4D Einstein--Gauss--Bonnet gravity coupled to modified logarithmic nonlinear electrodynamics

Spherically symmetric solution in 4D Einstein--Gauss--Bonnet gravity coupled to modified logarithmic nonlinear electrodynamics (ModLogNED) is found. This solution at infinity possesses the charged black hole Reissner--Nordstr\"{o}m behavior. We study the black hole thermodynamics, entropy, shadow, energy emission rate and quasinormal modes. It was shown that black holes can possess the phase transitions and at some range of event horizon radii black holes are stable. The entropy has the logarithmic correction to the area law. The shadow radii were calculated for variety of parameters. We found that there is a peak of the black hole energy emission rate. The real and imaginary parts of the quasinormal modes frequencies were calculated. The energy conditions of ModLogNED are investigated.


Introduction
Nowadays, there are many theories of gravity that are alternatives to Einstein's theory [1,2]. The motivation of generalisations of Einstein's theory of General Relativity (GR) is to resolve some problems in cosmology and astrophysics. One of important modification of GR is the Einstein-Gauss-Bonnet event horizon, and the shadow radii. The black hole energy emission rate is investigate in Sect. 5. In Sect. 6 we study quasinormal modes and find complex frequencies. Section 7 is a summary. In Appendix A energy conditions of ModLogNED model are investigated.

4D EGB model
The action of EGB gravity coupled to nonlinear electrodynamics (NED) in D-dimensions is given by where G is the Newton's constant, α has the dimension of (length) 2 . The Lagrangian of ModLogNED, proposed in [20], is where we use Gaussian units. The parameter β (β ≥ 0) possesses the dimension of (length) 2 , F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor, and F = (1/4)F µν F µν = (B 2 − E 2 )/2, where B and E are the induction magnetic and electric fields, correspondingly. Making use of the limit β → 0 in Eq. (2), we arrive at the Maxwell's Lagrangian L M = −F /(4π). The GB Lagrangian has the structure By varying action (1) with respect to the metric we have EGB equations where T µν is the stress (energy-momentum) tensor. To obtain the solution of field equations we need to use an ansatz for the interval. But the validity of Birkhoff's theorem [30] for our case of 4D EGB gravity coupled to ModLogNED model is not proven. Therefore, to simplify the problem we consider magnetic black holes with the static spherically symmetric metric in D dimension. In addition, we assume that components of the interval are restricted by the relation g 11 = g −1 00 . Thus, we suppose that the metric has the form The dΩ 2 D−2 is the line element of the unit (D − 2)-dimensional sphere. By following [13] we replace α by α → α/(D − 4) and taking the limit D → 4. We study the magnetic black holes and find F = q 2 /(2r 4 ), where q is a magnetic charge. Then the magnetic energy density becomes [20] At the limit D → 4 and from Eq. (4) we obtain By virtue of Eq. (7 ) one finds where m M is the black hole magnetic mass. Making use of Eqs. (9) and (10) we obtain the solution to Eq. (8) where m is the constant of integration (the Schwarzschild mass) and the total black hole mass is M = m+ m M which is the ADM mass. At the limit β → 0 one has lim β→0 h(r) = m M − q 2 /2r.
Then making use of Eq. (11), for the negative branch, we obtain lim β→0,α→0 that corresponds to GR coupled to Maxwell electrodynamics (the Reissner-Nordström solution). It is worth mentioning that for spherically symmetric D-dimensional line element (6), the Weyl tensor of the D-dimensional spatial part becomes zero [17]. Therefore, solution (11) corresponds to the consistent theory [14,15,16]. By introducing the dimensionless variable x = r/ √ βq, Eq. (11) is rewritten in the form where We will use the negative branch in Eqs. (11) and (12) with the minus sign of the square root to have black holes without ghosts. As α → 0, r → ∞ the metric function f (r) (11), for the negative branch, becomes showing, at infinity, the Reissner−Nordström behavior of the charged black holes. The plot of function (12) for a particular chose of parameters, A = 15, C = 1 (as an example), is depicted in Fig. 1. The expansion (14) was observed in other models (see, for example, [31]). According to Fig. 1 there can be two horizons or one (the extreme) horizon of black holes.

The black hole thermodynamics
To study the black hole thermal stability we will calculate the Hawking temperature where r + is the event horizon radius (f (r + ) = 0). From Eqs. (12) and (15) one finds the Hawking temperature is represented in Fig. 2. Figure 2 shows that the Hawking temperature is positive for some interval of event horizon radii. We will calculate the heat capacity to study the black hole local stability where M(x + ) is the black hole gravitational mass as a function of the event horizon radius. Making use of equation f (x + ) = 0 we obtain the black hole mass With the help of Eqs. (16) and (18) one finds .
In accordance with Eq. (17) the heat capacity has a singularity when the Hawking temperature possesses an extremum (∂T H (x + )/∂x + = 0). Equations (16) and (17) show that at one point, x + = x 1 , the Hawking temperature and heat capacity become zero and the black hole remnant mass is formed. In another point x + = x 2 with ∂T H (x + )/∂x + = 0, the heat capacity has a singularity where the second-order phase transition occurs. Black holes in the range x 2 > x + > x 1 are locally stable but at x + > x 2 black holes are unstable. Making use of Eqs. (17), (19) and (20) the heat capacity is depicted in Fig. 3 at C = 1. The Hawking temperature and heat capacity From Eqs. (16), (19) and (21) one finds the entropy with the integration constant Const. The integration constant can be chosen in the form Then making use of Eqs. (22) and (23) we obtain the black hole entropy with S 0 = πr 2 + /G being the Bekenstein-Hawking entropy and with the logarithmic correction but without the coupling β. One can find same entropy (24) in other models [33,34,35].

Black holes shadows
The light gravitational lensing leads to the formation of black hole shadow and a black circular disk. The Event Horizon Telescope collaboration [36] observed the image of the super-massive black hole M87*. A neutral Schwarzschild black hole shadow was studied in [37]. We will consider photons moving in the equatorial plane, ϑ = π/2. With the help of the Hamilton−Jacobi method one obtains the equation for the photon motion in null curves [38] where p µ is the photon momentum (ṙ = ∂H/∂p r ). The photon energy and angular momentum are constants of motion, and they are E = −p t and L = p φ , correspondingly. We can represent Eq. (25) as Photon circular orbit radius r p can be found from equation V (r p ) = V ′ (r) |r=rp = 0. Making use of Eq. (26) we find where ξ is the impact parameter. For a distant observer as r 0 → ∞, the shadow radius becomes r s = r p / f (r p ) (r s = ξ). By virtue of Eq. (12) and equation f (r + ) = 0 we obtain parameters A, B and C versus x + A = 1 + 2Cx 2 with x + = r + / √ βq. The functions (28) plots are depicted in Fig. 4. In accordance with Fig. 4, Subplot 1, event horizon radius x + increases when parameter A increases and Subplot 2 indicates that if parameter B increases, the event horizon radius decreases. According to Subplot 3 of Fig. 4, when parameter C increases the event horizon radius x + also increases. The photon sphere radii (x p ), the event horizon radii (x + ), and the shadow radii (x s ) for A = 15 and C = 1 are presented in Table 1. It is worth noting that the null geodesics radii x p correspond to the maximum of the potential V (r) (V ′′ ≤ 0) and belong to unstable orbits. Table 1 shows that when  It is worth mentioning that currently there is not unique calculation of the shadow radius of M87* or SgrA* black holes within ModLogNED because our model possesses four free parameters M, α, β and q (or M, A, B and C) but from observations one knows only two values: the black hole mass and the shadow radius.

Black holes energy emission rate
The black hole shadow, for the observer at infinity, is connected with the high energy absorption cross section [25,39]. At very high energies the absorption cross-section σ ≈ πr 2 s oscillates around the photon sphere. The energy emission rate of black holes is given by where ω is the emission frequency. By using dimensionless variable x + = r + / √ βq the black hole energy emission rate (29) becomes withT H (x + ) = √ βqT H (x + ) and ̟ = √ βqω. The radiation rate versus the dimensionless emission frequencyω for C = 1, A = 15 and B = 9, 14, 19, is depicted in Fig. 5. Figure 5 shows that there is a peak of the black hole energy emission rate. When parameter B increases, the energy emission rate peak becomes smaller and corresponds to the lower frequency. The black hole has a bigger lifetime when parameter B is bigger.

Quasinormal modes
The stability of BHs under small perturbations are characterised by quasinormal modes (QNMs) with complex frequencies ω. When Im ω < 0 modes are stable but if Im ω > 0 modes are unstable. Re ω, in the eikonal limit, is linked with the black hole radius shadow [40,41]. Around black holes, the perturbations by scalar massless fields are described by the effective potential barrier with l being the multipole number l = 0, 1, 2.... Equation (31) can be rewritten in the form Dimensionless variable V (x)βq is depicted in Fig. 6 for A = 15, B = 10, C = 1 (Subplot 1) and for A = 15, C = 1, l = 5 (Subplot 2). According to Figure 6, Subplot l, the potential barriers of effective potentials have maxima. For l increasing the height of the potential increases. Figure 6, Subplot 2, shows that when the parameter B increases the height of the potential also increases. The quasinormal frequencies are given by [40,41] Re ω = l r s = l f (r p ) where r s is the black hole shadow radius, r p is the black hole photon sphere radius, and n = 0, 1, 2, ... is the overtone number. The frequencies, at A = 15, C = 1, n = 5, l = 10, are given in Table 2. Because the imaginary parts of the frequencies in Table 2 are negative, modes are stable. The real part Re ω gives the oscillations frequency. In accordance with Table 2 when parameter B increasing the real part of frequency √ βqRe ω increases and the absolute value of the frequency imaginary part | √ βqIm ω | decreases. Therefore, when the parameter B increases the scalar perturbations oscillate with greater frequency and decay lower.

Summary
The exact spherically symmetric solution of magnetic black holes is obtained in 4D EGB gravity coupled to ModLogNED. We studied the thermodynamics and the thermal stability of magnetically charged black holes. The Hawking temperature and the heat capacity were calculated. The phase transitions occur when the Hawking temperature has an extremum. Black holes are thermodynamically stable at some range of event horizon radii when the heat capacity and the Hawking temperature are positive. The heat capacity has a discontinuity where the second-order phase transitions take place. The black hole entropy was calculated which has the logarithmic correction. We calculated the photon sphere radii, the event horizon radii, and the shadow radii. It was shown that when the model parameter B increases the black  hole energy emission rate decreases and the black hole possesses a bigger lifetime. We show that when the parameter B increases the scalar perturbations oscillate with greater frequency and decay lower. Other solutions in 4D EGB gravity coupled to NED were found in [33,34,35].