Nonlinear electromagnetic fields and symmetries

Abstract

We extend the classical results on the symmetry inheritance of the canonical electromagnetic fields, described by the Maxwell's Lagrangian, to a much wider class of models, which include those of the Born-Infeld, power Maxwell and the Euler-Heisenberg type. Symmetry inheriting fields allow the introduction of electromagnetic scalar potentials and these are proven to be constant on the Killing horizons. Finally, using the relations obtained along the analysis, we generalize and simplify the recent proof for the symmetry inheritance of the 3-dimensional case, as well as give the first constraint for the higher dimensional electromagnetic fields.

Generalizations of the Smarr formula for black holes with nonlinear electromagnetic fields

“…Let us look at the details of the electromagnetic field in this spacetime. First, the theorem 5.6 from [11], generalized for the NLE in [47], implies that F (k, X (i) ) = 0 and * F (k, X (i) ) = 0 for any of the three Killing vector fields X a (i) that generate the SO(3) isometry. The electric 1-form E = −i k F and the magnetic 1-form B = i k * F (more convenient at this point than the 1-form H = i k * Z), we can write the electromagnetic 2-form as…”

Spacetimes dressed with stealth electromagnetic fields

“…This example demonstrates how NLE fields may evade some well-known no-go theorems. First, nontrivial null electromagnetic fields cannot exist in a static spacetime [63] and the extension of this theorem [64] holds in NLE models under the assumption that L F = 0, which is broken by the stealth fields. Second, a linear electromagnetic field inherits symmetries in general spherically symmetric spacetime [65][66][67], but this does not necessarily hold in NLE models [64].…”

“…First, nontrivial null electromagnetic fields cannot exist in a static spacetime [63] and the extension of this theorem [64] holds in NLE models under the assumption that L F = 0, which is broken by the stealth fields. Second, a linear electromagnetic field inherits symmetries in general spherically symmetric spacetime [65][66][67], but this does not necessarily hold in NLE models [64]. Although the points with vanishing L F are completely out of the scope of the analysis in [64], the field (20) nevertheless obeys the same constraints on breaking of the symmetry inheritance: for any Killing vector K a of the metric (19) the Lie derivative £ K F ab is some linear combination of F ab and its Hodge dual * F ab .…”

On symmetry inheritance of nonminimally coupled scalar fields

“…As a consequence of the assumptions about the tensor E ab we know that the existence of a Killing vector field K a , such that £ K g ab = 0, implies £ K E ab = 0. For a minimally coupled field this would immediately lead to the condition £ K T ab = 0, a crucial equation in the recent detailed analyses of the symmetry inheritance for real and complex scalar fields [9,11], as well as the electromagnetic field [10,12]. However, as we are looking at nonminimally coupled fields, we have to resort to other strategies.…”

“…The earliest works on this topic have appeared in the mid-1970s [1][2][3][4][5][6][7][8], after which we have had only occasional bursts of activity, separated by relatively dormant phases. The most recent series of papers [9][10][11][12] have solved some long standing open problems and opened several new ones. However, all the results on the symmetry inheritance, both in the old as well as in the more recent papers, are about matter fields minimally coupled to gravity.…”