Space-time as a structured relativistic continuum

Abstract

It is well known that there are various models of gravitation: the metrical Hilbert-Einstein theory, a wide class of intrinsically Lorentz-invariant tetrad theories (of course, generally-covariant in the space-time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, ${\rm GL}(n,\mathbb{R})$, ${\rm SU}(2,2)$, ${\rm GL}(4,\mathbb{C})$, and so on). One believes usually in gauge models and we also do it. Nevertheless, it is an interesting idea to develop the class of ${\rm GL}(4,\mathbb{R})$-invariant (or rather ${\rm GL}(n,\mathbb{R})$-invariant) tetrad ($n$-leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus idea of affine symmetry. Formally, the obtained scheme is a generally-covariant tetrad ($n$-leg) model, but it turns out that generally-covariant and intrinsically affinely-invariant models must have a kind of non-accidental Born-Infeld-like structure. Let us also mention that they, being based on tetrads ($n$-legs), have many features common with continuous defect theories. It is interesting that they possess some group-theoretical solutions and more general spherically-symmetric solutions. It is also interesting that within such framework the normal-hyperbolic signature of the space-time metric is not introduced by hand, but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born-Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones.