M ar 1 99 4 A Periodic Analog of the Schwarzschild Solution

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 → ∞. In this article, we present a new exact solution of Einstein's equations in vacuo (for a comprehensive review of exact solutions, see [1]). This solution constitutes a periodic analog of the Schwarzschild solution [2], and is likewise free of singularities. Asymptotically, it behaves like the Kasner solution. Before describing the new solution, we would like to remind the reader of the link between meromorphic functions on the Riemann sphere CP 1 and on the torus T = C/{L 1 , L 2 } (L 1 and L 2 are the two periods on the lattice defining T). Given some meromorphic function f 0 (ξ), ξ ∈ C, we can consider the formal expression f (ξ) = ∞ m,n=−∞ f 0 (ξ + mL 1 + nL 2) + a mn (1) Provided we can choose the constants a mn in such a way that this series converges, the function f (ξ) is meromorphic on the torus T with the same number and positions of poles as the function f 0 (ξ) on CP 1. In this case, f (ξ) is a doubly periodic analog of the original function f 0. The simplest example of a function f 0 (ξ) admitting an appropriate choice of convergence generating constants a mn is f 0 (ξ) = ξ −2 with a mn = −(mL 1 + nL 2) −2 for (m, n) = (0, 0) and a 00 = 0, leading to the well-known Weierstrass ℘-function. (For f 0 (ξ) = ξ −1 , on the other hand, no choice of a mn will render the series (1) convergent since this would imply the existence of a meromorphic function on the torus having only one pole in contradiction to the Riemann-Roch theorem.) The same procedure works, of course, for any real harmonic function ω(ξ, ¯ ξ) which can be represented as ω = Ref (ξ) for some (locally) holomorphic …