Equiaffine braneworld

TL;DR

This paper explores the use of equiaffine hypersurface theory as a potentially more natural framework for higher-dimensional braneworld models, emphasizing the indefinite nature of the spacetime metric and its geometric implications.

Contribution

It introduces equiaffine hypersurface theory as an alternative to the traditional isometric immersion approach in braneworld models, highlighting its mathematical simplicity and physical relevance.

Findings

Proposes equiaffine hypersurface theory as a natural geometric framework.

Shows the spacetime metric is conformal to the second fundamental form.

Suggests potential physical benefits of this geometric approach.

Abstract

Higher dimensional theories, wherein our four dimensional universe is immersed into a bulk ambient, have received much attention recently, and the direction of investigation had, as far as we can discern, all followed the ordinary Euclidean hypersurface theory's isometric immersion recipe, with the spacetime metric being induced by an ambient parent. We note in this paper that the indefinite signature of the Lorentzian metric perhaps hints at the lesser known equiaffine hypersurface theory as being a possibly more natural, i.e., less customized beyond minimal mathematical formalism, description of our universe's extrinsic geometry. In this alternative, the ambient is deprived of a metric, and the spacetime metric becomes conformal to the second fundamental form of the ordinary theory, therefore is automatically indefinite for hyperbolic shapes. Herein, we advocate investigations in this direction by identifying some potential physical benefits to enlisting the help of equiaffine differential geometry.