Multi-Centered Metric on (n+1)-Dimensional Static Spacetimes

摘要

In this paper, we construct explicitly a multi-centered metric in (n+1)-dimensional static spacetimes which belong to a class of (pseudo)-Riemannian manifolds. Our starting point is to embed an n-dimensional complete manifold into an (n+1)-dimensional manifold which admits a timelike Killing vector. Then, the scalar curvature depends on the Laplace-Beltrami operator of the n-dimensional submanifold. This operator is set to be equals to the linear combination of the Dirac-delta function on the submanifold which can be thought of as remove points on (n+1)-dimensional manifold. Using completeness of the n-dimensional submanifold it can be shown that the solution does indeed exist. As an example we give the explicit form for flat Euclidean geometries, sphere, and hyperbolic geometries.