Algebraic geometry approach in gravity theory and new relations between the parameters in type I low-energy string theory action in theories with extra dimensions

• Published in: Annalen der Physik 2010-04, Vol.19 (3-5), p.254-257
• Main Author: Dimitrov, B.G.
• Format: Article
• Language: English
• Subjects: differential geometry and topology; differential geometry and topology; gravity in more than four dimensions; General relativity and gravitation; General relativity and gravitation; geometry; geometry; gravity in more than four dimensions; Kaluza-Klein theory; unified field theories
• ISSN: 0003-3804; 1521-3889
• DOI: 10.1002/andp.201010422

On the base of the distinction between covariant and contravariant metric tensor components, a new (multivariable) cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been derived and parametrized with complicated non‐elliptic functions, depending on the (elliptic) Weierstrass function and its derivative. This is different from standard algebraic geometry, where only two‐dimensional cubic equations are parametrized with elliptic functions and not multivariable ones. Physical applications of the approach have been considered in reference to theories with extra dimensions. The s.c. “length function” l(x) has been introduced and found as a solution of quasilinear differential equations in partial derivatives for two different cases of “compactification + rescaling” and “rescaling + compactification”. New physically important relations (inequalities) between the parameters in the action are established, which cannot be derived in the case l = 1 of the standard gravitational theory, but should be fulfilled also for that case.