Charged black holes in the Einstein-Maxwell-Weyl gravity

We construct charged asymptotically flat black hole solutions in Einstein-Maxwell-Weyl(EMW) gravity. These solutions can be interpreted as generalizations of two different groups: Schwarzschild black hole (SBH) and non-Schwarzschild black hole (NSBH) solutions. In addition, we discuss the thermodynamic properties of two groups of numerical solutions in detail, and show that they obey the first law of thermodynamics.

black hole (NSBH) in the region of 0.876 < r 0 < 1.143 for horizon radius r 0 in the EW gravity, besides the Schwarzschild black hole (SBH). Different from SBH, this NSBH has some remarkable properties: i) it admits positive and negative values of black hole mass; ii) when the horizon radius r 0 approaches some extremal value, this new black hole approaches the massless state [7]. In terms of the continued fractions, Kokkotas et.al [8] further constructed this NSBH solution in the analytical form. Very recently, the new black hole solutions have been also derived by adopting a new form of the metric in the pure Einstein-Weyl gravity [9]. Under test scalar field perturbation, the stabilities of NSBH and SBH have been also separately investigated [10,11], which recovered that the quasinormal modes for NSBH share larger real oscillation frequency and larger damping rate than the SBH branch. In particular, undamped oscillations (called quasi-resonances) also emerged for the NSBH, if perturbed scalar field possesses sufficiently large masses.
Inspired by above issues, Lin et.al further explored the Einstein-Maxwell-Weyl(EMW) gravity consisting of pure EW term and electromagnetic field [12], where it presented two groups (Groups I and II) of new charged black holes with fixed β = 0, γ = 1 and α = 1/2. Actually, the two groups solutions could be separately viewed as a charged generalization of the higher derivative curvature for SBH and a charged generalization of NSBH. However, Reissner-Nordström (RN) black hole solution is not a solution in the EMW gravity.
Comparing with the NSBH in the bound 0.876 < r 0 < 1.143 [4,5], new NSBH solutions have been obtained within an extended bound 0.363 < r 0 < 1.143 in pure EW gravity [7]. With regard to the EMW gravity, we can also construct new charged black hole solutions according to the new non-charged 'seed' solutions. Then, we will further discuss the thermodynamic properties of these new charged black holes in the EMW gravity. This paper is organized as follows. In Sec. II, we numerically derive new charged asymptotically flat black hole solutions. Then, some related thermodynamic properties of these new charged black holes are explored in Sec. III. Finally, we end the paper with conclusions and discussions in Sec. IV.

II. EINSTEIN-MAXWELL-WEYL GRAVITY AND NUMERICAL SOLUTIONS
The Einstein-Maxwell-Weyl action including pure Einstein-Weyl term and electromagnetic field is given by [12] where F µν = ∇ µ A ν − ∇ ν A µ is the electromagnetic tensor. Then, the equations of motion are obtained as [12] where the trace-free Bach tensor B µν and energy-momentum tensor of electromagnetic field T µν are defined as In order to construct new charged black hole solution, we assume a new metric ansatz with h(r) = f (r)e −2δ(r) and f (r) = 1 − 2m(r) r . Substituting the ansatz [eq.(5)] into eq.(3), one get where the prime ( ′ ) denotes differentiation with respect to r, and parameter Q denotes the electric charge.
Influenced by the functional form of the electric charge in the metric, the charged black holes have more than one horizon in general. For example, RN black hole has one event horizon and one Cauchy horizon. However, we suppose that the spacetime has only one horizon to make it easier for the expansion of m(r) and δ(r) around the event horizon r 0 .
Substituting these expansions into eq.(3), the coefficients δ i , A ti (for i = 1) and m i (for i = 2) can be solved in terms of the three non-trivial free parameters r 0 , m 1 and δ 0 . For example, m 2 , A t1 and δ 1 can be obtained as Here the coefficients of expansion can be also presented by m 1 = − δ 2 , and δ 0 = 1 2 ln( 1+δ cr 0 ) in refs. [4,5].
At the other asymptotic regime, that of radial infinity (r → 1), the metric functions and the Maxwell field may be expanded in power series, this time in terms of 1/r. The metric components reduce to where the parameter M and Q are associated with the mass and charge of black hole, and Φ is the electric potential.
Firstly, we reconsider non-Schwarzschild black hole solution (Q = 0) in the EW gravity.
Throughout this paper, we also take α = 1 2 and κ = 1 for simplicity [4,5]. The signal for a good black hole solution is that the functions f (r) and h(r) should approach very close to 1 as r increases. Comparing with the previous bound (0.876 < r 0 < 1.143) for the horizon radius r 0 [4,5], we numerically derive an extended bound 0.363 < r 0 < 1.143. The values 0.363 and 1.143 of horizon radius r 0 denote the disappearance of temperature T = 0 and massless state M = 0 for the non-Schwarzschild black hole, respectively. If r 0 is smaller than 0.363, the temperature of non-Schwarzschild black hole is negative. When r 0 is larger than 1.143, the black hole mass would become negative [4]. The corresponding bound for the parameter m 1 is 0.46 > m 1 > −0.422. At r 0 ≈ 0.876, the Schwarzschild and the non-Schwarzschild black holes 'coalesce' with m 1 = 0. In other words, for any selected value of r 0 in the above extended bounded interval, there exists only one value of m 1 that allows for a healthy non-Schwarzschild black hole. In Fig.1, we plot the metric functions f and h for the NSBH with horizon radius r 0 = 0.6 and 1. Now, we turn to discuss the charged black hole in the Einstein-Maxwell-Weyl gravity. It's worth noticing that the ref. [12] asserted the metric functions f (r) and h(r) for the charged black hole solutions in the Group II presented a peak outside the event horizon. It implied the presence of a unphysical negative effective mass [ Fig.3 in ref. [12]]. Here we reconstruct four different

III. THERMODYNAMIC PROPERTIES OF CHARGED BLACK HOLES
With the numerical charged black holes of Groups I and II, it's interesting to study the thermodynamic properties of these solutions. In order to do this, we need to collect the numerical results for a sequence of charged black-hole solutions with different values of Q.
In Fig. 4 The similar phenomenon occurs for charged black holes in the Group II. We construct charged black holes starting from a sequence of non-Schwarzschild black holes with horizon radius in the region of 0.363 < r 0 < 1.143. see Fig. 4   It is important to explore the free energy F = M − T S as a function of temperature. According to the left-hand side of joint point (T ≈ 0.091) in Fig.7(b), it can be seen that free energies of the charged black holes of Group II are always larger than that of non-Schwarzschild black hole, but smaller than that of Schwarzschild black hole at a given temperature T . Nevertheless, the charged black holes located on the right-hand side of joint point (T ≈ 0.091) always possess larger values for fixed T . The charged black holes in Group I display more complicated properties in Fig.7(a).  corresponding value m 1 = m * 1 that yields the charged black hole solution as the increase of charge Q without a singularity at spatial infinity. Later, the thermodynamic proprieties of these charged black holes were discussed in detail, and show that they obey the first law of thermodynamics.
Notice that the quasinormal modes of non-Schwarzschild solutions have been investigated in refs. [10,11], which shows that the non-Schwarzschild black hole is stable. Therefore it is necessary to investigate the quasinormal modes and stability of the charged black hole solutions in the Einstein-Maxwell-Weyl theory. Another interesting possibility is (Anti-) de Sitter charged black hole solutions in Einstein-Maxwell-Weyl gravity. They have shown in the Einstein-Hilbert theory of gravity with additional quadratic curvature terms [17]. Beside the analytic Schwarzschild (Anti-) de Sitter solutions, two groups of non-Schwarzschild (Anti-) de Sitter solutions were also obtained numerically. Their thermodynamic properties of the two groups of numerical solutions deserve a new work in future.