Chinese Acad Sci, Inst High Energy Phys, Beijing 100039, Peoples R China
; Chinese Acad Sci, Inst Theoret Phys, Beijing 100080, Peoples R China
; Max Planck Inst Gravitat Phys, D-14476 Golm, Germany
The Noether charge formalism on the covariant phase space is used and developed. For a stationary axisymmetric black hole in vacuum general relativity, the mass formula is the vanishing Noether charge with respect to the Killing vector field xi(K)(a) = t(K)(a) + Omega(H)phi(K)(a) on an entire Cauchy surface. Its differential form gives the first law of black hole mechanics. For a large class of vector fields, the central term of the reduced algebra near the horizon is proved to be zero by use of the null tetrad formalism. It implies that the microscopic interpretation of the black hole entropy based on the symmetry analysis proposed by Carlip is debatable.