# Mode-sum Regularization

The mode-sum regularization approach to self-force calculations relies on using an analytically computed approximation to the Detweiler-Whiting singular field to derive so-called mode-sum regularization parameters. On this page, I will provide a collection of codes and Mathematica scripts which are useful for mode-sum regularization calculations.

## Singular field of a point particle

The derivation of regularization parameters requires expansions of the singular field in the distance from the world-line. These expansions involve expressions which are too long to be included in a paper. Instead, we give them here as Mathematica code. For a full description of how these are calculated, see Phys. Rev. D 82, 104023 (2012).

For each case, we give a bzipped Mathematica package file containing the definition of a symbol of the same name. Tensors of rank n are given as n-dimensional lists, the elements of which correspond to the t, r, w1 and w2 components of the tensor. Each element is a SeriesData object in powers of ε, starting at order 1/ε. For example, to obtain the order 1/ε contribution to the t,t component of the singular metric perturbation:

```Import["hS.m.bz2"];
Coefficient[hS[[1, 1]], \[Epsilon], -1]```

The following expressions are available:

• hS.m.bz2: Singular metric perturbation for a generic geodesic in Schwarzschild spacetime written in Riemann normal coordinates to order ε^4.

## Regularization parameters for Schwarzschild spacetime

High order regularization parameters for the Schwarzschild spacetime were derived in Phys. Rev. D 82, 104023 (2012). for the case of a generic geodesic orbit. The expressions are given in Section V of the paper. However, as they are quite long, we are providing them here in a more convenient electronic form as a Mathematica notebook. We also include regularization parameters for other cases not explicitly given in the paper.

## Regularization parameters for Kerr spacetime

High order regularization parameters for the Kerr spacetime were derived in Phys. Rev. D 89, 024030 (2014) for the case of a generic equatorial geodesic orbit. Some expressions are given in the paper. However, as they are quite long, we are providing them here in a more convenient electronic form as a Mathematica notebook. We also include regularization parameters for other cases not explicitly given in the paper.

## One thought on “Mode-sum Regularization”

1. Abhay Shah says:

Hey maan!
Thanks for the Mathematica files.