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Exponential solutions of Euler-Lagrange equations for fields of complex linear frames on real space-time manifolds. (English) Zbl 1237.70133

Summary: We investigate a model of the field of complex linear frames on the product manifold \(M = \mathbb R\times G\), where \(G\) is a real semisimple Lie group. The model is invariant under the natural action of the group \(\text{GL}(n, \mathbb C)\) \((n = \dim M)\). It results in a modified Born-Infeld-type nonlinearity of field equations. We find a family of solutions of the Euler-Lagrange equations. These solutions are bases for the Lie algebra of left-invariant vector fields on \(\mathbb R\times G\) “deformed” by a \(\text{GL}(n, \mathbb C)\)-valued mapping of the exponential form. Each solution induces a pseudo-Riemannian metric on \(M = \mathbb R\times G\). The normal-hyperbolic signature (in the physical case where \(n=4\)) of this metric is not something aprioric and absolute, introduced “by hand” into our model but it is an intrinsic feature of solutions we found.

MSC:

70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
58E30 Variational principles in infinite-dimensional spaces
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