Dynamic black-hole entropy

We consider two non-statistical definitions of the entropy of dynamic (non- stationary) black holes in a spherical symmetry. The first is analogous to the original Clausius definition of thermodynamic entropy: there is a first law containing an energy-supply term which equals surface gravity times a total differential. The second is Wald's Noether-charge method, adapted to dynamic black holes by using the Kodama flow. Both definitions give the sam e value for the Einstein gravity: one-quarter the area of the trapping hori zon. (C) 1999 Elsevier Science B.V. All rights reserved.