The Dynamics of the Field of Linear Frames and Gauge Gravitation

Abstract

The paper is motivated by gauge theories of gravitation and condensed matter, tetrad models of gravitation and generalized Born-Infeld type nonlinearity. The main idea is that any generally-covariant and $\mathrm{GL}(n, \mathbb R)$-invariant theory of the n-leg field (tetrad field when $n=4$) must have the Born-Infeld structure. This means that Lagrangian is given by the square root of the determinant of some second-order twice covariant tensor built in a quadratic way of the field derivatives. It is shown that there exist interesting solutions of the group-theoretical structure. Some models of the interaction between gravitation and matter are suggested. It turns out that in a sense the space-time dimension $n=4$, the normal-hyperbolic signature and velocity of light are integration constants of our differential equations.