A static Einstein metric that generalizes the Schwarzschild metric

Abstract

A static Einstein metric that generalizes the Schwarzschild metric is considered. The event horizon is not necessarily a sphere and the term $dt\sp2$ is a function on such horizon. That the metric is Einstein establishes a relation between its terms. One demonstrates that the scalar curvature of the horizon is constant, and that the term $dt\sp2$ gives rise to (i) the metric of the horizon being Einstein, or (ii) the scalar curvature of the horizon being proportional to an eigenvalue of the Laplace operator.