Pseudo-Riemannian Metric with Neutral Signature Induced by Solutions to Euler–Lagrange Equations for a Field of Complex Linear Frames

Abstract

We investigate a model of the field of complex linear frames E on the product manifold M = R × G, where G is a real semisimple Lie group. The model is invariant under the natural action of the group GL( n , C ) ( n = dim M ). It results in a modified Born–Infeld-type nonlinearity of Euler–Lagrange equations. We analyse a family of solutions to Euler–Lagrange equations. Each solution E belonging to this family induces a pseudo-Riemannian metric γ[ E ] on M = R × G. In the physical case where n = dim M = 4, among these solutions there exist ones for which the signature of γ[ E ] is neutral (++− −). The existence of solutions leading to the neutral signature of γ[ E ] is interesting in itself. Additionally, it can shed new light onto the theory of generally-relativistic spinors and the conformal U(2, 2)-symmetry.