Einstein–Gauss–Bonnet Gravity with Nonlinear Electrodynamics: Entropy, Energy Emission, Quasinormal Modes and Deﬂection Angle

: The logarithmic correction to Bekenshtein–Hawking entropy in the framework of 4D Einstein–Gauss–Bonnet gravity coupled with nonlinear electrodynamics is obtained. We explore the black hole solution with the spherically symmetric metric. The logarithmic term in the entropy has a structure similar to the entropy correction in the semi-classical Einstein equations. The energy emission rate of black holes and energy conditions are studied. The quasinormal modes of a test scalar ﬁeld are investigated. The gravitational lensing of light around BHs was studied. We calculated the deﬂection angle for some model parameters.


Introduction
A valuable fundamental quantum theory of gravity should be renormalizable and unitary but general relativity (GR) is perturbatively nonrenormalizable, but unitary. The renormalizaton of the theory requires higher derivatives in the action and the presence of quadratic terms in curvature can give a renormalizeable theory of quantum gravity. Therefore, a modification of Einstein GR by including higher order curvature terms in action is of interest. It was shown in [1] that quantum corrections in quadratic gravity lead to unstable resonance which does not appear in the asymptotic spectrum. This proves that such theories also are unitary and stable to all orders. The inclusion of terms with four derivatives in the action makes the theory renormalizable, but leads to the presence of ghosts making the perturbative unitarity of the theory questionable. The authors of [2] showed that such spin-2 ghosts do not indicate a violation of unitarity. Higher-derivative corrections in string theory were discussed in [3]. Other applications of models, with higher-order derivatives, in astrophysics and cosmology, were studied in [4].
Another way to take into account higher order curvature terms is to add the Gauss-Bonnet (GB) Lagrangian to Einstein-Hilbert action. Such theory, 4D Einstein-Gauss-Bonnet (4D EGB) gravity in four dimension, recently attracted much attention [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). It was shown by Glavan and Lin [5] that in 4D EGB theory the GB term, which is a topological invariant before regularization, after regularization it is no longer a topological invariant and it contributes to the equation of motion. In [22,23], the authors obtained a solution of the semi-classical Einstein equations with conformal anomaly that also is a solution in the 4D EGB gravity. Recently, the scheme of Glavan and Lin was debated in [24][25][26][27][28][29][30]. In particular, it was shown in [27,28] that solutions in the 4D EGB theory differ from GR because they are due to extra infinitely strongly coupled scalar. It is worth noting that the theory of [31][32][33] gives the solution, in the spherically-symmetric metrics, that is a solution in the framework of rescaling procedure of [5] (see [34]). We explore a BH solution in the 4D EGB gravity theory coupled to nonlinear electrodynamics (NED) proposed in [35] making use of the theory of [31][32][33].
Here, we investigate the optical properties of BH by using the solution obtained in 4D EGB gravity coupled to NED. This paper is the continuation of the work [36]. NED considered here possesses the attractive features such as the absence of singularities and simplicity (the solution contains only elementary functions). In addition, at the weak field limit our NED is converted into Maxwell electrodynamics. It worth noting that the solution of well-known Born-Infeld electrodynamics in 4D EGB gravity contains special hypergeometric function [37]. The specific NED can give different astrophysical characteristics: the shadow radius of a charged BH, the BH energy emission rate, and the deflection angle of light from the BH. Therefore it is of interest to test solutions of BHs in 4D EGB gravity coupled to different NED which effect on astrophysical characteristics. Thus, several BH solutions in 4D EGB gravity coupled to NED were studied . The quasinormal modes, deflection angle, shadows of BHs and the Hawking radiation were studied in [41][42][43][44][45][46][47]. In this paper we analyse the shadow, the energy emission rate, quasinormal modes and the light deflection angle of the magnetically charged BH by using NED proposed in [35].
The paper is organized as follows. In Section 2, we consider the BH spherically symmetric solution in the framework of the 4D EGB gravity. It is shown that at infinity we have the Reissner-Nordström behavior of the charged BH. We obtain the logarithmic correction to Bekenstein-Hawking entropy in Section 3. In Section 4, we study the BH energy emission rate. The energy conditions are investigated in Section 5. It is shown that WEC, DEC and SEC are satisfied. In Section 6, we investigate the BH quasinormal modes and obtain the corresponding frequencies. We study the light defection angle by the BH solution in Section 7. Section 8 is a conclusion.

The Model
The EGB gravity action in D-dimensions coupled to NED is with α possessing the dimension of (length) 2 and the NED Lagrangian, proposed in [35] is given by where the parameter β (β ≥ 0) has the dimension of (length) 4 , F = (1/4)F µν F µν = (B 2 − E 2 )/2, F µν is the strength tensor and the GB Lagrangian is The field equation corresponding to action (1) reads where T µν is the energy-momentum tensor and The spherically symmetric D-dimensional line element is given by where dΩ 2 D−2 is the line element of the unit (D − 2)-dimensional sphere. The Equations (1), (3)-(5) are defined in D dimensions. In the following, we consider rescaling α → α/(D − 4) and the limit D → 4. Following [48] we consider a magnetically charged BH. The magnetic BH represents a magnetic monopole with the magnetic field B = q m /r 2 (q m is a magnetic charge) and F = q 2 m /(2r 4 ). The magnetic energy density is given by The static and spherically symmetric spacetime metric will be used. At the limit D → 4 by exploring the scheme of [5], the tt component of Equation (4) reads The Equation (8) holds for any 4D EGB gravity model with the static and spherically symmetric metric. The general solution to Equation (8) is given by where M is the integration constant. For Maxwell electrodynamics the energy density is ρ = q 2 /(2r 4 ) and Equation (9) leads to the metric function obtained in [6]. But at the limit r → 0 that solution leads to the non-physical complex value of the metric function. To have the stable BH [49] we will use the sign minus (the negative branch) before square root in Equation (9). For 4D EGB gravity coupled to NED (2) with the energy density (7), the solution (9) for the negative branch, gives the metric function [36] f (r) = 1 + r 2 2α It should be noted that the limit β → 0 for the last term in the square root, by using the L'Hôpital rule, becomes zero. The Weyl tensor for the D-dimensional spatial part of the spherically symmetric D-dimensional line element (6) vanishes. Therefore, the new solution (10) found in the context of [5] is also a new solution for the consistent theory [31][32][33]. Here we consider pure classical theory and the logarithmic correction in Equation (10) is due to the GB term in the action. With the dimensionless variable x = r/ 4 βq 2 m , Equation (10) is rewritten as where we use the dimensionless parameters The asymptotic of the metric function f (r) (10), for the negative branch, at r 1 is given by It follows from Equation (13) that M is a magnetic mass of the BH. This equation shows the Reissner-Nordström behavior of the charged BH at large r, and the metric becomes flat at infinity (r → ∞). The asymptotic of the metric function f (r), for the positive branch, does not correspond to the BH with the Reissner-Nordström behavior at infinity. It is worth noting that the condition x 4 + x(a + b ln(x/(x + 1))) > 0 should be satisfied to have the real metric function f (x). This requirement gives the restriction on the radius.
The nontrivial solution x 0 to the equation x 4 + x(a + b ln(x/(x + 1))) = 0 leads to the limitation x > x 0 . One can verify that curvature invariants, Ricci and Kretschmann scalars, have singularities at x = x 0 . Thus, the spacetime developed a spacetime singularity at x = x 0 .

Thermodynamics and BH Entropy
By using the expression for Hawking temperature T H (r + ) = f (r + )/(4π), where r + is the event horizon radius, f (r + ) = 0, and the prime means the derivative with respect to the argument, we obtain [36] where According to the first law of BH thermodynamics dM(x + ) = T H (x + )dS + φdq (φ is an electromagnetic potential) the entropy S at the constant charge q is given by With the help of Equations (14)- (16) we find the entropy were C is the integration constant. It is worth noting that there is uncertainty in the choice of C [50]. It is convenient to use the integration constant C in the form With Equations (17) and (18) one obtains the entropy where S 0 = πr 2 + /G is the Bekenstein-Hawking entropy. In accordance with Equation (19) the entropy includes the Bekenstein-Hawking entropy and the logarithmic correction. The attractive feature of entropy (19) is that it does not depend on the NED parameter β. Thus, there is not singularity of the entropy in the case β = 0. It should be noted that the entropy, similar to Equation (19) with the logarithmic term, appears in GR containing a conformal anomaly, and also in loop quantum gravity and in the theory of strings [22,23,50,51]. When α → 0 the entropy (19) becomes the Bekenstein-Hawking entropy. For the big event horizon radiuses the leading term of the entropy is the area law and for small x + the logarithmic correction is dominant. It has to be noted that at some parameters α, β, q m and r + the entropy vanishes. The entropy (17) (with C = 0) is zero when the event horizon radius obeys the equation r 4 + exp(r 2 + /α) = βq 2 m which is r 0 = 2αW 0 ( βq m /(2α)), where W 0 (x) is the Lambert function. The similar solution r 0 to Equation (19) for S = 0 is given by r 0 = 2αW 0 (G/(2πα)). At r + < r 0 the entropy is negative and such a situation takes place in the model of [52].

The Energy Emission Rate
The shadow of BHs is linked with the high energy absorption cross section σ seen for the observer at infinity [53][54][55] (see also [56]). The absorption cross-section, at very high energies, oscillates about an approximate value of the photon sphere σ ≈ πr 2 s , where x s defines the BH shadow radius r s = x s 4 βq 2 m (the impact parameter). The shadow radius can be expressed through the radius of the photon sphere r p by the relation r s = r p / f (r p ) for a distant observer, and r p is the solution of the equation f (r p )r p − 2 f (r p ) = 0 [36]. The energy emission rate in the high energy is given by Here, ω is the emission frequency and T H is the Hawking temperature. With the help of the dimensionless variables x = r/ 4 βq 2 m ,T H (x + ) = 4 βq 2 m T H (x + ), and Equation (20), we obtain and the Hawking temperature is given by Equation (14) and = β 1/4 √ q m ω. For some parameters b at a = 5, c = 1 we obtain the event horizon radius, the photon sphere radius and the shadow radius (expressed via the dimensionless variables) presented in Table 1 (see also [36]). It is worth noting that the parameters a, b and c are connected with M, α, β and q m by Equation (12). Because there are many parameters in the model, we use in Table 1 some set as an example. Making use of the data given in Table 1 we depicted the plot of the emission rate in Figure 1 for c = 1, a = 5 and b = 1.5, 2, 2.5. According to Figure 1 there is a peak of the energy emission rate for the BH depending on model parameters. When the parameter b increases, the maximum of the peak decreases and possesses the low frequency. Thus, the BH has a bigger lifetime at a bigger parameter b. One can investigate the dependence of the energy emission rate on parameters α, β, q m and M, putting some numerical values for them.

The Energy Conditions
The symmetrical energy-momentum tensor with the spherically symmetry leads to T t t = T r r and the radial pressure is given by p r = −T r r = −ρ. The tangential pressure is defined as where the prime means the derivative with respect to the argument. The weak energy condition (WEC) is satisfied when ρ ≥ 0 and ρ + p k ≥ 0 (k = 1,2,3) [58]. This guarantees that the energy density is non-negative as measured by any local observer. In accordance with Equation (7) ρ ≥ 0. Making use of Equation (7) one finds As a result, we have ρ ≥ 0, ρ + p r = 0, ρ + p ⊥ ≥ 0 and WEC holds. The dominant energy condition (DEC) takes place when [58] ρ ≥ 0, ρ + p k ≥ 0, ρ − p k ≥ 0. These conditions include WEC and we have to verify the condition ρ − p ⊥ ≥ 0. From Equations (7), (22) and (23) we obtain Thus, DEC is satisfied. As a result, the sound speed cannot exceed the speed of light. The strong energy condition (SEC) requires the condition ρ + ∑ 3 k=1 p k ≥ 0 [58]. With the help of Equations (7), (22) and (23) we obtain and SEC is satisfied.

Quasinormal Modes
Quasinormal modes (QNMs) are characterised by complex frequencies ω which give an information about the stability of BHs under small perturbations and they do not depend on the initial conditions. The outgoing boundary condition is imposed at infinity and the ingoing boundary condition at the event horizon. If Im ω > 0 the mode is unstable, otherwise it is stable. It was shown that Re ω in the eikonal limit is linked with the radius of the BH shadow [59,60]. In addition, the real and imaginary parts of QNMs frequencies are connected with the angular velocity and Lyapunov exponent of unstable circular null geodesics [61]. The perturbations by a scalar massless field around BHs are characterized by the effective potential barrier where l being the multipole number 0, 1, 2.... Equation (26) can be represented in the terms of dimensionless variable x = r/ 4 βq 2 m as The effective potential is plotted in Figure 2 for a = 5, b = 2, c = 1 and l = 1, 2, 3 and for a = 5, c = 1, l = 1 and b = 1, 2, 3. According to Figure 2, Subplot l, shows that the effective potentials represent a potential barrier with a maximum. The height of the potential increases when the l increases. In accordance with Figure 2, Subplot 2, the height of the potential increases if the parameter b increases. The real and imaginary parts of quasinormal frequencies are given by [59,60] Re ω = l r s = l f (r p ) where r s is the BH shadow radius (the impact parameter), r p is the radius of the BH photon sphere, n = 0, 1, 2, ... is the overtone number. The real and imaginary parts of the frequencies versus the parameter b at a = 5, c = 1, n = 1, l = 5 are given in Table 2.
The imaginary parts of the frequencies in Table 2 are negative, and therefore, the modes are stable and the real part represents the frequency of oscillations. According to Table 2 when the parameter b increases the real part of the reduced frequency 4 βq 2 m Re ω increases, but absolute value of the imaginary part of the reduced frequency | 4 βq 2 m Im ω | decreases. In other words, increasing the parameter b the scalar perturbations oscillate with greater frequency and decay slowly. To study the dependence of frequencies on parameters α, β, M, q m one has to put numerical numbers for these parameters in Equation (28).

Deflection Angle
Let us study the light deflection angle by the BH solution (10). We can determine the total deflection angle ∆ϕ by the formula [62] (see also [63]) where r p is the photon sphere radius. Taking into account that r p / f (r p ) = r s is the shadow sphere radius (r s = ξ is the impact parameter) one can represent Equation (29) in terms of the dimensionless variable as Making use of data in Table 1 we obtain the deflection angles from Equation (30) represented in Table 3. According to Table 3 when the parameter b increasing, with fixed a and c, the deflection angle is decreased. One can also study the dependence of the deflection angle on parameters β, α, q m and M by taking the numerical values for these parameters and putting them in Equation (10), finding the solution for r p : 2 f (r p ) − r p f (r p ) = 0 (r s = r p / f (r p )), and calculating the integral (29).

Conclusions
We use the exact spherically symmetric and magnetically charged BH solution in 4D EGB gravity coupled to NED obtained in [36] for further investigations. It is shown that we have the Reissner-Nordström behavior of the charged BH at infinity. The logarithmic correction to the Bekenstein-Hawking entropy is obtained from first law of BH thermodynamics. Similar corrections to the area law are appeared in quantum gravity. We obtain the solution for the event horizon radius when the entropy becomes zero. For the light BHs the logarithmic correction is important while for massive BHs (for a big event horizon radius) such correction is small. Then, the energy emission rate of BHs has been studied. We showed that the BH energy emission rate decreases with increasing the model parameter b and the BH has a bigger lifetime. To verify that the energy density is positive as measured by any local observer and the sound speed does not exceed the light speed, we investigate the energy conditions. It has been demonstrated that WEC, DEC and SEC are satisfied. The quasinormal modes that describe small perturbations around BHs are investigated. We have been studied the dependence of the hight of the effective potential barrier on the multipole number l and model parameter b. The height of the potential increases when the l or b increases. Complex frequencies, where the real part represents the frequency of oscillations and imaginary part characterises the oscillation decay, are calculated. We demonstrate that increasing the parameter b the scalar perturbations oscillate with greater frequency and decay slowly. Then, the gravitational lensing of light around BHs is studied by calculating the deflection angle ∆ϕ for some parameters. The deflection angle depends on the photon sphere radius r p , shadow radius r s and model parameters. It is shown that ∆ϕ is decreased if the parameter b increasing at fixed a and c.