Analog Hawking Effect: A Master Equation

We consider further the problem of the analog Hawking radiation. We propose a fourth order ordinary differential equation, which allows us to discuss the problem of Hawking radiation in analog gravity in a unified way, encompassing fluids and dielectric media. In a suitable approximation, involving weak dispersive effects, Wentzel-Kramers-Brillouin solutions are obtained far from the horizon (turning point), and furthermore an equation governing the behavior near the horizon is derived, and a complete set of analytical solutions is obtained also near the horizon. The subluminal case of the original fluid model introduced by Corley and Jacobson and the case of dielectric media are discussed. We show that in this approximation scheme there is a mode which is not directly involved in the pair-creation process. Thermality is verified and a framework for calculating the gray-body factor is provided.