Modifications to the Entropy of a Rotating Bardeen Black Hole Due to Magnetic Charge

Abstract

Applying the Parikh–Wilczek method and based on the thermodynamics laws of black holes, we investigate the structure of the entropy of rotating Bardeen black holes. We find that entropy includes three terms and thus violates the area law. The first two terms depend on all of the black hole characteristics, while the third one is solely dependent on the charge of a magnetic monopole arising from nonlinear electrodynamics. The existence of the additional term means that the entropy of regular black holes has a different structure from that of classical ones, so it cannot be considered as a constant and disregarded, as was implemented in the previous literature.

1. Introduction

Since the pioneering work of Bekenstein [1] on black hole entropy and Hawking [2] on black hole radiation, research on thermodynamic properties such as black hole entropy and phase transition has been intense and continues to grow, most of which focuses on classical singularity black holes [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. However, there has been an increasing focus on research on regular black holes recently [19,20,21,22,23,24,25]. The core idea behind regular black holes is to modify classical black hole solutions within the framework of classical general relativity and its extensions by introducing new fields or altering the gravitational interaction to remove their singularities and preserve fundamental characteristics such as the event horizon. Research on regular black holes may provide crucial clues and a testing platform for the construction of quantum gravity theory. The regular black hole was first proposed by Bardeen in 1968 [26], when he obtained a static, spherically symmetric regular black hole space–time by replacing the singularity within the event horizon with a de Sitter core. Since then, diverse regular black hole solutions have been proposed and their properties have been analyzed [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. These studies have collectively revealed distinctive thermodynamic behaviors in regular black holes. For instance, phase transitions resembling the van der Waals liquid–gas system were found in Bardeen and Hayward black holes [33,36], while nonlinear electrodynamics (NED) was shown to resolve singularities while introducing entropy corrections beyond the area law [27]. Rotation further enriches this landscape: it modifies Hawking radiation spectra [34] and influences black hole stability [37], with the rotating Bardeen solution [40] offering a unique platform to explore magnetic charge and spin synergy. Crucially, entropy in such spacetimes often exhibits logarithmic or topological terms [20,25], contrasting sharply with classical black holes—a prelude to our finding of a magnetic monopole-dependent entropy component.

In this paper, we extend the work of Parikh–Wilczek [4] to the rotating Bardeen black hole to investigate the corrected radiation rate of particles from the event horizon of the black hole. Then, we derive the modified entropy.

2. Black Hole Metric, Temperature, and Rotating Angular Velocity

In Boyer–Lindquist coordinates, the metric of a rotating Bardeen black hole reads [40]

$$\begin{array}{l}d{s}^2=-{\displaystyle \frac{\Delta -a^2{\sin }^2\theta }{\Sigma }}d{t}^2+{\displaystyle \frac{\Sigma }{\Delta }}d{r}^2+\Sigma d{\theta }^2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \frac{{(r^2+a^2)}^2-\Delta a^2{\sin }^2\theta }{\Sigma }}{\sin }^2\theta d{\varphi }^2-{\displaystyle \frac{4aMrh(r){\sin }^2\theta }{\Sigma }}dtd\varphi \end{array}$$

(1)

with

$$\Sigma =r^2+a^2{\cos }^2\theta \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\begin{array}{cc}& \end{array}\Delta =r^2+a^2-2Mrh(r)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\begin{array}{cc}& \end{array}h(r)={\left({\displaystyle \frac{r^2}{r^2+g^2}}\right)}^{3/2}$$

(2)

where M and a are the ADM mass and angular momentum per unit mass, respectively, and g describes the charge of a magnetic monopole due to nonlinear electrodynamics, self-consistently generated through a nonlinear field equation, and its field strength tensor $F_{\mu \upsilon }$ satisfies the NED equations with a magnetic ansatz: $F_{\theta \varphi }=g\sin \theta$ [27,40,41].

The horizons are decided by

$$\Delta (r)=r^2-2Mrh(r)+a^2=0$$

(3)

Due to the nonlinearity of h(r), Equation (3) has no analytical solution; the horizon radii can only be solved numerically. We use $r_H$ to denote the event horizon and factorize $\Delta (r)$ into the form of

$$\Delta (r)=(r-r_H)\eta (r)$$

(4)

In the dragging coordinates, the space–time line element is

$$d{s}^2={\tilde{g}}_{00}d{t}^2+{\displaystyle \frac{\Sigma }{\Delta }}d{r}^2+\Sigma d{\theta }^2$$

(5)

where

$${\tilde{g}}_{00}=g_{00}-{\displaystyle \frac{g_{03}^2}{g_{33}}}=-{\displaystyle \frac{\Delta \Sigma }{{(r^2+a^2)}^2-\Delta a^2{\sin }^2\theta }}$$

(6)

The black hole temperature is

$$T_H={\displaystyle \frac{\kappa }{2\pi }}={\displaystyle \frac{1}{2\pi }}\underset{r\to r_H}{Lim}\left(-{\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{g^{11}}{{\tilde{g}}_{00}}}}{\displaystyle \frac{d{\tilde{g}}_{00}}{dr}}\right)={\displaystyle \frac{\eta (r_H)}{4\pi (r_H^2+a^2)}}$$

(7)

and the dragging angular velocity is

$${\Omega }_H={\left.-{\displaystyle \frac{g_{03}}{g_{33}}}\right|}_{r=r_H}={\displaystyle \frac{a}{r_H^2+a^2}}$$

(8)

3. Impurely Thermal Spectrum and Radiation Rate

To obtain the Painlevé-type line element, which is well-behaved at the horizon, we need to make a further transformation like in Ref. [42]

$$dt=dT+f(r,\theta )dr+g(r,\theta )d\theta$$

(9)

with the integrability condition

$${\displaystyle \frac{\partial f(r,\theta )}{\partial \theta }}={\displaystyle \frac{\partial g(r,\theta )}{\partial r}}$$

(10)

and

$${\displaystyle \frac{\Sigma }{\Delta }}+f^2{\tilde{g}}_{00}=1$$

(11)

Then, the line element (5) is rewritten as

$$\begin{array}{l}d{s}^2={\tilde{g}}_{00}d{T}^2+2\sqrt{{\tilde{g}}_{00}\left(1-{\displaystyle \frac{\Sigma }{\Delta }}\right)}dTdr+d{r}^2+\left(\Sigma +g^2{\tilde{g}}_{00}\right)d{\theta }^2\\ \begin{array}{cc}& \end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+2{\tilde{g}}_{00}gdTd\theta +2g\sqrt{{\tilde{g}}_{00}\left(1-{\displaystyle \frac{\Sigma }{\Delta }}\right)}drd\theta \end{array}$$

(12)

It is suitable for describing particle tunneling. Setting $d{s}^2=d\theta =0$, we obtain the radial outgoing null geodesic

$$\dot{r}={\displaystyle \frac{dr}{dT}}={\displaystyle \frac{(\Sigma -\sqrt{\Sigma (\Sigma -\Delta )})}{\sqrt{{(r^2+a^2)}^2-\Delta a^2{\sin }^2\theta }}}$$

(13)

Considering the energy and angular momentum conservation of the black hole–particle system, when a particle of energy $\omega$ and angular momentum $j=a\omega$ is emitted, the black hole’s energy and angular momentum will become $M-\omega$ and $J=a(M-\omega )$; all of the equations related to $r_H(M)$ should be used with $M\to M-\omega$. To retain the axisymmetry of the black hole, we regard the outgoing particle as an ellipsoid shell of energy and angular momentum during the tunneling process. Meanwhile, we note that $\phi$ is an ignorable coordinate in the Lagrange function, for Equation (13) does not include it. After excluding the freedom of $\phi$, we have the action of the outgoing particle, which crosses the horizon outwards from $r_i$ to $r_j$

$$\begin{array}{l}Z={\displaystyle {\int }_{ti}^{tj}(L-P_{\varphi }\dot{\varphi })dt}\\ \hspace{0.17em} ={\displaystyle {\int }_{ri}^{rj}{P}_{r}dr}-{\displaystyle {\int }_{\varphi i}^{\varphi j}{P}_{\varphi }d\varphi }={\displaystyle {\int }_{ri}^{rj}{\displaystyle {\int }_0^{Pr}d{P}_r}dr}-{\displaystyle {\int }_{\varphi i}^{\varphi j}{\displaystyle {\int }_0^{P\varphi }d{P}_{\varphi }}d\varphi }\end{array}$$

(14)

where $P_r$ and $P_{\phi }$ are canonical momenta conjugate to $r$ and $\phi$, respectively, $r_i$ is the initial radius of the black hole, and $r_j$ is the final one.

According to Hamilton’s equation

$${\displaystyle \frac{dr}{dt}}={\left.{\displaystyle \frac{dH}{d{P}_r}}\right|}_{ (r,\phi ,P_{\phi })};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\begin{array}{cc}& \end{array}{\displaystyle \frac{d\phi }{dt}}={\left.{\displaystyle \frac{dH}{d{P}_{\phi }}}\right|}_{ (r,\phi ,P_r)}$$

(15)

with

$${\left.{\left.dH\right|}_{(r,\varphi ,P_{\varphi })}=dM;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}dH\right|}_{(r,\varphi ,P_r)}={\Omega }_{H}dJ;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}J=aM$$

(16)

Substituting Equations (8), (15), and (16) into (14) yields

$$Z={\displaystyle {\int }_{ri}^{rj}{\displaystyle {\int }_M^{M-\omega }\frac{r_H^2}{r_H^2+a^2}\frac{(\Sigma +\sqrt{\Sigma (\Sigma -\Delta )})\sqrt{{(r^2+a^2)}^2-\Delta a^2{\sin }^2\theta }}{\Sigma \Delta }dM}dr}$$

(17)

Obviously, $r=r_H$ is a coordinate singularity. The integral (17) can be performed by deforming the contour around the pole, so as to ensure that the positive energy solution decays in time. Finishing the $r$ integral and using Equation (7) yields

$$Z=-{\displaystyle \frac{i}{2}}{\displaystyle {\int }_M^{M-\omega }\frac{r_H^2}{T_H(r_H^2+a^2)}dM}$$

(18)

Of course, it has been noted that the action (14) is not invariant under canonical coordinate transformations, which makes it not a good quantum observable [43,44]. Meanwhile, it is argued that the action should include not only spatial but also temporal contributions. The temporal contribution to the action was first discussed in [45,46], which proved that it gives the correct Hawking temperature in all cases, regardless of whether one uses Painleve–Gullstrand coordinates, Schwarzschild coordinates, etc.

Based on the first law of black hole thermodynamics

$$\mathrm{d}M=T_H\mathrm{d}S+{\Omega }_H\mathrm{d}J$$

(19)

we can rewrite Equation (18) as

$$Z=-{\displaystyle \frac{i}{2}}{\displaystyle {\int }_{S(M)}^{S(M-\omega )}\mathrm{d}S}=-{\displaystyle \frac{i}{2}}\Delta S$$

(20)

where $\Delta S=S(M-\omega )-S(M)$ is the increment of the black hole entropy before and after. Adopting the Wentzel–Kramers–Brillouin approximation, the tunneling probability of the particle is related to the imaginary part of the action as $\mathrm{\Gamma }~\exp (-2\mathrm{Im}Z)$ [47]. So

$$\mathrm{\Gamma }~\exp (\Delta S)$$

(21)

The result (21) is compatible with the results obtained earlier [3,4,5,6,7,8,9,10] and shows that the radiation spectrum deviates from a truly thermal one.

4. Modified Entropy

From the horizon Equation (3), we have

$$M={\displaystyle \raisebox{1ex}{$\left(r_H^2+a^2\right)$}\!\left/ \!\raisebox{-1ex}{$2r_{H}h(r_H)$}\right.}$$

(22)

and

$$dM={\displaystyle \frac{\eta (r_H)}{2r_{H}h(r_H)}}d{r}_H$$

(23)

Substituting Equation (23) into (18) and comparing Equation (20) yields

$$dS={\displaystyle \frac{2\pi r_H}{h(r_H)}}d{r}_H$$

(24)

Therefore, the entropy

$$\begin{array}{l}S=2\pi {\displaystyle \int \frac{r_H}{h(r_H)}d{r}_H}\\ \hspace{0.17em}\hspace{0.17em}=2\pi \left[\sqrt{r_H^2+g^2}\left({\displaystyle \frac{r_H}{2}}-{\displaystyle \frac{g^2}{r_H}}\right)+{\displaystyle \frac{3g^2}{2}}\ln \left(r_H+\sqrt{r_H^2+g^2}\right)-{\displaystyle \frac{3g^2}{2}}\ln g\right]+S_0\end{array}$$

(25)

where S₀ is an integration constant with the dimension of [length]². We note that the entropy of the rotating Bardeen black hole does not obey the law of area. However, it was shown in the previous section that this entropy satisfies the first law.

5. Conclusions and Discussion

We have calculated the radiation rate and entropy of rotating Bardeen regular black holes.

Equation (21) indicates that the radiation rate depends on the entropy change of the black hole before and after particle emission from the horizon. This implies that the energy spectrum is not purely thermal, and yet it aligns with an underlying unitary theory. In fact, the first law of black hole thermodynamics (19) merges the energy conservation law $\mathrm{d}M=\mathrm{d}{Q}_h+{\Omega }_{H}dJ$ (where $Q_h$ is the heat quantity and no forces act) and the second law of thermodynamics $\mathrm{Td}S=\mathrm{d}{Q}_h$. The energy conservation is obeyed in any process, but the equation $\mathrm{Td}S=\mathrm{d}{Q}_h$ is only suitable for a reversible process ($\mathrm{d}S>\mathrm{d}{Q}_h/T$ for an irreversible process). So, the emission process is considered reversible in the Parikh–Wilczek tunneling framework.

Not considering the integration constant, Equation (25) is rewritten as

$$S=2\pi \sqrt{r_H^2+g^2}\left({\displaystyle \frac{r_H}{2}}-{\displaystyle \frac{g^2}{r_H}}\right)+3\pi g^2\ln \left(r_H+\sqrt{r_H^2+g^2}\right)-3\pi g^2\ln g$$

(26)

The entropy (26) includes three terms, and the conventional area law is invalid. When $g\to 0$, the second and third terms disappear so that the entropy reduces to the Kerr case and becomes $S=\pi r_H^2$ and then the area law is naturally obeyed. The second one is a logarithmic term and depends on the hole characteristics M, a, and g. The third one is solely determined by the charge g; its existence makes the form of the entropy of regular black holes differ from that of classical ones.

On the other hand, the expression (26) is applicable to a Bardeen black hole (a = 0), for it is not an explicit function of a. Thus, the entropy of Bardeen black holes, as well as rotating Bardeen ones, has completely the same form, and this result is different from that in Refs. [33,48] because of the presence of the additional third term. Therefore, the term just depending on the charge g should not be treated as a constant and ignored.

The additional entropy term (especially the pure g-dependent term in Equation (26)) does not challenge the holographic principle but reveals unique holographic structures in regular black holes. Because g and a are intrinsic, entropy modifications reflect new bulk degrees of freedom (e.g., nonlocal microstates tied to g). The corrected entropy–area relation (26) remains holographic but implies enhanced boundary encoding of bulk information. This is compatible with the generalized holographic framework of Bredberg et al. [49].

While magnetic charge g and rotation a could be interpreted as extrinsic in certain contexts (e.g., astrophysically induced flux or AdS/CFT boundary potentials), such perspectives are incompatible with self-consistent regular black holes. Here, g arises topologically from NED to ensure singularity resolution, and a modifies local horizon geometry. The entropy corrections in Equation (26) thus reflect intrinsic degrees of freedom, distinct from external decorations. This intrinsic view is necessitated by the solution’s mathematical structure [26,40] and aligns with holography when boundary states encode bulk topology [20,25].