Cubic Algebraic Equations in Gravity Theory, Parametrization with the Weierstrass Function and Non-Arithmetic Theory of Algebraic Equations

Details Cubic Algebraic Equations in Gravity Theory, Parametrization with the Weierstrass Function and Non-Arithmetic Theory of Algebraic Equations Cubic Algebraic Equations in Gravity Theory, Parametrization with the Weierstrass Function and Non-Arithmetic Theory of Algebraic Equations

2001-07-27

arxiv.org/abs/hep-th/0107231v2

J.Math.Phys. 44 (2003) 2542-2578

Physics

High Energy Physics

High Energy Physics - Theory

v.2; submitted to Journ.Math.Phys.(October 2001); Latex (Sci.Word,amsmath style), 77 pages, no figures, 4 appendixes; Sect.III

Scientific paper

10.1063/1.1560855

A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle.It was suggested that such an equation may find application in gravity theory, brane, string and Rundall-Sundrum theories. The obtained algebraic equation was brought by means of a linear-fractional transformation to a parametrizable form, expressed through the elliptic Weierstrass function, which was proved to satisfy the standard parametrizable form, but with $g_{2}$ and $g_{3}$ functions of a complex variable instead of the definite complex numbers (known from the usual arithmetic theory of elliptic functions and curves). The generally divergent (two) infinite sums of the inverse first and second powers of the poles in the complex plane were shown to be convergent in the investigated particular case, and the case of the infinite point of the linear-fractional transformation was investigated. Some relations were found, which ensure the parametrization of the cubic equation in its general form with the Weierstrass function.

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