Elliptic Curves, Algebraic Geometry Approach in Gravity Theory and Some Applications in Theories with Extra Dimensions I

TL;DR

This paper develops an algebraic geometry approach using elliptic functions to find exact solutions to Einstein's equations, explores parametrization of multi-variable cubic curves, and investigates implications for theories with extra dimensions and string theory.

Contribution

It introduces a novel parametrization method for multi-variable cubic curves and extends algebraic solutions for metric components in gravity and string theories.

Findings

Parametrization of multi-variable cubic curves beyond standard algebraic geometry.

Derived algebraic solutions for contravariant metric tensor components.

Formulated quasilinear differential equations for string theory parameters.

Abstract

Motivated by the necessity to find exact solutions with the elliptic Weierstrass function of the Einstein's equations (see gr-qc/0105022),the present paper develops further the proposed approach in hep-th/0107231, concerning the s.c. cubic algebraic equation for effective parametrization. Obtaining an ''embedded'' sequence of cubic equations, it is shown that it is possible to parametrize also a multi-variable cubic curve, which is not the standardly known case from algebraic geometry. Algebraic solutions for the contravariant metric tensor components are derived and the parametrization is extended in respect to the covariant components as well. It has been speculated that corrections to the extradimensional volume in theories with extra dimensions should be taken into account, due to the non-euclidean nature of the Lobachevsky space. It was shown that the mechanism of exponential "damping" of the physical mass in the higher-dimensional brane theory may be more complicated due to the variety of contravariant metric components for a spacetime with a given constant curvature. The invariance of the low-energy type I string theory effective action is considered in respect not only to the known procedure of compactification to a four-dimensional spacetime, but also in respect to rescaling the contravariant metric components. As a result, instead of the simple algebraic relations between the parameters in the string action, quasilinear differential equations in partial derivatives are obtained, which have been solved for the most simple case. In the Appendix, a new block structure method is presented for solving the well known system of operator equations in gravity theory in the N-dimensional case.