用量子弦探测黑洞奇点

中文摘要 AI 辅助

本文研究黑洞经典奇点附近的量子行为，采用Kantowski-Sachs度规描述黑洞内部，在该区域引入传播的弦。该方法的动机在于，经典奇点附近时空曲率达到普朗克或弦尺度，需用弦论自由度描述。弦动力学通过Polyakov作用量表述，对整个系统进行正则量子化以推导对应的Wheeler-DeWitt方程。尽管该方程在径向和弦的方位角部分可分离，但剩余的几何和弦自由度存在非平凡耦合。为解决这一问题，采用微扰方法，得到一阶近似下的波函数。零阶近似中，角部分由一般Heun函数描述，几何部分由贝塞尔函数表示；一阶波函数由包含零阶解的源项决定，通过余函数和特积分得到其显式表达式。我们确认零阶和一阶近似下，波函数在奇点和视界处均具有正则性。视界处的正则性对一个本征值施加了离散化条件，另一个本征值则因波函数的方位角周期性而量子化。重要的是，零阶和一阶解均满足DeWitt边界条件，这意味着在该量子框架内黑洞奇点得到解决。

英文摘要

In this paper, we investigate the quantum behaviour near the classical singularity of a black hole. We describe the black-hole interior by the Kantowski-Sachs metric, with a string propagating in this region. This approach is motivated by the fact that, near the classical singularity, spacetime curvature reaches Planck or string scales, necessitating a description in terms of string-theoretic degrees of freedom. The string dynamics is formulated via the Polyakov action, and the combined system is canonically quantized to derive the corresponding Wheeler-DeWitt equation. Although the equation is separable in the radial and azimuthal string sectors, the remaining geometric and string degrees of freedom are nontrivially coupled. To address this, we employ a perturbative approach and obtain the wave function up to first order. At the zeroth-order, the angular part is described by the general Heun function, while the geometric part is expressed in terms of the Bessel function. The first-order wave function is governed by a source term containing the zeroth-order solution, and we obtain its explicit expression in terms of the complementary function and the particular integral. We confirm regularity of the wave function at both zeroth and first order, at the singularity as well as at the horizon. Regularity at the horizon imposes a discretization condition on one eigenvalue, while another eigenvalue is quantized due to the azimuthal periodicity of the wave function. Importantly, both zeroth- and first-order solutions satisfy the DeWitt boundary condition, implying resolution of the black hole singularity within this quantum framework.

发表机构

• Indian Institute of Technology Guwahati（印度理工学院古瓦哈提分校）

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