The outer event horizon r + of a Reissner-Nordstrom black hole is shown to satisfy the conditions of a Non-Expanding Horizon and the inner horizon r − is assumed to be dynamically evolving in area as no restrictions are imposed. An area evolution law is formulated, using the expansion scalar, keeping in mind the imposed conditions. It is later shown that the expansion scalar is the black hole analogue to the thermodynamic heat flow, just as area of a horizon is the black hole analogue of entropy. It is shown that the expansion scalar and the thermodynamic heat of the horizon form a canonical pair and the Lagrangian and Hamiltonian equations are formulated for this condition. From the equations, the expansion is quantized and the Heisenberg Inequaltity for this system is formulated.

中文翻译：

---

Reissner-Nordstrom 度量的扩展标量的规范量化

Reissner-Nordstrom 黑洞的外部事件视界 r + 被证明满足非扩展视界的条件，而内部视界 r - 被假定为在区域中动态演化，因为没有强加任何限制。使用扩展标量制定区域演化规律，同时牢记强加的条件。后来证明，膨胀标量是与热力学热流类似的黑洞，就像视界的面积是熵的黑洞类似物一样。结果表明，膨胀标量和视界的热力学热形成正则对，并针对这种情况制定了拉格朗日方程和哈密顿方程。根据方程，扩展被量化，并且该系统的海森堡不等式被公式化。