Note on electrical and thermodynamic properties of isolated horizons

The electrical laws and Carnot cycle of isolated horizons (IH) are investigated in this paper. We establish Ohm's law and Joule's law of isolated horizons and find that the conceptual picture of black holes (membrane paradigm) can also apply to this kind of quasilocal black holes. We also investigate the geometrical properties near nonrotating IHs and find that under the first-order approximation of $r$, there exist a Killing vector $\frac{\ensuremath{\partial}}{\ensuremath{\partial}u}$ and a Hamiltonian conjugate to it, so this vector can be thought to be a physical observer. We calculate the energy as measured at infinity of a particle at rest outside a nonrotating IH, and we use this result to construct a reversible Carnot cycle with the isolated horizon as a cold reservoir, which confirms the thermodynamic nature of isolated horizons.