The Geometrodynamics of Sine-Gordon Solitons

TL;DR

This paper explores how multi-soliton solutions of the Euclidean sine-Gordon equation relate to Lorentzian black holes in Jackiw-Teitelboim gravity, revealing explicit solutions, coordinate transformations, and implications for black hole thermodynamics.

Contribution

It establishes a novel connection between sine-Gordon solitons and black hole geometries, providing explicit solutions and coordinate mappings in dilaton gravity.

Findings

Black hole mass is non-negative for all soliton parameters.

Explicit solutions for one- and two-soliton cases are derived.

Coordinates cover the entire black hole interior and exterior regions.

Abstract

The relationship between N-soliton solutions to the Euclidean sine-Gordon equation and Lorentzian black holes in Jackiw-Teitelboim dilaton gravity is investigated, with emphasis on the important role played by the dilaton in determining the black hole geometry. We show how an N-soliton solution can be used to construct ``sine-Gordon'' coordinates for a black hole of mass M, and construct the transformation to more standard ``Schwarzchild-like'' coordinates. For N=1 and 2, we find explicit closed form solutions to the dilaton equations of motion in soliton coordinates, and find the relationship between the soliton parameters and the black hole mass. Remarkably, the black hole mass is non-negative for arbitrary soliton parameters. In the one-soliton case the coordinates are shown to cover smoothly a region containing the whole interior of the black hole as well as a finite neighbourhood outside the horizon. A Hamiltonian analysis is performed for slicings that approach the soliton coordinates on the interior, and it is shown that there is no boundary contribution from the interior. Finally we speculate on the sine-Gordon solitonic origin of black hole statistical mechanics.