Boundary terms and Noether current of spherical black holes - art. no. 084004.

We consider two proposals for defining black hole entropy in spherical symm etry, where the horizon is defined locally as a trapping horizon. The first case! boundary:terms in a dual-null form df the reduced action in two dime nsions, gives a result that is proportional to the area. The second case, W ald's Noether current method, is generalized to dynamic black holes giving an entropy that is just the area of the trapping horizon. These results are compared with a generalized first law of thermodynamics. [S0556-2821(99)03 918-1].