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

摘要

Motivated by the necessity to find exact solutions with the ellipticWeierstrass function of the Einstein's equations (see gr-qc/0105022),thepresent 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 ispossible to parametrize also a multi-variable cubic curve, which is not thestandardly known case from algebraic geometry. Algebraic solutions for thecontravariant metric tensor components are derived and the parametrization isextended in respect to the covariant components as well. It has been speculatedthat corrections to the extradimensional volume in theories with extradimensions should be taken into account, due to the non-euclidean nature of theLobachevsky space. It was shown that the mechanism of exponential "damping" ofthe physical mass in the higher-dimensional brane theory may be morecomplicated due to the variety of contravariant metric components for aspacetime with a given constant curvature. The invariance of the low-energytype I string theory effective action is considered in respect not only to theknown procedure of compactification to a four-dimensional spacetime, but alsoin respect to rescaling the contravariant metric components. As a result,instead of the simple algebraic relations between the parameters in the stringaction, quasilinear differential equations in partial derivatives are obtained,which have been solved for the most simple case. In the Appendix, a new blockstructure method is presented for solving the well known system of operatorequations in gravity theory in the N-dimensional case.