New Conformally Invariant Born–Infeld Models and Geometrical Currents

A new conformally invariant gravitational generalization of the Born–Infeld (BI) model is proposed and analyzed from the point of view of symmetries. Taking a geometric identity involving the determinant functions detfBμν, Fμν with the Bach Bμν and the electromagnetic field Fμν tensors (with the 4-dimensional Greek letter indexes), two characteristic geometrical Lagrangian densities (Lagrangians) are derived: the first Lagrangian being the square root of the determinant function detBμν+Fμν (reminiscent of the standard BI model) and the second Lagrangian being the fourth root gdetBαγBβγ+FαγFβγ4. It is shown, after explicit computation of the gravitational equations, that the square-root model is incompatible with the inclusion of the electromagnetic tensor, consequently forcing the nullity of Fμν. In sharp contrast, the traceless fourth-root model is fully compatible and a natural ansatz of the type BμρBνρ∝Ωxgμν (conformal-Killing), with Ω the conformal factor and x the 4-coordinate, can be considered. Among other essential properties, the geometrical conformal Lagrangian of the fourth-root type is self-similar with respect to the determinant g of the metric tensor gμν and can be extended to non-Abelian fields in a way similar to the model developed by the author earlier. This self-similarity is related to the conformal properties of the model, such as the Bach currents or flows presumably of a topological origin. Possible applications and comparisons with other models are briefly discussed.