arXiv 2006.160172020-06gr-qc

紧双曲坐标下Kerr几何中Dirac方程可分性的挑战

On Challenges to Separability of the Dirac Equation in Kerr Geometry under Compact Hyperboloidal Coordinates

Aditya Tamar

备注：: 10 pages, Submitted to IJP. Comments on the validity of the result are especially encouraged

摘要：

狄拉克方程控制自旋为1/2的粒子的行为。该方程的分离成解耦的径向和角向微分方程是一个关键步骤，在分析和数值计算的数量，如本征值，准正规模式和束缚态。然而，这种分离是在坐标系中进行的，该坐标系在$r \rightarrow r_{horizon}$、$r \rightarrow r_\infty$或两者的任一限制区域中表现不佳。特别是，广泛使用的Boyer-Lindquist坐标包含了$r_{horizon}$和$r_\infty$时空几何的非物理特征。因此，最近发展的紧凑的双曲面坐标系统的克尔黑洞，是很好的行为，在这些限制区域的激励下，我们尝试的Dirac方程的分离。我们首先构造了一个适合于Newman-Penrose形式主义下的可分性分析的零四分体。然后，我们发现了一个意想不到的结果，即在这种四元数下，使用基于模的标准可分性方法，Dirac方程不会解耦为径向和角度方程，并讨论了这种现象的可能原因以及证明可分性对各种计算的重要性。

英文摘要：

The Dirac equation governs the behaviour of spin-1/2 particles. The equation's separability into decoupled radial and angular differential equations is a crucial step in analytical and numerical computations of quantities like eigenvalues, quasinormal modes and bound states. However, this separation has been performed in co-ordinate systems that are not well-behaved in either limiting regions of $r \rightarrow r_{horizon}$, $r \rightarrow r_\infty$ or both. In particular, the extensively used Boyer-Lindquist co-ordinates contains unphysical features of spacetime geometry for both $r_{horizon}$ and $r_\infty$. Therefore, motivated by the recently developed compact hyperboloidal co-ordinate system for Kerr Black Holes that is well behaved in these limiting regions, we attempt the separation of the Dirac equation. We first construct a null tetrad suitable for the separability analysis under the Newman-Penrose formalism. Then, an unexpected result is shown that by using the standard separability procedure based on the mode ansatz under this tetrad, the Dirac equation does not decouple into radial and angular equationsPossible reasons for this behaviour as well as importance of proving separability for various computations are discussed.

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