Ernst equation, Fay identities and variational formulas on hyperelliptic curves

We present a unified approach to theta-functional solutions of the stationary axisymmetric Einstein equations in vacuum. Using Fay's trisecant identity and variational formulas on hyperelliptic Riemann surfaces, we establish formulas for the metric functions, the Ernst potential and their derivatives.

[1]  P. Negi,et al.  Exact Solutions of Einstein's Field Equations , 2004, gr-qc/0401024.

[2]  Malcolm MacCallum,et al.  Exact Solutions of Einstein's Field Equations , 2003 .

[3]  V. Matveev,et al.  Theta Function Solutions of the Schlesinger System and the Ernst Equation , 2000 .

[4]  C. Klein,et al.  Exact Relativistic Gravitational Field of a Stationary Counterrotating Dust Disk , 1999, gr-qc/9908042.

[5]  V. Matveev,et al.  Solutions of Schlesinger system and Ernst equation in terms of theta-functions , 1998 .

[6]  C. Klein,et al.  Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces , 1998, gr-qc/9806051.

[7]  Neugebauer,et al.  General relativistic gravitational field of a rigidly rotating disk of dust: solution in terms of ultraelliptic functions. , 1995, Physical review letters.

[8]  D. Korotkin.  Finite-gap solutions of the stationary axisymmetric Einstein equation in vacuum , 1988 .

[9]  Takahiro Shiota,et al.  Characterization of Jacobian varieties in terms of soliton equations , 1986 .

[10]  V. Zakharov,et al.  Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions , 1978 .

[11]  D. Maison.  Are the stationary, axially symmetric Einstein equations completely integrable? , 1978 .

[12]  Sergei Petrovich Novikov,et al.  NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES , 1976 .

[13]  John D. Fay.  Theta Functions on Riemann Surfaces , 1973 .

[14]  H. Rauch.  Weierstrass points, branch points, and moduli of riemann surfaces , 1959 .

[15]  Igor Krichever,et al.  Method of averaging for two-dimensional "integrable" equations , 1988 .

[16]  Alan Weinstein,et al.  Progress in mathematics , 1979 .