Analytic treatment of black-hole gravitational waves at the algebraically special frequency
Analytic treatment of black-hole gravitational waves at the algebraically special frequency
复制标题
代数特殊频率黑洞引力波的解析处理
DOI：
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发表时间：
2000
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通讯作者：
A. M. V. D. Brink
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文献类型：
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作者：
A. M. V. D. Brink
我们研究了Regge-Wheeler（RWE）和Zerilli方程（ZE）在“代数特殊频率”Omega，这些方程承认一个精确解（在这里详细说明），产生它们之间的超对称（SUSY）关系。超对称性发生器的物理意义和一般的解决方案在欧米茄阐明如下。RWE根本没有（准正规或全透射）模式;然而，Ω仍然是“特殊的”，因为（a）对于进入视界的输出波，由于指数势尾，预期的发散有一个“奇迹般的”抵消，（B）在Ω = Ω处的分支切割不连续性在输出波中消失到无穷大。此外，（a）和（B）是相关的。对于ZE，它的唯一模式是逆超对称发生器，它同时是一个准正规模式和传播到无穷远的全透射模式。这些发现的微妙之处（对未来研究负假想欧米茄轴上或附近的方程具有普遍意义）可能有助于解释为什么这种情况有时会引起争议。对于有限的黑洞旋转，代数特殊模式被证明是完全传输的，并澄清了史瓦西极限的隐含奇异性。该分析借鉴了最近在开放系统中对超对称性的详细研究[math-ph/9909030]。
We study the Regge-Wheeler (RWE) and Zerilli equations (ZE) at the "algebraically special frequency" Omega, where these equations admit an exact solution (elaborated here), generating the supersymmetry (SUSY) relationship between them. The physical significance of the SUSY generator and of the solutions at Omega in general is elucidated as follows. The RWE has no (quasinormal or total-transmission) modes at all; however, Omega is nonetheless "special" in that (a) for the outgoing wave into the horizon one has a "miraculous" cancellation of a divergence expected due to the exponential potential tail, and (b) the branch-cut discontinuity at omega = Omega vanishes in the outgoing wave to infinity. Moreover, (a) and (b) are related. For the ZE, its only mode is the-inverse-SUSY generator, which is at the same time a quasinormal mode and a total-transmission mode propagating to infinity. The subtlety of these findings (of general relevance for future study of the equations on or near the negative imaginary omega-axis) may help explain why the situation has sometimes been controversial. For finite black-hole rotation, the algebraically special modes are shown to be totally transmitting, and the implied singular nature of the Schwarzschild limit is clarified. The analysis draws on a recent detailed investigation of SUSY in open systems [math-ph/9909030].