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Nonlinear electromagnetic generalization of the Kerr-Newman solution with cosmological constant

Oscar Galindo-Uriarte, Nora Breton

Abstract

We present the two exact solutions of the Einstein-Nonlinear electrodynamics equations that generalize the Kerr-Newman solution. We determined the generalized electromagnetic potentials using the alignment between the tetrad vectors of the metric and the eigenvectors of the electromagnetic field tensor. It turns out that there are only two possible nonlinear electromagnetic generalizations of the Kerr-Newman geometry, corresponding to different electromagnetic potentials. The new solutions possess horizons and satisfy physical energy conditions such that they can represent black holes with nonlinear electromagnetic charges, characterized by the parameters of mass, angular momentum, charge, and one nonlinear parameter; the nonlinear parameter resembles the effect of a cosmological constant, negative or positive, such that the solutions are asymptotically AdS or dS. The canonical form of the electromagnetic nonlinear energy-momentum tensor is analyzed in relation with the energy conditions; it is shown that the conformal symmetry is broken by the electromagnetic nonlinear matter; the corresponding nonlinear electromagnetic Lagrangian as a function of the coordinates is presented as well.

Nonlinear electromagnetic generalization of the Kerr-Newman solution with cosmological constant

Abstract

We present the two exact solutions of the Einstein-Nonlinear electrodynamics equations that generalize the Kerr-Newman solution. We determined the generalized electromagnetic potentials using the alignment between the tetrad vectors of the metric and the eigenvectors of the electromagnetic field tensor. It turns out that there are only two possible nonlinear electromagnetic generalizations of the Kerr-Newman geometry, corresponding to different electromagnetic potentials. The new solutions possess horizons and satisfy physical energy conditions such that they can represent black holes with nonlinear electromagnetic charges, characterized by the parameters of mass, angular momentum, charge, and one nonlinear parameter; the nonlinear parameter resembles the effect of a cosmological constant, negative or positive, such that the solutions are asymptotically AdS or dS. The canonical form of the electromagnetic nonlinear energy-momentum tensor is analyzed in relation with the energy conditions; it is shown that the conformal symmetry is broken by the electromagnetic nonlinear matter; the corresponding nonlinear electromagnetic Lagrangian as a function of the coordinates is presented as well.

Paper Structure

This paper contains 19 sections, 77 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Kerr-Newman metric
  3. Alignment conditions for the Kerr-like metric
  4. Nonlinear electromagnetic equations
  5. The key equation for the NLE potentials $A_{\mu}$ and ${}^{\star} P_{\mu}$
  6. The nonlinear electromagnetic generalizations of the Kerr-Newman metrics
  7. Case 1. The cubic vector potential
  8. Case 2. The quartic vector potential
  9. The quartic vector potential NLE-KN
  10. Horizons and ergosphere without cosmological constant
  11. The static limit of the NLE Quartic vector potential solution
  12. The nonlinear electromagnetic energy momentum tensor
  13. Energy conditions
  14. Weak Energy Condition (WEC)
  15. Dominant Energy Condition (DEC)
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Graphics of the behavior of the metric function $\Delta_r (r)$ for the cases Schwarzschild, RN, Kerr, KN, NLE-RN and NLE-KN.
  • Figure 2: The event horizon for KN (blue) and KN NLE (black), as well as the ergosphere of KN (orange) and the one of KN NLE (red) are displayed. The BH parameters are set to $Q_e = 0.8m$, $a = 0.6m$ and $\xi = -0.14/m^3$.