AdS and Lifshitz black hole solutions in conformal gravity sourced with a scalar field

Abstract

In this paper we obtain exact asymptotically anti-de Sitter black hole solutions and asymptotically Lifshitz black hole solutions with dynamical exponents z = 0 and z = 4 of four-dimensional conformal gravity coupled with a self-interacting conformally invariant scalar field. Then, we compute their thermodynamical quantities, such as the mass, the Wald entropy and the Hawking temperature. The mass expression is obtained by using the generalized off-shell Noether potential formulation. It is found that the anti-de Sitter black holes as well as the Lifshitz black holes with z = 0 have zero mass and zero entropy, although they have non-zero temperature. A similar behavior has been observed in previous works, where the integration constant is not associated with a conserved charge, and it can be interpreted as a kind of gravitational hair. On the other hand, the Lifshitz black holes with dynamical exponent z = 4 have non-zero conserved charges, and the first law of black hole thermodynamics holds. Also, we analyze the horizon thermodynamics for the Lifshitz black holes with z = 4, and we show that the first law of black hole thermodynamics arises from the field equations evaluated on the horizon. Furthermore, we study the propagation of a conformally coupled scalar field on these backgrounds and we find the quasinormal modes analytically in several cases. We find that for anti-de Sitter black holes and Lifshitz black holes with z = 4, there is a continuous spectrum of frequencies for Dirichlet boundary condition; however, we show that discrete sets of well defined quasinormal frequencies can be obtained by considering Neumann boundary conditions.

Spectroscopy of z=0 Lifshitz Black Hole

“…The Lifshitz BH solutions exist in the CG theory for both = 4 and = 0 [26,43]; however, when = 3 and = 4, Lifshitz BHs appear in the Horava-Lifshitz gravity [26,44,45]. Now, we focus on the 0LBH of the CG theory whose metric is given by [26]:…”

“…In the above metric, which is conformal to (A)dS (AdS if + 2 < 1 and dS if + 2 > 1) [26], = 0, = 1, and = −1 stand for 2-torus ( 2 + 2 ), unit 2-sphere ( 2 + sin 2 2 ), and unit hyperbolic plane ( 2 + sinh 2 2 ), respectively [45]. The metric solution has a curvature singularity at = 0, which becomes naked for = 0.…”

“…The metric solution has a curvature singularity at = 0, which becomes naked for = 0. There is an event horizon for = ±1 solution expressed as follows [26,45]:…”

“…Asymptotic limits of the potential (36) show that the effective potential may diverge beyond the BH horizon for the massive scalar particle; thus, in the far region, the QNMs might not be perceived by the observer. Hence, following a particular method [23,[32][33][34][35], we focus our analysis on the near-horizon (NH) region and impose the boundary conditions (45): ingoing waves at the event horizon and no wave at spatial infinity (since at infinity the effective potential of Schrödinger-like wave equation is divergent). After getting NH form of the SLE, we show that the radial equation is reduced to a confluent hypergeometric (CH) differential equation [36].…”

Mass and angular momentum of charged rotating Gödel black holes in five-dimensional minimal supergravity

“…Consequently, in some sense, the offshell generalized ADT formalism could also be viewed as the cousin of both the BBC and phase space approaches. Till now, there have existed a series of developments and applications of the off-shell ADT formulation in a range of gravity theories regardless of the asymptotic behavior of the spacetime metrics [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].…”