The Hawking temperature in the context of dark energy for Reissner–Nordstrom and Kerr background

Abstract

For emergent gravity metrics, the presence of dark energy modifies the Hawking temperature. We show that for the spherically symmetric Reissner-Nordstrom background metric, the emergent metric can be mapped into a Robinson-Trautman black hole. Allowed values of the dark energy density follow from rather general conditions. For some allowed values of the dark energy density this black hole can have zero Hawking temperature, i.e. the black hole does not radiate. For a Kerr background along θ = 0, the emergent black hole metric satisfies Einstein's equations for large r and always radiates. Our analysis is done in the context of emergent gravity metrics having k-essence scalar fields φ with a Born-Infeld type lagrangian. In both cases the scalar field φ(r, t) = φ 1 (r )+φ 2 (t) also satisfies the emergent gravity equations of motion for r → ∞ and θ = 0.

Thermodynamics for the k-essence emergent Reissner–Nordstrom–de Sitter spacetime

“…Note that the solution φ(r, t) (18) is satisfied emergent gravity equation of motion (7) at r → ∞:Ḡ 00 ∂ 2…”

“…Barriola-Velenkin, Robinson-Trautman (RT) type spacetimes [6,7] and for the Kerr, Kerr-Newman and Kerr-Newman-AdS background, the emergent metrics are satisfying Einstein equation for large r along θ = 0 [7,8]. Also, in the above references, the authors have established the modification of the Hawking temperature [30]- [44] in the presence of dark energy in an emergent gravity scenario.…”

Time-like geodesic structure for the emergent Barriola–Vilenkin type spacetime

“…where L denotes the angular momentum about an axis normal to the invariant plane. By the equation (13) and (16), we haveṫ = E β− 2M r andΦ = L r 2 . Substituting these values in (11), the Lagrangian can be rewritten as…”

The Hawking temperature in the context of dark energy for Kerr–Newman and Kerr–Newman–AdS backgrounds

“…We shall take the Lagrangian as L = L(X) = 1 − V √ 1 − 2X with V is a constant. This is a particular case of the DBI lagrangian [2][3][4]…”

“…The Lagrangian for k−essence scalar fields contains non-canonical kinetic terms. The general form of the Lagrangian for k−essence model is: L = −V (φ)F (X) where X = 1 2 g µν ∇ µ φ∇ ν φ and it does not depend explicitly on φ to start with [2,3,5,6].Relativistic field theories with canonical kinetic terms have the distinction from those with non-canonical kinetic terms associated with k−essence, since the nontrivial dynamical solutions of the k-essence equation of motion not only spontaneously break Lorentz invariance but also change the metric for the perturbations around these solutions. Thus the perturbations propagate in the so called emergent or analogue curved spacetime [5] with the metric different from the gravitational one.…”