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arXiv 2108.11274gr-qchep-th
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极端Kerr黑洞时空中的Hawking效应

Hawking effect in an extremal Kerr black hole spacetime

Saumya Ghosh, Subhajit Barman

备注：
Modified version, to appear in Phys. Rev. D
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摘要：

众所周知，极值黑洞不具有霍金辐射，这通常是通过从非极值情形中取极值极限来实现的。然而，人们不能从极端黑洞本身开始使用Bogoliubov变换方法来感知相同的现象，即，而不考虑限制情况。在这种情况下，Bogoliubov系数不满足所需的归一化条件。在正则公式中，它密切模仿Bogoliubov变换方法，人们可以一致地再现霍金量子的消失数密度的极值克尔黑洞。在这种方法中，空间近零坐标之间的关系，理解霍金效应是必要的，近似为线性和逆项的总和。在目前的工作中，我们首先表明，一个可以达到相同的结论，在规范制定，即使与近零坐标，其中包含一个额外的对数项之间的完整关系。值得一提的是，在非极值情况下，类似的对数项单独导致热霍金辐射。其次，我们研究了关系中只有逆项的情况（即，当与过去和未来观察者相关联的空间近零坐标彼此反向相关时），以理解它是否是消失数密度中的主要贡献项。第三，对于定性的实现，我们考虑一个简单的思想实验来理解相应的霍金温度，并得出结论，逆项确实在消失数密度中起着至关重要的作用。

英文摘要：

It is well known that extremal black holes do not Hawking radiate, which is usually realized by taking an extremal limit from the nonextremal case. However, one cannot perceive the same phenomenon using the Bogoliubov transformation method starting from an extremal black hole itself, i.e., without the limiting case consideration. In that case, the Bogoliubov coefficients do not satisfy the required normalization condition. In canonical formulation, which closely mimics the Bogoliubov transformation method, one can consistently reproduce the vanishing number density of Hawking quanta for an extremal Kerr black hole. In this method, the relation between the spatial near-null coordinates, imperative in understanding the Hawking effect, was approximated into a sum of linear and inverse terms only. In the present work, we first show that one can reach the same conclusion in canonical formulation even with the complete relationship between the near-null coordinates, which contains an additional logarithmic term. It is worth mentioning that in the nonextremal case, a similar logarithmic term alone leads to the thermal Hawking radiation. Secondly, we study the case with only the inverse term in the relation (i.e., when the spatial near-null coordinates associated to the past and future observers are inversely related to each other) to understand whether it is the main contributing term in vanishing number density. Third, for a qualitative realization, we consider a simple thought experiment to understand the corresponding Hawking temperature and conclude that the inverse term indeed plays a crucial role in the vanishing number density.

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