTRACT

The interaction of a charged particle with its own field results in the "self-force" on the particle, which includes but is more general than the radiation reaction force. In the vicinity of the particle in curved spacetime, one may follow Dirac and split the retarded field of the particle into two parts, (1) the singular source field, \sim q/r, and (2) the regular remainder field. The singular source field exerts no force on the particle, and the self-force is entirely caused by the regular remainder. We describe an elementary multipole decomposition of the singular source field which allows for the calculation of the self-force on a scalar-charged particle orbiting a Schwarzschild black hole.

Motivation & Objective

• To develop a physically intuitive method for computing the scalar self-force on a particle in generic orbit around a Schwarzschild black hole by adapting Dirac’s field splitting approach.
• To analytically compute the multipole moments of the singular source field, known as regularization parameters, which are essential for mode-sum self-force calculations.
• To express these regularization parameters in terms of complete elliptic integrals of the first and second kind for efficient numerical evaluation.
• To ensure consistency with known results by verifying agreement with Barack and Ori (2002) in the limit of circular orbits and specific parameter regimes.
• To provide a framework that avoids non-differentiable tail terms by using a regularized field decomposition that mirrors Dirac’s radiation field analogy.

Proposed method

• Adopt the Detweiler-Whiting regularization scheme, splitting the retarded scalar field into a singular source field (ψ^S) and a regular remainder field (ψ^R), where ψ^R is responsible for the self-force.
• Perform a spherical harmonic multipole decomposition of both ψ^ret and ψ^S to extract mode-by-mode components, enabling the mode-sum method for self-force evaluation.
• Derive analytical expressions for the multipole moments of ψ^S (the regularization parameters) using the particle’s orbital parameters and the background curvature invariants.
• Express the regularization parameters in terms of hypergeometric functions and complete elliptic integrals (K(α), E(α)) via angular averaging over the orbit.
• Utilize the external multipole moments of the Riemann tensor (E_IJ, B_IJ, E_IJK, B_IJK) evaluated on the worldline to construct the regularization parameters.
• Apply symmetry and trace-free (STF) tensor techniques to ensure proper transformation properties and consistency with the background vacuum geometry.

Experimental results

Research questions

• RQ1 How can the self-force on a scalar particle in a general orbit around a Schwarzschild black hole be computed using a physically transparent regularization scheme?
• RQ2 What are the analytical expressions for the regularization parameters (multipole moments of the singular field) in terms of orbital parameters and curvature invariants?
• RQ3 Can the regularization parameters be expressed in terms of standard special functions like complete elliptic integrals to enable efficient numerical evaluation?
• RQ4 How do the derived regularization parameters compare with existing results in the literature, particularly for circular orbits or known limiting cases?
• RQ5 Does the proposed method avoid the differentiability issues associated with tail terms in the standard self-force formalism?

Key findings

• The regularization parameters for the scalar self-force are derived analytically in terms of complete elliptic integrals K(α) and E(α), where α = J²/(r₀² + J²).
• The expressions for the non-zero B-terms (B_t, B_r, B_ϕ) are explicitly given in terms of K(α) and E(α), matching the results of Barack and Ori (2002) for circular orbits.
• The derivation confirms that the regularization parameters scale appropriately with orbital radius r₀ and angular momentum J, with B_t ∝ 1/r₀², B_r ∝ 1/r₀², and B_ϕ ∝ 1/r₀.
• The method ensures that the regular remainder field is a homogeneous solution of the scalar wave equation, consistent with Dirac’s radiation field analogy.
• The use of angular averaging via 〈χ^(-p)〉 = ₂F₁(p, ½; 1; α) enables exact analytical treatment of the multipole moments without numerical averaging.
• The final expressions for the regularization parameters are dimensionally consistent and tracefree, reflecting the vacuum nature of the Schwarzschild background.