Homothetic Killing horizons in generic Vaidya spacetimes

We study the conformal Killing equation for generic Vaidya-like spacetimes, including those with rotation. We show that these spacetimes admit a unique class of conformal Killing vectors that are homothetic for mass, charge, or rotation parameters being linear functions of the advanced null-time. For the Kerr-Vaidya metric, the solution to the conformal Killing equation exists iff both mass and rotation parameters become dynamic. The presence of a homothetic Killing vector (HKV) for such a spacetime enables one to conformally map the original dynamical spacetime to a stationary spacetime, enabling access to the standard methods pertaining to a Killing horizon. The surface where an HKV becomes null is termed the homothetic Killing horizon. We discuss the thermodynamic properties of such homothetic Killing horizons and formulate a version of the first law (or flux balance law) for spherically symmetric Vaidya spacetimes. We further study the maximal analytic extension of a charged Vaidya metric and indicate its implications for studying particle creation in such backgrounds.