[Paper Review] Radiation reaction in curved even-dimensional spacetime

D.V. Gal’tsov, Pavel Spirin|arXiv (Cornell University)|Dec 14, 2010

Cosmology and Gravitation Theories9 references3 citations

TL;DR

This paper presents a novel local method to compute radiation reaction for point particles coupled to massless scalar and vector fields in even-dimensional curved spacetime, bypassing non-local tail integrals via Green's function analysis on the worldline. The key result is the discovery of a Riemann tensor-dependent term in the local reaction force in six dimensions—absent in four dimensions—alongside curvature-dependent higher-derivative counterterms that generalize flat-space renormalization.

ABSTRACT

We develop a new method of computing radiation reaction for a point particle interacting with massless scalar and vector fields in curved space-time. It is based on the analysis of field Green's functions with both points lying on the particle world-line and does not require integration of the field stresses outside the world line as was used in the DeWitt-Brehme approach, thus leading to a substantial simplification of the problem. We start with space-time of an arbitrary dimension and show that the Hadamard expansion of the massless scalar and vector Green's functions contain only integer inverse powers of the Synge world function in even dimensions and only half-integer in the odd dimensions. The even-dimensional case then is treated in detail. We analyze divergencies, calculate higher-derivative counterterms, and find a recurrent formula for the local parts of the reaction force in neighboring dimensions. Higher-dimensional curved space counterterms are not simply the covariant generalizations of the flat ones, but contain additional curvature-dependent terms. We illustrate our formalism in four and six dimensions. In the first case we rederive the results of DeWitt-Brehme-Hobbs in a simpler way, in the second case we give a covariant generalization of the Kosyakov equation. The local part of the reaction force is found to contain a term proportional to the Riemann tensor which is absent in four dimensions.

Motivation & Objective

• To develop a simplified, purely local approach to compute radiation reaction in curved spacetime, avoiding the non-local tail integrals of the DeWitt-Brehme method.
• To analyze the structure of massless scalar and vector Green's functions in arbitrary even dimensions using the Hadamard expansion.
• To identify and compute higher-derivative counterterms in curved spacetime that renormalize the radiation reaction force.
• To derive a recurrent formula for the local part of the reaction force across neighboring even dimensions.
• To demonstrate the emergence of Riemann tensor-dependent terms in the local force in six dimensions, absent in four.

Proposed method

• The method is based on analyzing the Hadamard expansion of massless scalar and vector Green's functions with both points on the particle worldline.
• It avoids integration of field stresses outside the worldline, unlike the DeWitt-Brehme approach, leading to a significant simplification.
• The analysis shows that in even dimensions, the Green's function expansion contains only integer inverse powers of the Synge world function.
• Divergences are regularized using higher-derivative counterterms derived from the curvature structure of spacetime.
• A recurrence relation is derived for the local part of the reaction force in neighboring even dimensions.
• Explicit calculations are performed in four and six dimensions to verify the formalism and derive covariant expressions.

Experimental results

Research questions

• RQ1 Can radiation reaction in curved spacetime be computed without non-local tail integrals by focusing only on the worldline?
• RQ2 How do the divergences in the Green's function expansion differ between even and odd dimensions?
• RQ3 What are the curvature-dependent higher-derivative counterterms required for renormalization in higher-dimensional curved spacetime?
• RQ4 Does the local part of the radiation reaction force in six dimensions contain terms involving the Riemann tensor, as seen in four dimensions?
• RQ5 How do the counterterms in curved spacetime generalize the flat-space higher-derivative terms in higher dimensions?

Key findings

• The method successfully computes the radiation reaction force in a purely local manner, eliminating the need for non-local tail integrals.
• In even dimensions, the Hadamard expansion of Green's functions contains only integer inverse powers of the Synge world function.
• The local part of the reaction force in six dimensions includes a term proportional to the Riemann tensor, which is absent in four dimensions.
• Higher-dimensional curved space counterterms are not simple covariantizations of flat-space terms but include additional curvature-dependent terms.
• In four dimensions, the formalism reproduces the DeWitt-Brehme-Hobbs result in a simpler way, confirming consistency.
• In six dimensions, the formalism provides a covariant generalization of the Kosyakov equation, including Riemann tensor contributions to the finite force.