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arXiv 0905.2974gr-qchep-ph
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旋转黑洞时空中的大质量Dirac场:角解

The massive Dirac field on a rotating black hole spacetime: Angular solutions

Sam Dolan, Jonathan Gair

摘要:

Kerr-Newman背景下的有质量Dirac方程可用分离变量法求解。径向和角度方程耦合通过角特征值,这是确定从Cuprasekhar-Page(CP)方程。获得精确的角本征值是研究费米子场的散射、吸收和辐射的关键步骤。本文介绍一种求解CP方程的新方法。首先,我们介绍了一种新的表示自旋半球谐波。接下来,我们分解的CP方程(质量相关的自旋半球谐函数)的角解的球基。该方法产生一个三项递归关系,可以通过连续分数方法数值求解,或微扰获得一系列扩展的特征值。在$μ= \pm ω$(其中$ω$和$μ$是费米子的频率和质量)的情况下,我们得到了封闭形式的本征值和本征函数。我们研究了最大同旋模的本征值谱和零点。我们将我们的结果与以前的研究进行了比较,发现并纠正了文献中的一些错误。我们提供了一系列的扩展,表的特征值和数值拟合在很宽的参数范围内,并提出了选择的特征函数的图。希望这项研究能为所有对旋转黑洞背景下狄拉克方程感兴趣的研究人员提供有用的资源。

英文摘要:

The massive Dirac equation on a Kerr-Newman background may be solved by the method of separation of variables. The radial and angular equations are coupled via an angular eigenvalue, which is determined from the Chandrasekhar-Page (CP) equation. Obtaining accurate angular eigenvalues is a key step in studying scattering, absorption and emission of the fermionic field. Here we introduce a new method for finding solutions of the CP equation. First, we introduce a novel representation for the spin-half spherical harmonics. Next, we decompose the angular solutions of the CP equation (the mass-dependent spin-half spheroidal harmonics) in the spherical basis. The method yields a three-term recurrence relation which may be solved numerically via continued-fraction methods, or perturbatively to obtain a series expansion for the eigenvalues. In the case $μ= \pm ω$ (where $ω$ and $μ$ are the frequency and mass of the fermion) we obtain eigenvalues and eigenfunctions in closed form. We study the eigenvalue spectrum, and the zeros of the maximally co-rotating mode. We compare our results with previous studies, and uncover and correct some errors in the literature. We provide series expansions, tables of eigenvalues and numerical fits across a wide parameter range, and present plots of a selection of eigenfunctions. It is hoped this study will be a useful resource for all researchers interested in the Dirac equation on a rotating black hole background.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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