A static Einstein metric that generalizes the Schwarzschild metric

TL;DR

This paper explores a generalized static Einstein metric extending the Schwarzschild solution, revealing that the horizon's scalar curvature is constant and linking the metric's terms to Einstein or Laplacian eigenvalue properties.

Contribution

It introduces a new class of static Einstein metrics with non-spherical horizons and analyzes their geometric properties, extending classical black hole solutions.

Findings

Horizon scalar curvature is constant.

The metric term $dt^2$ relates to horizon Einstein conditions or Laplacian eigenvalues.

Horizon geometry can be non-spherical.

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.