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A Periodic Analog of the Schwarzschild Solution

D. Korotkin, H. Nicolai

Abstract

We construct a new exact solution of Einstein's equations in vacuo in terms of Weyl canonical coordinates. This solution may be interpreted as a black hole in a space-time which is periodic in one direction and which behaves asymptotically like the Kasner solution with Kasner index equal to $4M L^{-1}$, where $L$ is the period and $M$ is the mass of the black hole. Outside the horizon, the solution is free of singularities and approaches the Schwarzschild solution as $L \rightarrow \infty$.

A Periodic Analog of the Schwarzschild Solution

Abstract

We construct a new exact solution of Einstein's equations in vacuo in terms of Weyl canonical coordinates. This solution may be interpreted as a black hole in a space-time which is periodic in one direction and which behaves asymptotically like the Kasner solution with Kasner index equal to , where is the period and is the mass of the black hole. Outside the horizon, the solution is free of singularities and approaches the Schwarzschild solution as .

Paper Structure

This paper contains 4 theorems, 30 equations.

Key Result

Theorem 1

Let $\omega_0(x,\rho)$ be any solution of the Euler-Darboux equation corresponding to an asymptotically flat metric (m1), i.e. where $r=\sqrt{x^2 +\rho^2}$; $M=-\frac{1}{2} \beta$ is the mass. Let Then series (ps) is convergent for all $(x,\rho)$ except the points $(x_0+nL,\rho_0)$, where the function $\omega_0(x,\rho)$ is singular ($n\in {\bf Z}$), and defines a periodic function with period $L

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4