arXiv 0708.20132007-08-15gr-qc

黑洞蒸发中的量子振幅：自旋1和2

Quantum amplitudes in black-hole evaporation: Spins 1 and 2

A. N. St. J. Farley, P. D. D'Eath

发表信息：: AnnalsPhys.321:1334-1374,2006
DOI：: 10.1016/j.aop.2005.11.011

摘要：

$s=1$在麦克斯韦场和$s=2$线性引力波扰动的球对称爱因斯坦/无质量标量背景，描述引力坍缩到黑洞的量子振幅，处理类比与以前的治疗$s=0$标量场扰动的引力坍缩在后期。在这两个$s=1$和$s=2$的情况下，我们隔离合适的“坐标”变量，可以作为边界数据上的最终类空超曲面$Σ_F$。为了简单起见，我们采取的数据在一个初始的预塌缩表面$Σ_I$是精确的球对称。在空间无穷远处测量的$Σ_{I}, Σ_{F}$之间的（大）洛伦兹固有时间间隔用$T$表示。假设时间间隔$T$已旋转到复形中：$T\to{\mid}T{\mid}\exp(-iθ)$，对于$0<θ\leqπ/2$，则复杂化的经典边值问题是适定的。我们计算了二阶变分经典Lorenztian作用量$S ^{(2)}_{\rm class}$。在费曼的基础上，我们将半经典振幅$\exp(iS^{(2)}_{\rm class})$的极限作为$θ\to 0_+$，恢复了洛伦兹量子振幅。$ s=1$的边界数据涉及麦克斯韦磁场; $s=2$的数据涉及外尔曲率张量的磁性部分。磁边界条件相互关联，并通过超对称性与自然$s={1 \over 2}$边界条件关联。

英文摘要：

Quantum amplitudes for $s=1$ at Maxwell fields and for $s=2$ linearised gravitational wave perturbations of a spherically symmetric Einstein/massless scalar background, describing gravitational collapse to a black hole, are treated by analogy with a previous treatment of $s=0$ scalar-field perturbations of gravitational collapse at late times. In both the $s=1$ and $s=2$ cases, we isolate suitable 'co-ordinate' variables which can be taken as boundary data on a final space-like hypersurface $Σ_F$. For simplicity, we take the data on an initial pre-collapse surface $Σ_I$ to be exactly spherically symmetric. The (large) Lorentzian proper-time interval between $Σ_{I}, Σ_{F}$, measured at spatial infinity, is denoted by $T$. The complexified classical boundary-value problem is expected to be well-posed, provide that the time interval $T$ has been rotated into the complex: $T\to{\mid}T{\mid}\exp(-iθ)$, for $0<θ\leqπ/2$. We calculate the second-variation classical Lorenztian action $S ^{(2)}_{\rm class}$. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as $θ\to 0_+$ of the semi-classical amplitude $\exp(iS^{(2)}_{\rm class})$. The boundary data for $ s=1$ involve the Maxwell magnetic field; the data for $s=2$ involve the magnetic part of the Weyl curvature tensor. The magnetic boundary conditions are related to each other and to the natural $s={1 \over 2}$ boundary conditions by supersymmetry.

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