Travel Grant for the International Conference on "Finsler Extension of Relativity Theory"

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Classical general relativity is built in large part upon the extension of Riemannian Geometry to Lorentzian Geometry (more generally, semi-Riemannian Geometry). In fact, time independent questions are basically Riemannian in nature. For examples the classification of black holes is based on time independent solutions of the field equations; the (first) proof of the positive mass theorem is also based on Riemannian Geometry. At present researchers are interested in issues involved in the rigorous extension and generalization of the principal geometric and physical structures associated with the spaces of classical and relativistic physics (e.g. of Riemannian geometry) to Finsler spaces. One of the problems of basing Relativity on Riemannian Geometry is the difficulty with "casualty". This is due to the fact that geodesics in Riemannian Geometry are reversible. If a space-time admits a CTC (closed time-like curve) then one could travel back in time and all kinds of (murderous) paradoxes could occur. In contrast, there exist Finsler metrics for which the geodesics are irreversible hence, provides a more natural setting for dealing with "casualty" in Relativity. There are many issues, more technical in nature, involving Finsler Geometry that are useful in Relativity. Among these are: (a) spaces whose metric functions are symmetric polynomials depending on four variables of the third order (known as Chernov spaces in Finsler Geometry), (b) spaces whose metric functions are symmetric polynomials depending on four variables of the fourth order (known as Berwald-Moore spaces in Finsler Geometry), (c) multi-linear symmetric forms as Finsler extension of scalar product, (d) connection between Finsler spaces and hypercomplex numbers, (e) possibility of extending, in the Finsler setting, the Lorentzian splitting theorem, originally conjectured by S. T. Yau as the natural extension of the Riemannian splitting theorem of Cheeger and Gromov.