The center of some spherically symmetric spacetimes, such as Reissner–Nordström (RN) spacetime, is singular. The spacetime without any geometric singularity can be said to be regular. A black hole in regular spacetime is called regular black hole. The Bardeen regular black hole is the first black hole with magnetic monopole to be found in Einstein gravity coupled to nonlinear electrodynamics (NED).^[1,2] In the NED gravity, some regular black hole solutions have been proposed,^[3–26] the electromagnetic field of which behaves as the Maxwell field in the weak field limit.^[3–5] The NED solutions with multiple parameters given in Refs. [19]–[22] describe the regular NED black holes with electric charge. It is of great interest to generalize this class of the NED solutions.

The Smarr formula for the black hole with a central singularity, such as RN black hole, plays an important role. However, the usual Smarr formula is not applicable to the case of the NED black hole.^[27–29] Starting from the Komar integral, the Smarr-type formula may be derived, where the boundary of Komar integral is taken on the infinite surface.^[30] On the other hand, one can assume that the Smarr formula for the NED black hole is valid.^[31] From the presupposed Smarr formula for the NED black hole, the quantities similar to the temperature and electric potential of NED black hole can be obtained. The latter study does not seem to yield physically meaningful results. The thermodynamic properties of the regular NED black hole need to be further studied.

In this paper, we propose the NED solution with electric charge and study its thermodynamic properties. In Section 2, we give the NED solution with three parameters, determine the radius of the NED black hole and calculate its electric potential. In Section 3, we obtain the first law of thermodynamics of the NED black hole. In a special case, we show the Smarr-like formula for the mass of the NED spacetime, and obtain the approximate Smarr formula associated with the outer horizon of NED black hole. Lastly, we discuss some problems on the regularity parameter and the Smarr formula of NED spacetime.

The action for Einstein gravity coupling to nonlinear electrodynamics may be taken as

The metric function of the NED spacetime in Ref. [20] takes where M, q, α, β, σ, and λ are some constants. By considering regularity, the metric function can be simplified to with four parameters M > 0, β > 0, σ > 0, and λ > 0.^[20,21] For β = 1 and σ = 3, it yields the NED spacetime solution with electric charge. In order to describe the regular NED spacetime with electric charge, we consider the metric function with three parameters: the mass of spacetime, the electric charge parameter and the regularity parameter of spacetime. We propose the following metric function 8 or the mass function 9 with the mass M > 0, the electric charge parameter χ ⩾ 0, and the regularity parameter λ > 0. The regular NED spacetime (8) with a nonzero χ is not included in the ones given in Refs. [20] and [21]. For χ = 0, the metric function (8) describes the Hayward spacetime with magnetic monopole.^[8] For χ ≠ 0, it describes the NED spacetime with electric charge. In what follows, we will consider the regular NED spacetime with a nonzero electric charge. In the central region, the NED spacetime behaves as a de-Sitter spacetime, with an effective cosmological constant. For r ≫ χ and r ≫ λ, the metric function (8) is expanded as 10 where the term r⁻³ is absent, and is the electric charge of the NED spacetime.

The spacetime described by the metric function (8) can have two black hole horizons. Three solutions (r₁, r₂, and r₃) of the equation f = 0 satisfy the following relationships: 11 12 13 From Eq. (13), it can be seen that at least one of the three solutions is negative. Letting r₃ < 0, r₁ = r₊, r₂ = r₋ with r₊ ⩾ r₋, then two positive solutions of f = 0 are given as 14 15 with 16 where r₊ and r₋ denote radii of the outer and inner horizon of the NED black hole. As χ → 0 and λ → 0, radii of the outer and inner horizons of the NED black hole approach 2M and 0, respectively.

From , the surface gravity on the horizon of the NED black hole is obtained as 17 with r_H = r_±. From κ_H = 0, one obtains the equation satisfied by the extremal NED black hole 18 where r_E = r₊ = r₋ denotes radius of the extremal NED black hole. As λ → 0, r_E → χ; as χ → 0, r_E → 2^1/3 λ. For the four cases of λ/M = 0.1, 0.5, 0.75, 1.0, the allowable ranges of parameter χ are determined to be 0 < χ/M ⩽ 0.666, 0.582, 0.41, 0.102, the corresponding radii of extremal black hole are given as r_E/M = 0.668, 0.805, 1.004, 1.263, respectively (See Fig. 1). For λ ≈ 0, radius of the extremal black hole is r_E ≈ χ ≈ 0.667M. The maximum of parameter λ takes λ_max = 1.05827M corresponding to χ ≈ 0. In this case, r_E ≈ 1.333M is the largest radius of the extremal NED black hole.

Fig. 1.

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	Fig. 1. The relationship between radii of horizons and parameter χ with some fixed values of λ. Dashed horizontal line represents radius of extremal black hole rE corresponding to χ = χE. Panel (a): rE ≈ 0.668M and χE ≈ 0.666M for λ = 0.1M. Panel (b): rE ≈ 0.805M and χE ≈ 0.582M for λ = 0.5M. Panel (c): rE ≈ 1.004M and χE ≈ 0.41M for λ = 0.75M. Panel (d): rE ≈ 1.263M and χE ≈ 0.102M for λ = M.

Substituting Eq. (9) into Eqs. (6) and (7) gives the Lagrangian of the NED spacetime and its derivative as 19 20 As r → ∞, approaches −Mχ/4πr⁴ and is close to 4π. From Eq. (20), the strength of the electromagnetic tensor is given as 21 As r → ∞, F approaches −q²/16π² r⁴. From Eq. (5) (or ), the electrostatic field is obtained as 22 On the horizon of the NED black hole, the electrostatic field takes 23 with x_H = r_H/M, , and . As and , equation (23) reduces approximately to 24 For the case of , the electrostatic field takes approximately with r_H = x_HM and x_H ≈ 2. Assuming that is a nonzero constant c, then .

By integrating , the electrostatic potential of the NED spacetime is given as 25 with Φ(∞) = 0. On the horizon of the NED black hole, the electrostatic potential takes 26 As and , equation (26) approximately takes 27 For c ≈ 0, is approximately the electrostatic potential of the RN black hole. In order to be consistent with the usual definition of electrostatic potential of black hole, Φ should be rescaled as Φ → 4π Φ.

For a spherically symmetric spacetime with multiple horizons, one can define multiple temperatures associated with the surface gravities of the black hole horizons. From T_H = (1/2π) κ_H with κ_H = (1/2) f′(r_H), the horizon temperature of the NED black hole is obtained as 28 As χ → 0 and λ → 0, T_H approaches 1/8π M. In f(R) gravity with the action , the black hole entropy takes with A_H the horizon area of black hole and f′ = d f/d R.^[32,33] Clearly, the entropy of the NED regular black hole is still the Bekenstein–Hawking entropy.^[34]

Einstein quasi-local energy enclosed by the sphere of radius r in the spacetime (4) takes E = (1/2) r (1 − f),^[35] which is consistent with the mass function (9). Letting f = 0, one obtains the energy enclosed by the horizon of the NED black hole as E_H = (1/2) r_H. The first law of thermodynamics for the NED black hole reads^[36–38] 29 where p_H = p_r (r_H) denotes the radial pressure on the horizon of the NED black hole, is the entropy of the NED black hole, and is the volume enclosed by the horizon of the NED black hole. For the system enclosed by the horizon of the NED spacetime described by Eq. (8), the energy flux through the horizon and the work done by the horizon are 30 31 From Eqs. (30) and (31), one can check the first law of thermodynamics Eq. (29) with dE_H = (1/2)d r_H. The first law of thermodynamics Eq. (29) can be applied to the system enclosed by the outer horizon as well as to the one enclosed by the inner horizon with a negative temperature. This is similar to the case of the RN black hole.

From Eq. (9), one can see that the mass function (or energy) of the NED spacetime m monotonously increases with the increase of r. The mass function tends to M, as r tends to be infinite. So, the total energy of the NED spacetime is its mass M. From f(r_H) = 0, the mass of the NED spacetime is given as 32 with χ = q²/2M. From Eq. (10), one can see that there is an asymptotic term 2M(λ³ − χ³)/r⁴ in the expansion of the metric function f. This means that the NED spacetime described by Eq. (8) with λ ≠ χ probably contains a nonzero magnetic charge. For λ = χ, the mass of the NED spacetime is obtained as 33 where 34 35 Equations (33)–(35) show that the mass of the NED spacetime may be expressed as M = M(S_H,q). Mathematically, one can always define the quantities and . For the NED black hole, the two quantities are different from T_H and Φ_H (see Fig. 2). This means that there is not the exact Smarr formula for the NED black hole with a nonzero electric charge.

Fig. 2.

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	Fig. 2. The figure shows that the temperature of the NED black hole T+ and the ratio , with minimum ζ+min ≈ 1 and maximum ζ+max ≈ 1.1866 corresponding to q = 1.0525. It shows that the electrostatic potential of the NED black hole Φ+ and the ratio , with maximum ξ+max ≈ 1 and minimum ξ+min ≈ 0.9276 corresponding to q = 1.0525.

In terms of entropy S₊, temperature T₊ and electric potential Φ₊ of the outer horizon of the NED black hole, the mass of the NED spacetime may be expressed as 36 where ΔM₊/M is a small quantity. We call Eq. (36) the Smarr-like formula associated with the outer horizon of the NED black hole. For q = 0.1M, the value of ΔM₊ takes −5.9 × 10⁻¹¹M. For a larger value of q, it is still a very small quantity. Letting q = 0.5M, the value is ΔM₊ ≈ − 0.000028M. In the near extremal case (q ≈ q_m with q_m = 1.052526825), it is ΔM₊ ≈ − 0.079M. As a result, there is the approximate formula expressed in terms of the quantities defined on the outer horizon of the NED black hole 37 The above formula shows how is the total energy in the NED spacetime made up. The smaller the electric charge q in Eq. (37), the more accurate the approximate Smarr formula is. As q → 0, it gradually approaches an exact Smarr formula.

Assuming a non-zero cosmological constant, then the metric function of the NED spacetime takes 38 with l ≫ M being a constant and f given by Eq. (8). The mass of the NED spacetime may be expressed as with r_C ≫ M being the radius of the cosmological horizon. It can be checked that there exists an approximate Smarr formula with an extremely high accuracy 39 where Φ_C = Φ(r_C) with Φ given by Eq. (25) is the electric potential on the cosmological horizon, is the potential associated with l, is the entropy of the cosmological horizon, and is its temperature.

For the topological dilaton black hole with power-Maxwell field and a nonzero cosmological constant, there is the exact Smarr formula.^[39] However, the mass of the NED spacetime described by Eq. (38) cannot be exactly expressed as the form of the Smarr formula. This should be related to the regularity of the NED spacetime.

In the asymptotic expansion of the metric function Eq. (10), there is a term proportional to r⁻⁴. As is well known, there is also such a term in the asymptotic expansion of the metric function of the NED spacetime with magnetic monopole.^[8] So, the meaning of this term is not completely clear since the magnetic monopole can have a contribution of the same magnitude in the far region of the NED spacetime. As an NED spacetime with only electric charge, we have considered the situation of λ = χ. From Eq. (8), one can see that f → 1 − (2M/λ³)r² with r → 0. In the case of λ ≠ 0, there is a regular core in the center of the NED spacetime. In this sense, we call λ the regularity parameter of the NED spacetime.

The systems enclosed by the horizons of the NED spacetime described by Eq. (8) satisfy the first law of thermodynamics. The mass function (9) shows that M is not an integration constant of the energy of the NED spacetime. So, one can only define the mass of spacetime, but not the mass of black hole. For λ = χ, we give the Smarr-like formula for the mass of the NED spacetime and the approximate Smarr formula associated with the outer horizon of the NED black hole. For a nonzero cosmological constant, there is also the approximate Smarr formula associated with the cosmological horizon. The approximate Smarr formula is an important thermodynamic relation for the NED spacetime with electric charge.