Nonlinearly charged dilatonic black holes and their Brans–Dicke counterpart: energy dependent spacetime

Abstract

Regarding the wide applications of dilaton gravity in the presence of electrodynamics, we introduce a suitable Lagrangian for the coupling of dilaton with gauge field. There are various Lagrangians which show the coupling between scalar fields and electrodynamics with correct special situations. In this paper, taking into account conformal transformation of Brans-Dicke theory with an electrodynamics Lagrangian, we show that how scalar field should couple with electrodynamics in dilaton gravity. In other words, in order to introduce a correct Lagrangian of dilaton gravity, one should check at least two requirements: compatibility with Brans-Dicke theory and appropriate special situations. Finally, we apply the mentioned method to obtain analytical solutions of dilaton-Born-Infeld and Brans-Dicke-Born-Infeld theories with energy dependent spacetime.

Extended phase space thermodynamics and P–V criticality: Brans–Dicke–Born–Infeld vs. Einstein–Born–Infeld-dilaton black holes

“…It is a matter of calculation to show that there is no consistent conformal transformation between the actions (1) and (23) [74]. In other words, although the action (1) leads to the well-known action of Maxwell-dilaton gravity for β → ∞, it is not conformally related to the BD-BI action, and therefore, Eq.…”

“…Considering the conformally ill-defined action (1), in the next section, we follow the method of Ref. [74] to obtain a conformally well-defined action of BI-dilaton gravity.…”

Phase Transition of Black Holes in Brans–Dicke Born–Infeld Gravity through Geometrical Thermodynamics

“…The Einstein-BI-dilaton gravity action and its related field equations can be obtained from the BD-BI action and its related field equations by applying the mentioned conformal transformation [17]:…”

“…It was shown that the proper potential is a Liouville-type one with both topological and BI correction terms, as [17] V (Φ) = 2Λ exp ( 4 Φ − 1 )…”

“…To deal with this issue, one could apply conformal transformation on known solutions of other modified theories like dilaton gravity [16]. For instance, nonlinearly charged dilatonic black hole solutions and their BD counterpart in an energy dependent space-time have been obtained by applying a conformal transformation [17]. The first attempt for modifying the Maxwell theory to a consistent theory for describing point charges was made in 2 Advances in High Energy Physics 1912 by Mie [18,19].…”