von-Neumann stability and singularity resolution in loop quantized Schwarzschild black hole
von-Neumann stability and singularity resolution in loop quantized Schwarzschild black hole
复制标题
循环量子化史瓦西黑洞中的冯诺依曼稳定性和奇点分辨率
DOI：
10.1088/1361-6382/aaa18d
复制
发表时间：
2018
影响因子：
3.5
通讯作者：
Singh, Parampreet
中科院分区：
文献类型：
--
作者：
Yonika, Alec;Khanna, Gaurav;Singh, Parampreet
尽管几个时空的循环量子化已经通过数值模拟的显式状态演化展示了反弹的存在，但关于黑洞内部中心奇点是如何解决的问题仍然悬而未决。循环量子化中的量子哈密顿约束被证明是一个有限差分方程，了解它的稳定性对于深入了解基本量子化的可行性和由此产生的物理含义是很重要的。对于最近由Corichi和Singh给出的Schwarzschild内部的循环量子化，我们迈出了解决这些问题的第一步。利用解的可分性和全二维量子差分方程组进行了von-Neumann稳定性分析。这导致了黑洞的稳定性条件，与普朗克质量相比，黑洞的质量非常大。对于质量较小的黑洞，发现了数值不稳定性的证据。此外，宏观黑洞的稳定性分析导致了在数值演化中对允许态的选择的限制。没有按照这一约束急剧达到峰值的状态会导致不稳定性。在使用运动学范数的前提下，利用量子差分方程解出了尖峰高斯态，并得到了奇异性分辨率。发现其中一个三和弦变量存在反弹，但对于另一个三和弦变量，奇异性分辨率相当于通过零体积的非奇异通道。在奇点分辨前后的很长一段时间内，态都在经典轨道处达到峰值，并在零体积范围内保持其半经典性质。我们的主要结果是量子反弹发生在回路量子化的Schwarzschild内部，至少对于宏观黑洞是这样。小黑洞的不稳定性可能是使用运动学范数的结果，然而，这意味着需要进一步了解所考虑的量子化及其物理希尔伯特空间的可行性。
Though loop quantization of several spacetimes has exhibited existence of a bounce via an explicit evolution of states using numerical simulations, the question about the way central singularity is resolved in the black hole interior has remained open. The quantum Hamiltonian constraint in loop quantization turns out to be a finite difference equation whose stability is important to understand to gain insights on the viability of the underlying quantization and resulting physical implications. We take first steps towards addressing these issues for a loop quantization of the Schwarzschild interior recently given by Corichi and Singh. Von-Neumann stability analysis is performed using separability of solutions as well as a full two dimensional quantum difference equation. This results in a stability condition for black holes which have a very large mass compared to the Planck mass. For black holes of smaller masses evidence of numerical instability is found. In addition, stability analysis for macroscopic black holes leads to a constraint on the choice of the allowed states in numerical evolution. States which are not sharply peaked in accordance with this constraint result in instabilities. With the caveat of using kinematical norm, sharply peaked Gaussian states are evolved using the quantum difference equation and singularity resolution is obtained. A bounce is found for one of the triad variables, but for the other triad variable singularity resolution amounts to a non-singular passage through the zero volume. States are found to be peaked at the classical trajectory for a long time before and after the singularity resolution, and retain their semi-classical character across the zero volume. Our main result is that quantum bounce occurs in loop quantized Schwarzschild interior at least for macroscopic black holes. Instability of small black holes which can be a result of using kinematical norm nevertheless signifies the need of further understanding of the viability of the considered quantization and its physical Hilbert space.
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