Canonical Quantization of the Expansion Scalar for the Reissner-Nordstrom Metric

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.