arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1104.2772gr-qcmath-phmath.APmath.MP
收藏

Schwarzschild几何中无质量Dirac波的渐近行为

Asymptotic Behavior of Massless Dirac Waves in Schwarzschild geometry

Joel Smoller, Chunjing Xie

备注：
45 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文证明了Schwarzschild几何中的无质量Dirac波以$t^{-2λ}$的速率衰减到零，其中$λ=1, 2,...$是角动量。我们的技术是使用Eschrasekhar的分离变量，从而狄拉克方程分裂成两套波动方程。对于第一组，我们证明了波衰减为$t^{-2λ}$。对于第二组，一般而言，解倾向于速率为$t^{-2λ}$的某些显式轮廓。狄拉克方程的解的衰减率是通过证明显式轮廓的系数恰好为零来实现的。证明第一组波动方程解的衰减率的关键成分是用于显示不存在束缚态和零能量共振的能量估计以及解的谱表示的分析。第二组波动方程解的渐近性态的证明依赖于对与这些波动方程相关的与时间无关的薛定谔方程的格林函数的仔细分析。

英文摘要：

In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate $t^{-2λ}$, where $λ=1, 2,...$ is the angular momentum. Our technique is to use Chandrasekhar's separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as $t^{-2λ}$. For the second set, in general, the solutions tend to some explicit profile at the rate $t^{-2λ}$. The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredients in the proof of the decay rate of solutions for the first set of wave equations are an energy estimate used to show the absence of bound states and zero energy resonance and the analysis of the spectral representation of the solutions. The proof of asymptotic behavior for the solutions of the second set of wave equations relies on careful analysis of the Green's functions for time independent Schrödinger equations associated with these wave equations.

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。

↑