Charged spherically symmetric black holes in f(R) gravity and their stability analysis
aa r X i v : . [ g r- q c ] J u l Charged spherically symmetric black holes in f ( R ) gravity and their stability analysis G. G. L. Nashed , and S. Capozziello , , , Centre for Theoretical Physics, The British University in Egypt, P.O. Box 43,El Sherouk City, Cairo 11837, Egypt Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt Dipartimento di Fisica “E. Pancini“, Universit´a di Napoli “Federico II”,Complesso Universitario di Monte Sant’ Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli,Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy Gran Sasso Science Institute, Viale F. Crispi, 7, I-67100, L’Aquila, Italy. Laboratory for Theoretical Cosmology,Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia. (Dated: July 18, 2019)A new class of analytic charged spherically symmetric black hole solutions, which behave asymptoticallyas flat or (A)dS spacetimes, is derived for specific classes of f ( R ) gravity, i.e., f ( R ) = R − α √ R and f ( R ) = R − α √ R − Λ , where Λ is the cosmological constant. These black holes are characterized by the dimensionalparameter α that makes solutions deviate from the standard solutions of general relativity. The Kretschmannscalar and squared Ricci tensor are shown to depend on the parameter α which is not allowed to be zero.Thermodynamical quantities, like entropy, Hawking temperature, quasi-local energy and the Gibbs free energyare calculated. From these calculations, it is possible to put a constrain on the dimensional parameter α to have0 < α < .
5, so that all thermodynamical quantities have a physical meaning. The interesting result of thesecalculations is the possibility of a negative black hole entropy. Furthermore, present calculations show that fornegative energy, particles inside a black hole, behave as if they have a negative entropy. This fact gives rise toinstability for f RR <
0. Finally, we study the linear metric perturbations of the derived black hole solution. Weshow that for the odd-type modes, our black hole is always stable and has a radial speed with fixed value equalto 1. We also, use the geodesic deviation to derive further stability conditions.
I. INTRODUCTION
Challenging problems ranging from quantum gravity to dark energy (DE) and dark matter (DM) give support to searchfor other gravitational theories beyond the standard Einstein general relativity (GR). Actually, GR has many unsolved issueslike singularities, the nature of DE and DM, etc. All these issues encourage scientists to modify GR or extend it in view ofaddressing shortcomings at UV and IR scales [1]. In other words, viable modified/extended theories should be compatiblewith the current experimental constraints and should give motivations on issues in quantum gravity and cosmology. Thus, it isstraightforward to extend directly GR considering it as a limit of a more general theory of gravitation . Among the possibleextensions of GR, the so-called f ( R ) gravity generalizes the Einstein-Hilbert action by substituting the Ricci scalar R by ananalytic differentiable function. The fundamental reasons for this approach come out from the formulation of any quantumfield theory on curved spacetime [3]. f ( R ) gravity have some important applications, like the Starobinsky model, f ( R ) = R + α R , α > f ( R ) gravity is capableof explaining the observed cosmic acceleration without assuming the cosmological constant. Possible toy models have the form f ( R ) = R − β R n where β and n have positive values [7–9]. Nevertheless, this model suffers from instability problems becauseof the second derivative of the function f that has a negative value, i.e., f RR < f ( R ) [20, 21]. Finally f ( R ) models give interesting results for structure formation, as themodification of the spectra of galaxy clustering, CMB, weak lensing, etc. [22–30]. There are many applications of f ( R ) fromthe astrophysical point of view [31–38], for general reviews of f ( R ) gravity, see [1, 39–41].From the viewpoint of mathematics, modified/extended gravity poses the issue to establish or modified well-known facts ofGR like the stability of solutions, initial value problem and the problem of deriving new black hole solutions [42–46]. As it iswell-known, in addition to the cosmological solutions, there exist axially symmetric as well as spherical ones that could have amain role in several astrophysical problems spanning from black hole solutions to galactic nuclei. Modified gravitational theoriesmust include black hole solutions like Schwarzschild-like in order to be compatible with GR results and, in principle, must givenew black hole solutions that might have physical interest. According to this fact, the way to find out exact or approximate blackhole solutions is highly important to investigate if observations can be matched to modified/extended gravity [47, 48]. It is worth saying that modified gravity means that GR is not recovered but equivalent scheme as the Teleparallel Equivalent General Relativity (TEGR) canbe recovered. Extended gravity means that in a given limit, or for a given choice GR is recovered. For a discussion see [2].
In the framework of f ( R ) gravity there is a specific interest for spherically symmetric black hole solutions. They have beenderived using constant Ricci scalar [49]. Moreover, spherically symmetric black hole solutions, including perfect fluid matter,have been analyzed [50]. Additionally, by using the method of Noether symmetry, many spherically symmetric black holes havebeen derived [50]. Hollenstein and Lobo [51] derived exact solutions of static spherically symmetric spacetimes in f ( R ) coupledto non-linear electrodynamics. For the readers interested in the static black holes we refer to [52–87] and the references therein.Using the Lagrangian multiplier, new analytic solutions with dynamical Ricci scalar have been derived [88]. It is the purpose ofthe present study, by using the field equation of f ( R ), to generalize these black hole solutions [88] and derive new charged blackhole solutions with dynamical Ricci scalar asymptotically converging towards flat or (A)dS spacetimes.Gravitational stability of a black hole solution is considered as a main problem for checking the adequateness of any black holesolutions [89, 90]. However, the stability analysis appears not directly to be applicable to f ( R ) black hole solutions because itinvolves fourth-order derivative terms in the linearized equations [91, 92]. In that case, it is necessary that the black holes are freefrom tachyon and ghost instabilities that would come into the game as soon as one is considering f ( R ) gravity [93]. Therefore,one may transform f ( R ) gravity into the corresponding scalar-tensor theory to remove the fourth-order derivative terms [94]. Itwas suggested that the stability of black hole solutions does not rely on the frame due to the fact that it is a classical solutionthat is considered as the ground state [95]. It is well known that a non-minimally coupled scalar makes the linearized GR fieldequations around the black hole very intricate when compared to a minimally coupled scalar in the context of GR [96]. Due tothis intricacy, some people have used conformal transformations to find the corresponding theory in the Einstein frame wherea minimally coupled scalar appears. Taking into account these difficulties, several perturbation studies on the black holes indifferent modified gravitational theories have been developed. See, for example, [93, 97–100].The paper is organized as follows. In Sec. II, a summary of Maxwell- f ( R ) gravity is provided. In Sec. III, a sphericallysymmetric ansatz is applied to the field equation of Maxwell- f ( R ) theory and an exact solution is derived. In Sec. IV, thesame spherically symmetric ansatz is applied to the field equation of Maxwell- f ( R ) theory that includes a cosmological con-stant. Solving the resulting differential equations, we derive a new black hole solution that behaves as (A)dS. In Sec. V, thecharacteristic properties of these black holes are analyzed. In Sec. VII C, thermodynamical quantities like entropy, quasi-localenergy, Hawking temperature, Gibbs energy are calculated. We show that the entropy of the derived black hole solutions are notproportional to the horizon area and show some regions of the parameter space where the entropy becomes negative. The mainreason for this result is due to the parameter α related to the higher order correction. In Sec. VII we study the linear stabilityusing the odd perturbations to the black holes derived in Secs. III and IV. Furthermore, in Sec. VII, we derive the stabilityconditions considering the geodesic motion. In Sec. VIII, we discuss the main results of the present study and draw conclusions. II. MAXWELL– f ( R ) GRAVITY
The theory of gravity that will be considered in this work is the f ( R ) gravity which was first taken into account in [101]. Seealso [1, 7–9, 102]: S : = S g + S E . M . , (1)where S g is the gravitational action given by: S g : = κ Z d x √− g ( f ( R ) − Λ ) , (2)where Λ represents the cosmological constant, R is the Ricci scalar, κ is the gravitational constant, g is the determinant of themetric and f ( R ) is an analytic differentiable function. In this study S E . M . is the action of the non-linear electrodynamics fieldwhich takes the form: S E . M . : = − F s , (3)where s ≥ F = F µν F µν , where F µν = A [ µ,ν ] with A µ being the gauge potential 1-form and the comma denotes the ordinary differentiation [82].The field equations of f ( R ) gravitational theory can be obtained by carrying out the variations of the action given by Eq. (1)with respect to the metric tensor g µν and the strength tensor F that yield the following form of the field equations [103, 104]: I µν = R µν f R − g µν f ( R ) − g µν Λ + g µν ✷ f R − ∇ µ ∇ ν f R − π T µν ≡ , (4) The square brackets represent the anti-symmetrization, i.e. A [ µ,ν ] = ( A µ,ν − A ν,µ ) and the symmetric one is represented by A ( µ,ν ) = ( A µ,ν + A ν,µ ). ∂ ν (cid:16) √− g F µν F s − (cid:17) = , (5)with R µν being the Ricci tensor defined by R µν = R ρµρν = Γ ρµ [ ν,ρ ] + Γ ρβ [ ρ Γ βν ] µ , where Γ ρµν is the Christoffel symbols of second kind. The d’Alembert operator ✷ is defined as ✷ = ∇ α ∇ α where ∇ α V β is thecovariant derivatives of the vector V β and f R = d f ( R ) dR . In this study T µν is defined as T µν : = π s g ρσ F νρ F µσ F s − −
14 g µν F s ! , (6)which is the energy momentum-tensor of the non-linear electrodynamic field. When s =
1, we get the standard energy-momentum tensor of Maxwell field.The trace of equation (4) is:
R f R − f ( R ) − Λ + ✷ f R = T where T = F s ( sF − F s ) , (7)It is worth noticing that, for s =
1, it is T =
0. This property means that the Maxwell field is conformally invariant. In thefollowing, we are going to assume some form of the field Eqs. (4) without and with a cosmological constant to derive exactsolutions that asymptotically behave as flat or (A)dS spacetimes.
III. AN EXACT CHARGED BLACK HOLE SOLUTION
Let us derive a charged black hole solution adopting the model f ( R ) = R − α √ R . To this aim, we are going to use thefollowing spherically symmetric ansatz : ds = B ( r ) dt − dr B ( r ) − r d Ω , (8)where d Ω = d θ + sin θ is the line element on the unit sphere. The Ricci scalar of the metric (8) has the form R = − r B ′′ − rB ′ − Br . (9)Applying the ansatz (8) to Eqs. (4), (5) and (7), after using (9) and putting the parameter s =
1, we get the following non-vanishing field equations : The reason to take the ansatz (8) is to be able to find an exact solution. Other forms make the field equations very complicated and not easy to solve. Here and through all this study B ≡ B ( r ), B ′ = dB ( r ) dr , B ′′ = d B ( r ) dr , etc. Also in this application we put Λ = I tt = r √ R n r √ R [ r BB ′′′′ + B ′′′ (1 / r B ′ + r B ) + r B ′′ ( B + rB ′ ) − r B ′ + B ′ ( r − rB ) + B ( B − + r √ R h r B ′ + r RBB ′′′′ − r BB ′′′ + / r B ′′′ { r B ′′ ( rB ′ − B ) + B ′ (29 rB − r ) + B ( B − } + r B ′′ ( r q ′′ − B + rB ′ − + r B ′′ × [23 r B ′ + rB ′ (23 B + r q ′ − + B − B + r q ′ − + r B ′ + B ′ (34 r B + r q ′ − r ) + rB ′ ( B − × (7 B + r q ′ − + B − [ r q ′ − i + r R h B √ R [4 r B ′′ + r B ′′′ − rB ′ + − B )] + α n r RBB ′′′′ − / r BB ′′′ + / r B ′′′ [ r B ′′ ( rB ′ − B ) + r B ′ + B ′ (31 rB − r ) + B ( B − − r B ′′ − r B ′ + r B ′ [96 B − + B ′′ (cid:16) r − r B ′ − r B (cid:17) + r B ′′ [57 r B ′ + rB ′ [3 B − + + B (4 − B )] − rB ′ ( B − B − − B − (2 B − oio = , I rr = − r √ R n r √ R h ( rB ′ + B )[4(1 − B ) − rB ′ + r (4 B ′′ + rB ′′′ )] − (cid:16) r B ′′′ [ rB ′ + B ] + r B ′′ [5 B + r q ′ + rB ′ − + r B ′ + B ′ [16 r q ′ + rB − r ] + B − r q ′ − B − (cid:17)i − α r R (cid:16) r B ′′′ [4 B + rB ′ ] − r B ′′ + B ′′ h r ( B + − r B ′ i − r B ′ + rB ′ (15 − B ) − B − B − (cid:17)o = , I θθ = I φφ = − r √ R n r √ R h r BB ′′′′ + r B ′′′ ( rB ′ + B ) + r B ′′ (2 rB ′ − B ) + rB ′ (2 − B ) − r B ′ + B ( B − i − r √ R h r BRB ′′′′ − r BB ′′′ + r B ′′′ n r B ′′ ( rB ′ − B ) + r B ′ + rB ′ (13 B − + B ( B − o + r B ′′ − r B ′′ ( r q ′ + − rB ′ + B ) + rB ′′ (cid:16) r B ′ − rB ′ [14 + r q ′ − B ] + B − − B + r q ′ ] (cid:17) + r B ′ − rB ′ (3 + r q ′ ) + B ′ ( B − B − r q ′ ) − rq ′ ( B − i + r R h √ R ( B [( B − − r B ′′′ − r B ′′ + rB ′ ] ) + α (cid:16) r BB ′′′′ R + / r BB ′′′ − r B ′′′ [ r B ′′ { rB ′ − B } + r B ′ + rB ′ [14 B − + B ( B − + r B ′′ + r B ′′ [18 B + rB ′ − + r B ′′ [33 r B ′ + rB ′ (27 B − − B − B − + r B ′ + r B ′ (74 − B ) + rB ′ ( B − B − − { − B − B + B } (cid:17)io = , I = − r √ R n r √ R [ r BB ′′′′ + r B ′′′ ( rB ′ + B ) + r B ′′ ( B + rB ′ ) − r B ′ + rB ′ (1 − B ) + B ( B − + r √ R h r RBB ′′′′ + r BB ′′′ − r B ′′′ { r B ′′ [ rB ′ − B ] + r B ′ + rB ′ (15 B − + B ( B − } + / r B ′′ + r B ′′ (6 rB ′ − B − + r B ′′ { r B ′ + rB ′ (14 B − + B − B + } + / r B ′ + r B ′ (6 B − + rB ′ ( B − B − − / B − + B i − r R h B √ R (4 + r B ′′′ − B + r B ′′ − rB ′ ) − α (cid:16) r RBB ′′′′ + / r BB ′′′ − r B ′′′ n r B ′′ ( rB ′ − B ) + r B ′ + rB ′ (16 B − + B ( B − o + r B ′′ + r B ′′ { rB ′ + B − } + r B ′′ { r B ′ + rB ′ (16 B − + − B + B ) } + r B ′ − r B ′ (106 − B ) + rB ′ ( B − B − + B − B − (cid:17)io = , (10)where q is the gauge potential which is defined as A : = q ( r ) dt . (11)If we subtract I tt from I rr and solving the system I tt − I rr and I θθ , which is a closed system for the two unknown functions B ( r )and q ( r ), we get the following exact solution B ( r ) = − α r + α r , A = √ α r . (12)The analytic solution (12) satisfy the system of differential equations (10) including the trace equation I . Using Eq. (9) we getthe Ricci scalar in the form R = r , (13)which is also a consistency check for the whole procedure. The metric of the above solution takes the form ds = − α r + α r ! dt − − α r + α r ! − dr − r d Ω , (14)which asymptotically behaves as a flat space-time. We have to stress the fact that solution (12) is different from that obtained in[88] due to the fact that they derived their solution using the form f ( R ) = R + α √ R . Therefore, our solution is identical with theirone when we neglect the term 13 α r , which is responsible for the electric charge, and reverse the negative sign to be positive tosatisfy the field equation of f ( R ) = R + α √ R . We must stress on the fact that the dimensional parameter α must take a positivevalue so that solution (12) satisfies the field equations (4), (5) and (7). IV. AN EXACT (A)DS CHARGED BLACK HOLE SOLUTION
Let us derive now a charged (A)dS black hole solution for the model f ( R ) = R − α √ R − Λ . Applying the anzatz (8) to thefield Eqs. (4), (5) and (7), after using (9) and putting s =
1, we get the following non-vanishing field equations We define R = R − Λ . I tt = r √R n r p R [ r BB ′′′′ + B ′′′ (1 / r B ′ + r B ) + r B ′′ ( B + rB ′ ) − r B ′ + rB ′ (1 − B ) + B ( B − + r p R h r B ′ + r R BB ′′′′ − r BB ′′′ + / r B ′′′ { r B ′′ ( rB ′ − B ) + B ′ (29 rB − r + r Λ ) + B (5 B + r Λ − } + r B ′′ ( r q ′′ + r Λ − B + rB ′ − + r B ′′ h r B ′ + rB ′ (23 B + r q ′ − + r Λ ) + r q ′ ( B − + r Λ ) + B + B (8 r Λ − + r Λ (4 r Λ − + i + r B ′ + r B ′ (34 B + r q ′ + r Λ − + rB ′ ( B − + r Λ ) (cid:16) B + r q ′ − + r Λ (cid:17) + r q ′ ( B − + r Λ ) + B [14 r Λ − + B (16 r Λ − r Λ + + r Λ − r Λ − i + r R h B √R [4 r B ′′ + r B ′′′ − rB ′ + − B )] + α n r R BB ′′′′ − / r BB ′′′ + / r B ′′′ h r B ′′ ( rB ′ − B ) + r B ′ + rB ′ (31 B − + r Λ ) + B ( B − + r Λ ) i − r B ′′ − r B ′ + r B ′ [96 B − + r Λ ] + r B ′′ (cid:16) − rB ′ − rB + r Λ (cid:17) + r B ′′ [57 r B ′ + rB ′ [21 B − + r Λ ] + B (4 − B − r Λ ) + (4 r Λ − ] − rB ′ h B + B (47 r Λ − − − r Λ (88 r Λ − i − B − (2 B − + B r Λ + Br Λ (3 r Λ − + (4 r Λ − ] oio = , I rr = − r √R n r p R h ( rB ′ + B )[4(1 − B ) − rB ′ + r (4 B ′′ + rB ′′′ )] − (cid:16) r B ′′′ [ rB ′ + B ] + r B ′′ [5 B + r q ′ + rB ′ − + r Λ ] + r B ′ + rB ′ [16 r q ′ + B − + r Λ ] + B − + r Λ ) r q ′ + + r Λ − B − r Λ (1 + B ) (cid:17)i − α r R (cid:16) r B ′′′ [4 B + rB ′ ] − r B ′′ + r B ′′ [4( B + + r Λ − rB ′ ] − r B ′ + rB ′ (15 − B − r Λ ) − B − B − + r Λ (4 B + r Λ + (cid:17)o = , I θθ = I φφ = − r √R n r p R h r BB ′′′′ + r B ′′′ ( rB ′ + B ) + r B ′′ (2 rB ′ − B ) + rB ′ (2 − B ) − r B ′ + B ( B − i − r p R h r BRB ′′′′ − r BB ′′′ + r B ′′′ n r B ′′ ( rB ′ − B ) + r B ′ + rB ′ (13 B − + r Λ ) + B (9 B − + r Λ ) o + r B ′′ − r B ′′ ( r q ′ + − rB ′ + B − r Λ ) i + rB ′′ (cid:16) r B ′ − rB ′ [14 + r q ′ − B − r Λ ] + B − − r Λ ) r q ′ − B [10 B + r Λ − + r Λ [3 + r Λ ] + (cid:17) + r B ′ − rB ′ (3 + r q ′ − r Λ ) + B ′ h ( B − + r Λ ) r q ′ + B [3 B − + r Λ ] + r Λ [4 r Λ − − r [ q ′ ( B − + r Λ ) − B (5 B + r Λ − + r Λ − ] i + r R h √R B h B − − r B ′′′ − r B ′′ + rB ′ i + α (cid:16) r R BB ′′′′ + / r BB ′′′ − r B ′′′ h r B ′′ { rB ′ − B } + r B ′ + rB ′ [14 B − + r Λ ] + B (11 B − + r Λ ) i + r B ′′ + r B ′′ [18 B + rB ′ − + r Λ ] + r B ′′ h r B ′ + rB ′ (27 B − + r Λ ) − B − B − − r Λ { B + r Λ + } i + r B ′ + r B ′ (74 − B + r Λ ) + rB ′ h B − B + Br Λ + r Λ + − r Λ i − { − B − r Λ − B + B − r Λ + r Λ } + (8 r Λ − r Λ − (cid:17)io = , I = − r √R n r p R [ r BB ′′′′ + r B ′′′ ( rB ′ + B ) + r B ′′ ( B + rB ′ ) − r B ′ + rB ′ (1 − B ) + B ( B − + r p R h r R BB ′′′′ + r BB ′′′ − r B ′′′ { r B ′′ [ rB ′ − B ] + r B ′ + rB ′ (15 B − + r Λ ) + B (5 B − + r Λ ) } + / r B ′′ + r B ′′ (6 rB ′ − B − + r Λ ) + r B ′′ { r B ′ + rB ′ (14 B − + r Λ ) + B (6 B − + r Λ ) + (4 r Λ − } + / r B ′ + r B ′ (6 B − + r Λ ) + rB ′ ( B − + r Λ )(2 B − + r Λ ) − / B + B + B r Λ + r B Λ [8 r Λ − + / r Λ − i − r R h B √R (4 + r B ′′′ − B + r B ′′ − rB ′ ) − α (cid:16) r R BB ′′′′ + / r BB ′′′ − r B ′′′ n r B ′′ ( rB ′ − B ) + r B ′ + rB ′ (16 B − + r Λ ) + B ( B − + r Λ ) o + r B ′′ + r B ′′ { rB ′ + B − + / r Λ } + r B ′′ { r B ′ + rB ′ (16 B − + / r Λ ) + − B + B + / r Λ ) + / r Λ ( r Λ − } + r B ′ − r B ′ (106 − B − / r Λ ) + rB ′ (15 B + B [344 / r Λ − + r Λ / r Λ − + + B + B [34 / r Λ − + B [2 − / r Λ + / r Λ ] + r Λ − × (16 r Λ − (cid:17)io = . (15)If we subtract the component I tt from the component I rr and solve the system I tt − I rr and I θθ which is a closed system for thetwo unknowns functions B ( r ) and q ( r ), we get the exact solution B ( r ) = − r Λ − α r + α r , A = √ α r , (16)Using Eq. (16) in (9) we get the Ricci scalar in the form R = r Λ + r . (17)The metric of the above solution takes the form ds = − r Λ − α r + α r ! dt − − r Λ − α r + α r ! − dr − r d Ω . (18)which behaves asymptotically as (A)dS spacetime. Solution (16) is different from that derived in [88] due to the reason discussedfor solution (12). Same constrain put on the parameter α in the non charge case is also true here. V. PHYSICAL PROPERTIES OF THE BLACK HOLES
The metric of solution (12) can be rewritten in the form ds = − Mr + q r ! dt − − Mr + q r ! − dr − r d Ω , where M = α , q = √ α , (19)which shows clearly that the dimensional parameter α cannot be equal zero and, in that case, the line element coincides with theReissner-Nordstr¨om spacetime. Also the metric of solution (16) may be rewritten as ds = − r Λ − Mr + q r ! dt − − r Λ − Mr + q r ! − dr − r d Ω , where, again M = α and q = √ α . (20)which shows that line element coincides with the (A)dS Reissner-Nordstr¨om spacetime. Equations (19) and (20) show in a clearway that the dimensional parameter α must not equal zero.Let us study now the regularity of the solutions (12) and (16) when B ( r ) =
0. For solution (12), we evaluate the scalarinvariants and get R µνλρ R µνλρ = + r α + α r + r [ α + + r α r , R µν R µν = r α − α r + α r , R = r , (21)where R µνλρ R µνλρ , R µν R µν , R are the Kretschmann scalars, the Ricci tensor square, the Ricci scalar, respectively. Equations (21)show that the solutions, at r =
0, have true singularities and the dimensional parameter α ,
0. Also Eq. (12) as well as Eq. (14)show clearly that the dimensional parameter α cannot be equal to zero which insure that solution (12) cannot reduce to GR. Thismeans that this solution is a new exact charged one in the frame of f ( R ) gravitational theory.Using Eq. (16) we get the scalar invariants in the form R µνλρ R µνλρ = r Λ α + r Λ α + r α + α r + r [ α + + r + α r , R µν R µν = r Λ α + r Λ α + α r − α r + α r , R = r Λ + r . (22)The same considerations carried out for solution (12) can also be applied for solution (16) which insure also that solution (16) isa novel charged one in the framework of f ( R ) gravity that cannot reduce to GR. VI. BLACK HOLE THERMODYNAMICS
Now we are going to explore the thermodynamics of the new black hole solutions derived in the previous sections. TheHawking temperature is defined as [105–108] T + = B ′ ( r + )4 π , (23)where the event horizon is located at r = r + which is the largest positive root of B ( r + ) = B ′ ( r + ) ,
0. TheBekenstein-Hawking entropy in the framework of f ( R ) gravity is given as [105–110] S ( r + ) = A f R ( r + ) , (24)where A is the area of the event horizon. The form of the quasi-local energy in the framework of f ( R ) gravity is defined as[105–110] E ( r + ) = Z " f R ( r + ) + r + n f ( R ( r + )) − R ( r + ) f R ( r + ) o dr + . (25)At the horizon, one has the constraint B ( r + ) = r + Eq . (12) = α h + √ + α i , r − Eq . (12) = α h − √ + α i r + Eq . (16) = Root (4 x α Λ − α x + x + , (26)where Root (4 x α Λ − α x + x +
2) is the roots of the equation (4 x α Λ − α x + x + = α should not be equal zero to ensures that the black hole (12) has no analogy with GR. Moreover, Eqs. (26) tellus that the dimensional parameter α should be positive so that the horizons have a positive real value. Therefore we must put therestriction α >
0, otherwise we get a non-real value for the horizon. This constraint is consistent with the relation given by Eqs.(19) and (20) which allows the mass parameter to have the correct sign in the metric and the charge parameter has a real value.Moreover, if the parameter α takes a negative value, the solutions (12) and (16) do not satisfy the field Eqs. (4), (5) and (7).The relation between the radial coordinate r and the dimensional parameter α of the black hole (12) is represented in Figure1. From this figure we can see the root of B ( r ) defining the black hole outer event horizon r + [111]. We can continue the studyof thermodynamics assuming α > r + onlywhich is consistent with α > S + Eq . (12) = π α h + √ + α i h − √ + α i , S + Eq . (16) = π r + − α r + ] . (27)The first equation of Eq. (27) shows that, in order to have a positive entropy, the dimensional parameter α must take the value0 < α < .
5. The second equation of (27) tells us that we must have α < r + for positive entropy. Eqs. (27) are drawn in Figure2. As one can see, from Figure 2 (a), for 0 . > α > + ve entropy. For the black hole (16), as Figure 2(b) shows, we have a phase transition at 2 . − ve value for 0 < α < . + ve at r = . S is not proportionalto the area of the horizon due to Eq. (24). We should also note that the entropy S is proportional to the area if there is no Ricciscalar squared term i.e. f R =
1. The Hawking temperatures associated with the black hole solutions (12) and (16) are T + Eq . (12) = α (1 + √ + α + α )4 π (1 + √ + α ) , T + Eq . (16) = r + − α Λ r + + πα r + , (28) We substitute the value of α in terms of Λ using Eq. (16) through this section. Figure 1. Schematic plot of the radial coordinate r versus the dimensional parameter α that characterize the spherically symmetric black hole[12]. (a) The entropy of the black hole solution (12) (b) The entropy of the black hole solution (16) Figure 2. Schematic plot of the entropy of the two black holes (12) and (16) versus the dimensional parameter α and r + respectively. where T + is the Hawking temperature at the event horizon. We represent the Hawking temperature in Figure 3. Figure 3 (a),which is related to the black hole (12), shows that we have a positive temperature when the parameter α has the value 0 < α < . + ve value. From Eq. (25), the quasi-local energyof the two black holes (12) and (16) are calculated as E + Eq . (12) = + √ + α − α α , E + Eq . (16) = r + (cid:16) − α r + + Λ α r + (cid:17) . (29)The first equation of (29) shows that the dimensional parameter has to be α ,
0. We plot the energy in Figure 4 which shows0 (a) The Hawking temperature of the black hole solution (12) (b) The Hawking temperature of the black hole solution (16)
Figure 3. Schematic plot of the Hawking temperature of the two black holes (12) and (16) versus the dimensional parameter α and r + ,respectively. (a) The quasilocal energy of the black hole solution (12) (b) The quasilocal energy of the black hole solution (16) Figure 4. Schematic plot of the quasilocal energy of the black holes (12) and (16) versus the dimensional parameter α and r + , respectively. that for Figure 4 (a), the quasi-local energy has a + ve value when 0 < α < .
5. In the other case, we have a negative value forthe quasi-local energy till r + = G ( r + ) = E ( r + ) − T ( r + ) S ( r + ) (30)where E ( r + ), T ( r + ) and S ( r + ) are the quasilocal energy, the temperature and entropy at the event horizons, respectively. Using1 (a) The free energy of the black hole solution (12) (b) The free energy of the black hole solution (16) Figure 5. Schematic plot of the free energy of the black holes (12) and (16) versus the dimensional parameter α and r + , respectively. Eqs. (24), (26), (27) and (29) in (30) we get G + Eq . (12) = + √ + α − α α − α r + (1 + √ + α + α )(2 − √ + α )4(1 + √ + α ) , G + Eq . (16) = r + (4 − α r + Λ α r + )8 − ( r − Λ α r + − α r )48 α r + . (31)The behaviors of the Gibbs energy of our black holes are presented in Figures 5(a), 5(b) for particular values of the modelparameters. As Figure 5(a) shows, for the black hole solution (12), the Gibbs energy is positive when 0 < α < . r < .
73 and positive when 2 . < r . VII. THE STABILITY OF CHARGED BLACK HOLE SOLUTIONS IN f ( R ) GRAVITY
In order to study the stability of the above black hole solutions, it is better recast f ( R ) gravity in terms of the correspondingscalar- tensor theory. Discarding the cosmological term, the Lagrangian (2) can be rewritten as S = κ Z d x √− g [ φ R − V ( φ )] , (32)where φ is a scalar field coupled to the Ricci scalar R and V ( φ ) is the potential (see [1] for details). Here, we will discuss thebehavior of the perturbations about a static spherically symmetric vacuum background, whose metric is written as above, that is ds = g µν dx µ dx ν = B ( r ) dt − dr B ( r ) − r ( d θ + sin θ d φ ) . (33)where g µν is the background metric. Considering the above black hole solutions, we want to investigate whether these back-grounds are stable or not against linear perturbations, and what we can learn in terms of speed of propagation for the scalargravitational modes. For such a theory, the background equations of motion read V = − B φ ′ r − φ B ′ r − φ ′ B ′ + φ r − B φ r , φ ′′ = , R = dVd φ , (34)where ′ stands for differentiation with respect to r .2 A. Outline of the Regge-Wheeler-Zerilli formalism
Before studying the metric perturbation of static spherically symmetric spacetime of f ( R ) gravity, let us give a brief summaryof the formalism developed by Regge, Wheeler [113], and Zerilli [114] to decompose the metric perturbations according to theirtransformation properties under two-dimensional rotations. Although Regge, Wheeler and Zerilli considered the perturbationsof the Schwarzschild space-time in GR, the formalism depends on the properties of spherical symmetry and then can be appliedto f ( R ) gravity as well.Let us denote the metric slightly perturbed from a static spherically symmetric spacetime by g µν = g µν + h µν , where h µν represents infinitesimal quantities. In the lowest, linear approximation, the perturbations are supposed to be very smaller withrespect to the background, that is g µν >> h µν . Then, under two-dimensional rotations on a sphere, h tt , h tr and h rr transform asscalars, h ta and h ra transform as vectors and h ab transforms as a tensor ( a , b are either θ or φ ). Any scalar quantity Φ can beexpressed in terms of the spherical harmonics Y ℓ m ( θ, φ ) Φ ( t , r , θ, φ ) = X ℓ, m Φ ℓ m ( t , r ) Y ℓ m ( θ, ϕ ) . (35)In the spherically symmetric spacetimes the solution will be independent of the index m , therefore this subscript can be omittedand we take into account only the index ℓ which represents the multipole number, which arises from the separation of angularvariables by the expansion into spherical harmonics ∆ θ,φ Y ℓ ( θ, φ ) = − ℓ ( ℓ + Y ℓ ( θ, φ ) , (36)exactly in the same way as it happens for the hydrogen atom problem in quantum mechanics when dealing with the Schr¨odingerequation. Any vector V a can be decomposed into a divergence part and a divergence-free part as follows: V a ( t , r , θ, φ ) = ∇ a Φ + E ba ∇ b Φ , (37)where Φ and Φ are two scalars and E ab ≡ p det γ ǫ ab with γ ab being the two-dimensional metric on the sphere and ǫ ab being thetotally anti-symmetric symbol with ǫ θϕ =
1. Here ∇ a represents the covariant derivative with respect to the metric γ ab . Since V a is a two-component vector, it is completely specified by the quantities Φ and Φ . Then we can apply the scalar decomposition(35) to Φ and Φ to decompose the vector quantity V a into spherical harmonics.Finally, any symmetric tensor T ab can be decompose as T ab ( t , r , θ, φ ) = ∇ a ∇ b Ψ + γ ab Ψ +
12 ( E ac ∇ c ∇ b Ψ + E bc ∇ c ∇ a Ψ ) , (38)where Ψ , Ψ and Ψ are scalars. Since T ab has three independent components, Ψ , Ψ and Ψ completely specify T ab . Then wecan again apply the scalar decomposition (35) to Ψ , Ψ and Ψ to decompose the tensor quantity T ab into spherical harmonics.We refer to the variables accompanied by E ab by odd-type variables and the others by even-type variables. What makes thesedecompositions useful is that, in the linearized equations of motion (or equivalently, in the second order action) for h µν , odd-typeand even-type perturbations are completely decoupled. This fact reflects the invariance of the background spacetime under paritytransformations. Therefore, one can study odd-type perturbations and even-type ones separately as we will do in the following. B. Perturbations in f ( R ) gravity The odd modesIt is well known that there are two classes of vector spherical harmonics (polar and axial) which are build out of combinationsof the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics. The difference between thetwo families is their parity. Under the parity operator π a spherical harmonic with index ℓ transforms as ( − ℓ , the polar class ofperturbations transform under parity in the same way, as ( − ℓ and the axial perturbations as ( − ℓ + .Using the Regge-Wheeler formalism, the odd-type metric perturbations can be written as h tt = , h tr = , h rr = , (39) h ta = X ℓ, m h ,ℓ m ( t , r ) E ab ∂ b Y ℓ m ( θ, ϕ ) , (40) h ra = X ℓ, m h ,ℓ m ( t , r ) E ab ∂ b Y ℓ m ( θ, ϕ ) , (41) h ab = X ℓ, m h ,ℓ m ( t , r ) h E ca ∇ c ∇ b Y ℓ m ( θ, ϕ ) + E cb ∇ c ∇ a Y ℓ m ( θ, ϕ ) i . (42)3Using the gauge transformation x µ → x µ + ξ µ , where ξ µ are infinitesimal, we can show that not all the metric perturbationsare physical and some of them can be set to vanish. For the odd-type perturbation, we can consider the following gaugetransformation: ξ t = ξ r = , ξ a = X ℓ m Λ ℓ m ( t , r ) E ba ∇ b Y ℓ m , (43)where Λ ℓ m can always set h ,ℓ m to vanish (Regge-Wheeler gauge). By this procedure, Λ ℓ m is completely fixed and there is noremaining gauge degrees of freedom. Then, after substituting the metric into the action (32) and performing integrations byparts, we find that the action for the odd modes becomes S odd = κ X ℓ, m Z dt dr L odd = κ X ℓ, m Z dt dr j (cid:20) φ (cid:16) ˙ h − h ′ (cid:17) + h ˙ h φ r + h r " r φ ′ + φ + ( j − φ B − ( j − B φ h r (cid:21) , (44)where we neglect the suffix ℓ for the fields, and j = ℓ ( ℓ + h yields[ φ ( h ′ − ˙ h )] ′ = r " r φ ′ + j φ + ( j − φ B h + φ ˙ h r , (45)which cannot be solved for h . Let us now rewrite the above action as L odd = j φ ˙ h − h ′ + h r ! − j ( φ + r φ ′ ) h r + j h r " r φ ′ + φ + ( j − φ B − j ( j − B φ h r . (46)so that all the terms containing ˙ h are inside the first squared term. Using a Lagrange multiplier Q , we can rewrite Eq. (46) asfollows L odd = j φ " Q ˙ h − h ′ + h r ! − Q − j ( φ + r φ ′ ) h r + j h r " r φ ′ + φ + ( j − φ B − j ( j − B φ h r . (47)Eq. (47) shows that both fields h and h can be integrated out by using their own equations of motion, which can be written as h = − r ˙ Q ( j − B , (48) h = r φ ( j −
2) [( φ + r φ ′ ) Q + r φ Q ′ ] . (49)These relations link the physical modes h and h to the auxiliary field Q . Once Q is known also h and h are. After substitutingthese expressions into the Lagrangian and performing an integration by parts for the term proportional to Q ′ Q , one finds theLagrangian in the canonical form L odd = j r φ j − B ˙ Q − j B φ r j − Q ′ − µ Q , (50)where µ = j h j φ − Br φφ ′′ + B φ − r φφ ′ B ′ + r B φ ′ − φ − r φ B ′ i φ ( j − . (51)From Eq. (50), we can derive the no ghost conditions j ≥ , and B ≥ . For solutions proportional to e i ( ω t − kr ) with large k and ω , we have the radial dispersion relation ω = B k , where we made use of the background equations of motion. Finally the expression for the radial speed reads c odd = dr ∗ d τ ! = , where we used the radial tortoise coordinate ( dr ∗ = dr / B ) and the proper time ( d τ = B dt ).4 C. Black hole stability: Geodesic
The trajectories of a test particle in a gravitational field are described by the geodesic equations d x σ d λ + n σµν o dx µ d λ dx ν d λ = , (52)where λ is an affine parameter along the geodesic. The geodesic deviation takes the form [115] d ξ σ d λ + n σµν o dx µ d λ d ξ ν ds + n σµν o , ρ dx µ d λ dx ν d λ ξ ρ = , (53)with ξ ρ being the deviation 4-vector. Applying (52) and (53) into (8) we get for the geodesic equations d td λ = , B ′ ( r ) dtd λ ! − r d φ d λ ! = , d θ d λ = , d φ d λ = , (54)and for the geodesic deviation d ξ d λ + B ( r ) B ′ ( r ) dtd λ d ξ d λ − rB ( r ) d φ d λ d ξ d λ + (cid:16) B ′ ( r ) + B ( r ) B ′′ ( r ) (cid:17) dtd λ ! − (cid:0) B ( r ) + rB ′ ( r ) (cid:1) d φ d λ ! ξ = , d ξ d λ + B ′ ( r ) B ( r ) dtd λ d ζ d λ = , d ξ d λ + d φ d λ ! ξ = , d ξ d λ + r d φ d λ d ξ d λ = , (55)where B ( r ) is defined by the metric (19) or (20), B ′ ( r ) = dB ( r ) dr . Using the circular orbit θ = π , d θ d λ = , drd λ = , (56)we get d φ d λ ! = B ′ ( r ) r (2 B ( r ) − rB ′ ( r )) , dtd λ ! = B ( r ) − rB ′ ( r ) . (57)Eqs. (55) can be rewritten as d ξ d φ + B ( r ) B ′ ( r ) dtd φ d ξ d φ − rB ( r ) d ξ d φ + (cid:16) η ′ ( r ) + η ( r ) η ′′ ( r ) (cid:17) dtd φ ! − (cid:0) η ( r ) + r η ′ ( r ) (cid:1) ζ = , d ξ d φ + ξ = , d ξ d φ + B ′ ( r ) B ( r ) dtd φ d ξ d φ = , d ξ d φ + r d ξ d φ = . (58)The second equation of (58) shows that it is a simple harmonic motion which means that the motion in the plan θ = π/ ξ = ζ e i σφ , ξ = ζ e i σφ , and ξ = ζ e i σφ , (59)where ζ , ζ and ζ are constants, and the variable φ has to be determined. Substituting (59) in (58), we get3 BB ′ − ω B ′ − rB ′ + rBB ′′ B ′ > , (60)which is the stability condition for any charged static spherically symmetric spacetime. The condition (60) for the black holes(19) and (20) can be rewritten as r + q M > , r + M > , and 1 + r Λ M > , (61)which are the stability conditions according to the values of the parameters Λ , M and q .5 VIII. DISCUSSION AND CONCLUSIONS
Spherically symmetric spacetimes constitute an essential part of black hole physics because all the fundamental properties ofthe black holes can be explained and can further be used to recognize and hence generalize in any eligible more general scenario[116]. In this paper, we discussed two main issues. In the first part, we focused on a spherically symmetric spacetime in theframework of f ( R ) gravitational theories. We derived new black hole charged solutions for the specific forms f ( R ) = R − α √ R and f ( R ) = R − α √ R − Λ . The main merits of these black holes are the fact that they depend on the dimensional parameter α and have dynamical Ricci scalar, i.e., R = r for the first model of f ( R ) and R = r Λ+ r for the second one. These solutionsare new and cannot reduce to the standard solutions of GR due to the fact that the parameter α is not allowed to have a zerovalue. We calculate the scalar invariant of those black holes and found that the Kretschmann and Ricci tensor square invariantsdepending on the dimensional parameter α . All of the invariants show true singularity at r = f ( R ) gravity is the fact that entropy is not always proportional to the area of the horizon [117–119]. We have shown that, for some constraint on the parameter 0 < α < .
5, we have a positive value of the entropy. However,for the black hole solution (16) there is a region in which the entropy has a negative value [117–120]. This is not the first timethat a black hole with negative entropy is found. Several black holes with negative entropy have been found as well as in chargedGauss-Bonnet (A)dS gravity [117–119]. As our calculations show, negative entropy may be interpreted as a region where theparameter α has transitions into forbidden regions related to some phase transition. The complete understanding of gravitationalentropy of non-trivial solution in the framework of f ( R ) gravitational theories remains the subject of future research.We also calculated the thermodynamical quasi-local energy and showed that it has a positive value when 0 < α < . α . Also, we haveshown that the Hawking temperature has always positive value when 0 . > α > α > < α < .
5. However, the black hole (16) is not globally stable when r < .
73 and become stable when r > . r < .
73 as Fig. 2(b) shows. The results obtained here, together with other results in the literature, seem to indicate thatthe thermodynamical origin of f ( R ) gravitational theories, when horizons are present, has a broad of validation. To confirm thisstatement we need to know more about the novel black holes derived in this paper. This will be done in future studies.Finally, we have studied the linear perturbations around the static spherically symmetric charged spacetime derived in f ( R )gravity. Due to the fact that f ( R ) is a fourth order theory, we have rewritten its Lagrangian as a Ricci scalar coupled with a scalarfield to make the study of perturbation more practice. We have derived the gradient instability condition for our black holesusing the odd-type modes. Furthermore, we have calculated the radial propagation speed and showed that it is equal one. Tomake the picture more complete, we have derived the stability conditions using also the geodesic deviation for the black holes.These conditions are different with respect to the charged black hole of GR, the Reissner-Nordstr¨om spacetime. This differenceis due to the fact that the charged black hole derived in this study is a solution in the context of f ( R ) only and cannot be reducedto GR. ACKNOWLEDGMENTS
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G.G.L. Nashed
1, 2, ∗ and S. Capozziello
3, 4, † Centre for Theoretical Physics, The British University,P.O. Box 43, El Sherouk City, Cairo 11837, Egypt Mathematics Department, Faculty of Science, Ain Shams University, Cairo 11566, Egypt Dipartimento di Fisica “E. Pancini“, Universit´a di Napoli “Federico II”,Complesso Universitario di Monte Sant’ Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli,Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy. (Dated: July 18, 2019)We make a short comment on the results derived in [1]. Our black hole is characterized by twoparameters: one is the mass and the other is the charge. Both of them are linked to the dimensionalparameter α which is related to the f ( R ) by f ( R ) = R − α √ R − Λ. The thermodynamics of thisblack hole is studied as well as its stability.
In the paper [1], we have applied a spherically symmetric spacetime with a unknown function B ( r ) and derived anew non-trivial black hole in the framework of f ( R ). The function B ( r ) of this solution takes the form B ( r ) = 12 − αr + 13 αr . (1)The above equation has a typo in its last term, i.e instead of + 13 αr , it must be − αr . If we take into account thissign all the results presented in [1] are correct in contrast to the claim given in [2]. We thank the authors of Ref. [2]for pointing out this typo. [1] G. G. L. Nashed and S. Capozziello, Phys. Rev. D99 , 104018 (2019), 1902.06783.[2] C. Conde, C. Galvis, and E. Larra˜naga (2019), 1907.01050. ∗ Electronic address: [email protected] ††