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arXiv 1503.01774gr-qchep-th
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黑洞事件视界内部的量子窥视

A quantum peek inside the black hole event horizon

Sumanta Chakraborty, Suprit Singh, T. Padmanabhan

摘要:

我们求解标量场的克莱因-戈登方程,背景几何是坍缩形成黑洞的尘埃云,在(1+1)时空的任何地方:即事件视界内外,任意接近曲率奇点。这使我们能够确定正则化的应力张量期望值,在适当的量子态中的任何地方(即,(Unruh真空)。我们用它来研究的行为能量密度和流量测量的径向自由落体观测器在任何给定的事件在本地惯性系。在黑洞之外,随着坍缩的进行,能量密度和通量导致了从黑洞发出的霍金辐射所预期的标准结果。在坍缩的尘埃球内部,物质和标量场的能量密度在(1+1)和(1+3)时空维的奇点附近发散;但是场的能量密度超过经典物质的能量密度。在(1+3)维中,奇点周围的小空间体积内的总能量(标量场和经典物质)是有限的(并且随着区域的大小变为零而变为零),但是量子场的总能量仍然占主导地位。在事件视界内部,但\textit{outside}是坍缩的物质,自由下落的观察者发现能量密度和通量在奇点附近发散。在这个区域中,即使是包含奇点的小空间体积内的积分能量也会发散。这个结果在(1+1)和(1+3)维时空中都成立,在(1+3)维的一个小区域内总能量存在\emph{milder}发散。这些结果表明,即使在区域\emph{outside the matter but inside the event horizon},接近奇点的反反应效应是显着的。

英文摘要:

We solve the Klein-Gordon equation for a scalar field, in the background geometry of a dust cloud collapsing to form a black hole, everywhere in the (1+1) spacetime: that is, both inside and outside the event horizon and arbitrarily close to the curvature singularity. This allows us to determine the regularized stress tensor expectation value, everywhere in the appropriate quantum state (viz., the Unruh vacuum) of the field. We use this to study the behaviour of energy density and the flux measured in local inertial frames for the radially freely falling observer at any given event. Outside the black hole, energy density and flux lead to the standard results expected from the Hawking radiation emanating from the black hole, as the collapse proceeds. Inside the collapsing dust ball, the energy densities of both matter and scalar field diverge near the singularity in both (1+1) and (1+3) spacetime dimensions; but the energy density of the field dominates over that of classical matter. In the (1+3) dimensions, the total energy (of both scalar field and classical matter) inside a small spatial volume around the singularity is finite (and goes to zero as the size of the region goes to zero) but the total energy of the quantum field still dominates over that of the classical matter. Inside the event horizon, but \textit{outside} the collapsing matter, freely falling observers find that the energy density and the flux diverge close to the singularity. In this region, even the integrated energy inside a small spatial volume enclosing the singularity diverges. This result holds in both (1+1) and (1+3) spacetime dimensions with a \emph{milder} divergence for the total energy inside a small region in (1+3) dimensions. These results suggest that the back-reaction effects are significant even in the region \emph{outside the matter but inside the event horizon}, close to the singularity.

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