Production and decay of evolving horizons

We consider a simple physical model for an evolving horizon that is strongly interacting with its environment, exchanging arbitrarily large quantities of matter with its environment in the form of both infalling material and outgoing Hawking radiation. We permit fluxes of both lightlike and timelike particles to cross the horizon, and ask how the horizon grows and shrinks in response to such flows. We place a premium on providing a clear and straightforward exposition with simple formulae. To be able to handle such a highly dynamical situation in a simple manner we make one significant physical restriction—that of spherical symmetry—and two technical mathematical restrictions: (1) we choose to slice the spacetime in such a way that the spacetime foliations (and hence the horizons) are always spherically symmetric. (2) Furthermore, we adopt Painlevé–Gullstrand coordinates (which are well suited to the problem because they are nonsingular at the horizon) in order to simplify the relevant calculations. Of course physics results are ultimately independent of the choice of coordinates, but this particular coordinate system yields a clean physical interpretation of the relevant physics. We find particularly simple forms for surface gravity, and for the first and second law of black hole thermodynamics, in this general evolving horizon situation. Furthermore, we relate our results to Hawking's apparent horizon, Ashtekar and co-worker's isolated and dynamical horizons, and Hayward's trapping horizon. The evolving black hole model discussed here will be of interest, both from an astrophysical viewpoint in terms of discussing growing black holes and from a purely theoretical viewpoint in discussing black hole evaporation via Hawking radiation.