[Paper Review] A Periodic Analog of the Schwarzschild Solution

D. Korotkin, Hermann Nicolai|arXiv (Cornell University)|Jan 1, 1994

Relativity and Gravitational Theory2 references20 citations

TL;DR

This paper constructs a new exact solution to Einstein's vacuum equations by creating a periodic analog of the Schwarzschild black hole in a space-time that is periodic in one spatial direction. Using a convergence-improved infinite sum over translated copies of the Schwarzschild Ernst potential, the solution remains regular outside the horizon and asymptotically approaches a Kasner metric with index α = 4M/L, reducing to the standard Schwarzschild solution in the limit L → ∞.

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 $L ightarrow \infty$.

Motivation & Objective

• . The paper aims to construct a new exact solution of Einstein's equations that is periodic in one spatial direction, extending the Schwarzschild solution to a non-asymptotically flat but periodic spacetime.
• It seeks to generalize the method of constructing periodic solutions using doubly periodic functions and convergence-generating constants, adapted to the Euler-Darboux equation governing static, axisymmetric vacuum solutions.
• The objective includes verifying that the resulting metric remains regular outside the event horizon and free of conical singularities on the symmetry axis.
• It aims to analyze the asymptotic behavior of the solution, showing it approaches a Kasner metric with Kasner index α = 4ML⁻¹.
• The study investigates whether such a periodic black hole configuration can be stable and free of singularities, despite the absence of asymptotic flatness.

Proposed method

• . The method begins with the Ernst formalism for stationary axisymmetric vacuum solutions, reducing the Einstein equations to the Euler-Darboux equation when the solution is static.
• It applies a series construction: for a given static solution ω₀(x, ρ), the periodic analog is formed as ω(x, ρ) = ∑ₙ [ω₀(x + nL, ρ) + aₙ], with constants aₙ chosen to ensure convergence.
• For the Schwarzschild solution, the convergence is guaranteed by setting aₙ = −β/(L|n|) for n ≠ 0, with β = −2M, ensuring O(1/n²) decay in the series terms.
• The periodic Ernst potential is then defined as an infinite product: E(x, ρ) = E₀(x, ρ) ∏ₙ₌₁^∞ E₀(x + nL, ρ)E₀(x − nL, ρ) exp(4M/(nL)), which inherits periodicity E(x + L, ρ) = E(x, ρ).
• The conformal factor k(x, ρ) is reconstructed via the first-order equation kₓ = (ρ/2)ωₓωᵨ and kᵨ = (ρ/4)(ω²ᵨ − ω²ₓ), ensuring the metric remains periodic.
• The periodicity of k is verified by showing that the line integral of the k-differential form vanishes over a closed contour enclosing the horizon, due to symmetry and cancellation of singular contributions.

Experimental results

Research questions

• RQ1 . Can a periodic analog of the Schwarzschild solution be constructed that remains regular outside the horizon and preserves key physical features like the event horizon and symmetry?
• RQ2 How does the asymptotic behavior of the periodic solution compare to the standard Schwarzschild case, particularly in terms of the Kasner index?
• RQ3 Does the infinite chain of black holes in the periodic solution avoid conical singularities on the symmetry axis, and if so, why?
• RQ4 What is the role of the convergence-generating constants in the series construction, and how do they ensure global regularity?
• RQ5 How does the solution behave in the limit L → ∞, and does it recover the standard Schwarzschild solution?

Key findings

• . The constructed solution is a valid, regular vacuum solution of Einstein's equations that is periodic in the x-direction with period L and free of singularities outside the event horizon.
• The solution asymptotically approaches a Kasner metric with Kasner index α = 4M/L, where M is the black hole mass and L is the spatial period.
• The asymptotic behavior of the Ernst potential is E ∼ Cρ⁴ᴹ⁄ᴸ as ρ → ∞, confirming the Kasner-type scaling.
• The event horizon remains compact, coinciding with the segment ρ = 0, |x| ≤ M, and the solution is regular on the symmetry axis outside the horizon due to symmetry and proper choice of integration constants.
• In the limit L → ∞, the solution reduces pointwise to the standard Schwarzschild solution, confirming consistency with known results.
• The solution avoids conical singularities on the symmetry axis between black holes due to reflection and translation symmetry, and the absence of such singularities is confirmed by the vanishing of the line integral of the k-differential over a closed contour.