Recent investigations revealed that the near horizon Hamiltonian of a massless, chargeless outgoing particle, for its particular motion in static as well as stationary black holes, is effectively \(\sim xp\) kind. This is unstable by nature and has the potential to explain a few interesting physical phenomena. From the path integral kernel, we first calculate the density of states. Also, following the idea of Singh and Padmanabhan (Phys Rev D 85:025011, 2012. https://doi.org/10.1103/PhysRevD.85.025011. arXiv:1112.6279 [hep-th]) here, in the vicinity of the horizon, we calculate the effective path corresponding to its Schrodinger version of Hamiltonian through the path integral approach. The latter result appears to be complex in nature and carries the information of escaping the probability of the particle through the horizon. In both ways, we identify the correct expression of Hawking temperature. Moreover, here we successfully extend the complex path approach to a more general black hole like Kerr spacetime. We feel that such a complex path is an outcome of the nature of near horizon instability provided by the horizon and, therefore, once again bolstered the fact that the thermalization mechanism of the horizon may be explained through the aforesaid local instability.

最近的研究表明，无质量、无电荷的出射粒子的近视界哈密顿量，由于其在静态和静止黑洞中的特定运动，实际上是 \(\sim xp\ )类。这本质上是不稳定的，并且有可能解释一些有趣的物理现象。根据路径积分核，我们首先计算态密度。另外，遵循 Singh 和 Padmanabhan (Phys Rev D 85:025011, 2012. https://doi.org/10.1103/PhysRevD.85.025011. arXiv:1112.6279 [hep-th]) 的想法，在地平线附近，我们通过路径积分方法计算对应于其薛定谔版本的哈密顿量的有效路径。后一个结果看起来本质上很复杂，并且携带着粒子穿过视界逃逸概率的信息。通过这两种方式，我们都确定了霍金温度的正确表达式。此外，在这里我们成功地将复杂路径方法扩展到更普遍的黑洞，如克尔时空。我们认为，如此复杂的路径是地平线提供的近地平线不稳定性性质的结果，因此，再次支持了这样一个事实：地平线的热化机制可以通过上述局部不稳定性来解释。