论文标题
关于在紧凑型倍或坐标下KERR几何形状中Dirac方程可分离性的挑战
On Challenges to Separability of the Dirac Equation in Kerr Geometry under Compact Hyperboloidal Coordinates
论文作者
论文摘要
狄拉克方程控制了自旋1/2颗粒的行为。该方程的分离性在脱钩的径向和角度微分方程中是对特征值，准模式和结合状态等数量分析和数值计算的关键步骤。但是，这种分离是在$ r \ rightarrow r_ {horizon} $，$ r \ rightarrow r_ \ r_ \ infty $或两者兼而有之的任何一个限制区域中进行的坐标。特别是，广泛使用的Boyer-Lindquist坐标包含$ r_ {horizon} $和$ r_ \ infty $的时空几何特征。因此，由最近在这些限制区域中表现良好的Kerr黑洞的最近开发的紧凑型倍层叠加坐标系统的动机，我们尝试将Dirac方程的分离。我们首先构建一个适用于纽曼 - 芬罗形式主义下的可分离性分析的零四个。然后，出乎意料的结果表明，通过基于该四分法下的模式ANSATZ使用标准可分离性过程，狄拉克方程不会将这种行为的径向和角度方程式分解为径向和角度方程，并且讨论了对各种计算的可分离性的重要性。
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.
