Note on Legendre decomposition of the Pontryagin density in Kerr

TL;DR

This paper simplifies the Legendre polynomial decomposition of the Pontryagin density in Kerr black holes within Chern-Simons gravity, making the analytical expressions more compact and computationally accessible.

Contribution

It provides a concise, rational polynomial-based expression for the Legendre decomposition of the Pontryagin density, improving upon previous complex quadruple sum formulations.

Findings

Simplified the Legendre decomposition expression.

Expressed the decomposition using rational polynomials and special functions.

Facilitated easier analytical and numerical calculations.

Abstract

In arXiv:1406.0957v1 ("Scalar field excited around a rapidly rotating black hole in Chern-Simons modified gravity"), Konno and Takahashi have recently developed some analytical results for the scalar field about a Kerr black hole in the decoupling limit of dynamical Chern-Simons gravity. This involved a decomposition of the source (the Pontryagin density) in terms of Legendre polynomials. Here we give a two-line expression for this decomposition which simplifies their quadruple sum. Our expressions are rational polynomials multiplying Legendre functions of the second kind, or equivalently rational polynomials multiplying hypergeometric functions.