关于Kerr中Pontryagin密度的Legendre分解的注记
Note on Legendre decomposition of the Pontryagin density in Kerr
摘要：
在arXiv：1406.0957v1（“在Chern-Simons修正引力中围绕一个快速旋转黑洞激发的标量场”）中，Konno和Takahashi最近在动力学Chern-Simons引力的退耦极限中对Kerr黑洞周围的标量场进行了分析.这涉及到一个分解的来源（庞特里亚金密度）在勒让德多项式。在这里，我们给出了一个两行的表达式，这种分解简化了他们的四重和。我们的表达式是有理多项式乘以第二类勒让德函数，或等价的有理多项式乘以超几何函数。
英文摘要：
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.
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