[Paper Review] Solutions of Schlesinger system and Ernst equation in terms of theta-functions

D. Korotkin, V. B. Matveev|ArXiv.org|Oct 12, 1998

Nonlinear Waves and Solitons23 references7 citations

TL;DR

This paper establishes a direct correspondence between solutions of the Schlesinger system and the stationary axisymmetric Einstein equation via algebro-geometric methods, expressing all metric coefficients in terms of Riemann theta-functions. The key contribution is a complete theta-function parametrization of the Ernst potential and metric components, enabling explicit construction of vacuum gravitational solutions from algebraic curves and holomorphic differentials.

ABSTRACT

We discuss the relationship between Schlesinger system and stationary axisymmetric Einstein's equation on the level of algebro-geometric solutions. In particular, we calculate all metric coefficients corresponding to solutions of Ernst equation in terms of theta-functions.

Motivation & Objective

• To establish a rigorous connection between the Schlesinger system and the stationary axisymmetric Einstein equation in general relativity.
• To derive explicit expressions for all metric coefficients in terms of Riemann theta-functions for solutions of the Ernst equation.
• To provide a systematic algebro-geometric framework for constructing vacuum gravitational solutions from algebraic curves and meromorphic differentials.
• To generalize known theta-function solutions of integrable systems to the context of stationary axisymmetric gravity.
• To offer a constructive method for generating solutions of the Ernst equation using algebraic geometry and theta functions.

Proposed method

• Utilizes the algebro-geometric approach to integrable systems, particularly the finite-gap integration method.
• Constructs solutions of the Schlesinger system using a Riemann theta-function over a compact algebraic curve of genus g.
• Relates the monodromy data of the Schlesinger system to the period matrix of the curve and holomorphic differentials.
• Expresses the Ernst potential as a bilinear combination of theta-functions and their derivatives.
• Derives the metric components (e.g., g_tt, g_rr, g_phi_phi) explicitly from the Ernst potential using theta-function identities.
• Employs the Riemann bilinear relations and the Riemann theta characteristics to ensure consistency and reality conditions of the metric.

Experimental results

Research questions

• RQ1 How can solutions of the Schlesinger system be mapped to solutions of the stationary axisymmetric Einstein equation?
• RQ2 What is the explicit theta-function representation of the Ernst potential in terms of algebraic curves?
• RQ3 How are the metric coefficients of the stationary axisymmetric spacetime expressed using Riemann theta-functions?
• RQ4 What role do holomorphic differentials and the period matrix play in constructing these solutions?
• RQ5 Can the entire gravitational potential and metric be reconstructed from the theta-function data of a finite-genus algebraic curve?

Key findings

• All metric coefficients of the stationary axisymmetric spacetime are explicitly expressed in terms of Riemann theta-functions and their derivatives.
• The Ernst potential is shown to be a rational function of theta-functions and their logarithmic derivatives, derived from the Schlesinger system's solution.
• The construction yields real, regular solutions of the Einstein equation for a wide class of algebraic curves with appropriate reality conditions.
• The method provides a complete parametrization of vacuum solutions via the moduli space of algebraic curves and their holomorphic differentials.
• The solution framework generalizes known results for hyperelliptic curves and extends them to arbitrary genus curves.
• The approach confirms the integrability of the stationary axisymmetric Einstein system through the theta-function parametrization of its solutions.