An integral representation for the massive Dirac propagator in Kerr geometry in Eddington-Finkelstein-type coordinates
An integral representation for the massive Dirac propagator in Kerr geometry in Eddington-Finkelstein-type coordinates
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DOI：
10.4310/atmp.2018.v22.n1.a3
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发表时间：
2016-06
期刊：
影响因子：
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通讯作者：
F. Finster;Christian Roken
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文献类型：
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作者：
F. Finster;Christian Roken
在穿透视界的Eddington-Finkelstein型坐标下，在非极端Kerr几何中考虑了质量Dirac方程。我们得到了狄拉克传播子的积分表示，其中包含了在Chandrasekhar分离变量中出现的常微分方程组的解。这个积分表示描述了狄拉克波在事件视界之外和跨越事件视界直到柯西视界的动力学。作为证明，我们将狄拉克方程写成哈密顿形式。Dirac的主要缺点之一是时间演化不是么正的，因为波可能击中奇点。这个问题是通过在柯西视界内施加适当的Dirichlet型边界条件而不影响外部动力学来解决的。另一个主要困难是Dirac哈密顿量在视界处不是椭圆的。结合对称双曲组理论和边界附近的椭圆方法，我们构造了所得到的哈密顿量的自伴扩张。最后，我们将斯通公式应用于哈密顿量的谱测量，并用分离的微分方程组的解来表示预解。
The massive Dirac equation is considered in the non-extreme Kerr geometry in horizon-penetrating Eddington-Finkelstein-type coordinates. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's separation of variables. This integral representation describes the dynamics of Dirac waves outside and across the event horizon, up to the Cauchy horizon. For the proof, we write the Dirac equation in Hamiltonian form. One of the main di�culties is that the time evolution is not unitary, because the wave may \hit" the singularity. This problem is resolved by imposing suitable Dirichlet-type boundary conditions inside the Cauchy horizon, having no effect on the outside dynamics. Another main difficulty is that the Dirac Hamiltonian fails to be elliptic at the horizons. Combining the theory of symmetric hyperbolic systems with elliptic methods near the boundary, we construct a self-adjoint extension of the resulting Hamiltonian. We finally apply Stone's formula to the spectral measure of the Hamiltonian and express the resolvent in terms of solutions of the separated ODEs.