The D-dimensional Schwarzschild-de Sitter metric and Reissner-Nordstrom-de Sitter metric are derived directly by solving the Einstein and Einstein-Maxwell equations. The D-dimensional Kerr metric is rederived by using the complex coordinate transformation method and the D-dimensional Kerr-de Sitter metric is also given. The conjecture about the D-dimensional metric of a rotating, charged mass is given at the end of this paper.

We show that the charged rotating metric in de Sitter space, derived by Mallett and used by Koberlein, is incorrect. Mallett's metric and his energy-momentum tensor do not satisfy the Einstein-Maxwell field equations with a cosmological term, in the non-radiating and radiating Kerr-Newman-de Sitter cases. We give the corresponding correct metric and the radiating energy-momentum tensor.

Two important invariant identities about the products of the Riemann tensor RaByh and the Ricci tensor Ra8 and the scalar curvature R are derived by the technique of Weyl decomposition of the Riemann tensor and by the spinor formalism. These identities are very useful in four dimensions for simplifying the final expression of the a 3 coefficient of the scalar fields and for simplifying the evaluation of the vacuum-polarization energy-momentum tensor. This result is of relevance to the work of Jack and Parker on the summed form of the heat kernel.

The problem of solving the Dirac equation in a Kerr space-time has long been unresolved. In 1976, S. Chandrasekhar solved the decoupling problem of the Dirac equation with non-zero rest mass in a Kerr background, thus provided the possibility of attacking the problem of the Hawking evaporation of Dirac particles. In this paper, starting from Chandrasekhar's results, we find the solution of the Dirac equation with non-zero rest mass just out side the event horizon of a nearly extreme Kerr black hole and sucessfully derived the Hawking thermal formula.

Coupling between angular momentum and the cosmological constant does not produce distortion or deformation of apparent horizons. Only to a first-order perturbation or approximation can it affect the shape of event horizons.