New Black Hole Solution in f(R) Theory and Its Related Physics

Abstract

Recent observations suggest that General Relativity (GR) faces challenges in fully explaining phenomena in regimes of strong gravitational fields. A promising alternative is the $f\left(R\right)$ theory of gravity, where R denotes the Ricci scalar. This modified theory aims to address the limitations observed in standard GR. In this study, we derive a black hole (BH) solution without introducing nonlinear electromagnetic fields or imposing specific constraints on R or the functional form of $f\left(R\right)$ gravity. The BH solution obtained here is different from the classical Schwarzschild solution in GR and, under certain conditions, reduces to the Schwarzschild (A)dS solution. This BH is characterized by the gravitational mass of the system and an additional parameter, which distinguishes it from GR BHs, particularly in the asymptotic regime. We show that the curvature invariants of this solution remain well defined at both small and large values of r. Furthermore, we analyze their thermodynamic properties, demonstrating consistency with established principles such as Hawking radiation, entropy, and quasi-local energy. This analysis supports their viability as alternative models to classical GR BHs.

1. Introduction

Recent groundbreaking achievements, such as the direct detection of gravitational waves [1,2] and the imaging of matter dynamics near supermassive BHs [3,4], have provided a wealth of precise and complementary observational data. These advancements have enabled deeper investigations into self-gravitating systems and the nature of gravitational interactions. GR has successfully passed numerous observational tests (e.g., [5,6,7]), solidifying its status as the most reliable and thoroughly tested theory of gravity to date. Nevertheless, GR continues to face unresolved theoretical and observational challenges, including the mysteries of dark energy [8,9,10], the nature of dark matter [11,12], and the ongoing quest to unify gravity with the other fundamental forces in a complete quantum theory of gravity [13,14].

Numerous extensions and alternatives to Einstein’s theory have been proposed to address these limitations. These approaches aim to either relax certain foundational assumptions of GR or build upon them in various ways [15,16,17,18]. However, none of the proposed models have yet succeeded in resolving all the aforementioned challenges simultaneously. It is widely believed that the extreme gravitational fields near a BH may hold the key to uncovering new physics. These regions could either reinforce GR or offer compelling evidence for a new theoretical framework [4].

Therefore, improving our understanding of how theoretical predictions align with observational data is essential. To this end, some research efforts adopt a theory-agnostic approach, employing general BH parametrizations [19,20] to explore gravitational phenomena without committing to a specific underlying theory.

This approach utilizes generic metrics capable of describing BH geometries across a range of gravitational theories, which are parameterized by a small set of variables. The primary objective is to identify potential deviations from GR-based metrics, which are typically constrained by fitting observational data [21,22]. The methodology involves two key steps:

• Using astrophysical observations to determine whether significant deviations from GR exist;
• If such deviations are detected, the most appropriate BH solution is reconstructed based on the data. This solution can then serve as an indirect test of alternative gravitational theories.

To explore modifications of gravity, higher-order corrections are often introduced to the Einstein–Hilbert action. Among the simplest such extensions is $f\left(R\right)$ gravity, where R denotes the Ricci scalar [23]. This framework has shown significant cosmological potential; for instance, it can successfully model inflation [24] (see ref. [25] for discussions on cosmological density perturbations), explain the late-time acceleration of the universe [26,27,28,29,30,31,32,33,34,35], or even offer a unified description of both phenomena [36,37,38].

In this study, we employed a combination of astrophysical methods to accurately identify potential deviations from the Schwarzschild metric. To this end, we focused on $f\left(R\right)$ gravity, which extends the Einstein’s Hilbert action by introducing an arbitrary function of the Ricci scalar R [23,36,39,40,41,42,43,44]. Specifically, we utilized the order-reduction approach within $f\left(R\right)$ gravity [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66] to investigate deviations within this theoretical framework.

Investigating modified theories of gravity also provides a pathway to addressing spacetime singularities. While the singularity theorems [67,68] support their existence, singularities in Einstein’s GR can be avoided by introducing unconventional matter sources. For instance, Sakharov [69] and Gliner [70] proposed that a vacuum configuration producing a de Sitter core near the center could result in singularity-free BHs [71]. Bardeen [72] introduced a nonsingular BH solution, which was later shown by Ayón-Beato and García [73,74], and independently by Bronnikov [75] to arise from nonlinear electrodynamics (NLE) as the source. Hayward constructed a nonsingular BH by specifying an appropriate mass function [76]. Fan and Wang developed a family of nonsingular BH solutions within the framework of NLE, encompassing the Bardeen and Hayward solutions, along with new classes of solutions [77]. Further analysis of the parameters and physical relevance of these solutions was provided by Bronnikov [78] and Toshmatov et al. [79]. Lan et al. have also discussed the issue of regular BHs within Einstein’s GR [80].

Nonsingular BH solutions have also been identified in various modified gravity theories, including $f\left(R\right)$ gravity [81], Lovelock gravity [82], Palatini gravity [83], Born–Infeld gravity [84], and gravity theories incorporating corrections from the generalized uncertainty principle [85]. Although these solutions may be susceptible to instabilities [86], they can be interpreted as effective models that incorporate quantum gravity effects to smooth out the central region [87]. This study aims to derive a BH solution within $f\left(R\right)$ gravity that remains finite as $r\to 0$, without introducing electromagnetic fields or imposing specific forms for the Ricci scalar.

This study is organized as follows: In Section 2, we present the field equations for $f\left(R\right)$. The new BH solution, including its metric potentials and curvature invariants, is introduced in Section 3. The behavior of the solution for both large and small values of r is also analyzed in Section 3. In Section 5, we investigate the physical properties of the solution by computing its thermodynamic quantities, which are supported by both analytical and graphical analysis. Finally, in Section 5, we summarize and discuss the main findings of the study. Detailed calculations are provided in Appendix A.

2. The f(R) Theory and Its Field Equations

The action of $f\left(R\right)$ theory is defined as follows:

$$\begin{array}{c}\hfill {\mathbb{I}}_{f\left(R\right)}=\int d^{4}x\sqrt{-g}\left[f\left(R\right)+2{\kappa }^2{\mathcal{L}}_m\right].\end{array}$$

(1)

Here, g denotes the determinant of $g_{\mu \nu }$, $f\left(R\right)$ is an arbitrary function R, ${\mathcal{L}}_m$ represents the Lagrangian density of matter, and ${\kappa }^2=8\pi G/c^4$, where G and c are Newton’s constant and the speed of light, respectively.

This theory can be formulated using two main approaches. The first, known as the metric formalism, treats the metric and matter fields as the fundamental dynamical variables. The second, called the Palatini formalism, also considers the metric and matter fields as dynamical but treats the Levi–Civita connection as independent of the metric. In this work, we adopt the metric formalism.

By applying the variational principle with respect to the metric to the action (1), we obtain

$$\begin{array}{c}\hfill f_R{R}_{\nu }^{\mu }-{\displaystyle \frac{1}{2}}{\delta }_{\nu }^{\mu }f+\left({\delta }_{\nu }^{\mu }\square -g^{\mu \beta }{\nabla }_{\beta }{\nabla }_{\nu }\right)f_R={\kappa }^2{\Theta }_{\nu }^{\mu }.\end{array}$$

(2)

Here, $f_R$ is the first derivative of $f\left(R\right)$ w.r.t. the Ricci scalar, ${\nabla }_{\nu }$ is the covariant derivative, □ is the d’Alembertian operator, and ${\Theta }_{\mu \nu }$ is the energy–momentum tensor of matter.

Taking the trace of Equation (2), we obtain

$$\begin{array}{c}\hfill \mathbb{I}=3\square f_R+R{f}_R-2f\left(R\right)\equiv 0.\end{array}$$

(3)

From Equation (3) one can derive $f\left(R\right)$ as

$$\begin{array}{c}\hfill f\left(R\right)={\displaystyle \frac{1}{2}}\left[3\square f_{\mathcal{R}}+R{f}_{\mathcal{R}}\right].\end{array}$$

(4)

Using of Equation (4) in Equation (2) gives [45,46,47,48,49,50,51,52,53,66]

$$\begin{array}{c}\hfill {\mathbb{I}}_{\mu \nu }=R_{\mu \nu }{f}_{\mathcal{R}}-{\displaystyle \frac{1}{4}}{g}_{\mu \nu }R{f}_R+{\displaystyle \frac{1}{4}}{g}_{\mu \nu }\square f_R-{\nabla }_{\mu }{\nabla }_{\nu }{f}_R-{\kappa }^2{\Theta }_{\mu \nu }.\end{array}$$

(5)

We assume the spherically symmetric spacetime to have the form

$$\begin{array}{c}\hfill d{s}^2=-S\left(r\right)d{t}^2+{\displaystyle \frac{d{r}^2}{S_1\left(r\right)}}+r^2\left[d{\theta }^2+{sin}^2\theta d{\varphi }^2\right],\end{array}$$

(6)

where $S\left(r\right)$ and $S_1\left(r\right)$ are functions of r. From Equation (6), we obtain

$$\mathcal{R}\left(r\right)={\displaystyle \frac{r^2{S}_1{S}^{\prime 2}-r^{2}S{S}^{\prime }{S}_1^{\prime }-2r^{2}S{S}_1{S}^{″}-4rS\left[S_1{S}^{\prime }-S{S}_1^{\prime }\right]+4S^2(1-S_1)}{2r^2{S}^2}},$$

(7)

where (′) means the derivative w.r.t r. The field equations of $f\left(R\right)$ theory are then found by using the line element (6) and the field Equation (5) to yield

$$\begin{array}{cc}\hfill {{\mathbb{I}}^t}_t=& {\displaystyle \frac{1}{8S^2{r}^2}}\left(\mathfrak{F}{S}^{\prime 2}{S}_1{r}^2-\mathfrak{F}{S}^{\prime }{S}_1^{\prime }S{r}^2-2\mathfrak{F}{S}^{″}{S}_{1}S{r}^2-4\mathfrak{F}r{S}_1{S}^{\prime }S+4\mathfrak{F}r{S}_1^{\prime }{S}^2-4\mathfrak{F}{S}^2+4\mathfrak{F}{S}^2{S}_1-3S{r}^2{\mathfrak{F}}^{\prime }{S}_1{S}^{\prime }\right.\hfill \\ & \left.+2S^2{r}^2{\mathfrak{F}}^{″}{S}_1+S^2{r}^2{\mathfrak{F}}^{\prime }{S}_1^{\prime }+4S^{2}r{\mathfrak{F}}^{\prime }{S}_1\right)=0,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {{\mathbb{I}}^r}_r=& {\displaystyle \frac{1}{8S^2{r}^2}}\left(\mathfrak{F}{S}^{\prime 2}{S}_1{r}^2-2\mathfrak{F}{S}^{″}{S}_{1}S{r}^2-\mathfrak{F}{S}^{\prime }{S}_1^{\prime }S{r}^2-4\mathfrak{F}r{S}_1^{\prime }{S}^2+4\mathfrak{F}r{S}_1{S}^{\prime }S-4\mathfrak{F}{S}^2+4\mathfrak{F}{S}^2{S}_1+S{r}^2{\mathfrak{F}}^{\prime }{S}_1{S}^{\prime }\right.\hfill \\ & \left.-6S^2{r}^2{\mathfrak{F}}^{″}{S}_1-3S^2{r}^2{\mathfrak{F}}^{\prime }{S}_1^{\prime }+4S^{2}r{\mathfrak{F}}^{\prime }{S}_1\right)=0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {{\mathbb{I}}^{\theta }}_{\theta }=& {{\mathbb{I}}^{\varphi }}_{\varphi }={\displaystyle \frac{-1}{8S^2{r}^2}}\left(\mathfrak{F}{S}^{\prime 2}{S}_1{r}^2-4\mathfrak{F}{S}^2+4\mathfrak{F}{S}^2{S}_1-\mathfrak{F}{S}^{\prime }{S}_1^{\prime }S{r}^2-2\mathfrak{F}{S}^{″}{S}_{1}S{r}^2-S{r}^2{\mathfrak{F}}^{\prime }{S}_1{S}^{\prime }-2S^2{r}^2{\mathfrak{F}}^{″}{S}_1\right.\hfill \\ & \left.-S^2{r}^2{\mathfrak{F}}^{\prime }{S}_1^{\prime }+4S^{2}r{\mathfrak{F}}^{\prime }{S}_1\right)=0,\hfill \end{array}$$

(10)

Finally, the trace equation given by Equation (3) yields the form

$$\begin{array}{cc}\hfill \mathbb{I}=& {\displaystyle \frac{1}{S^2{r}^2}}\left(3S{r}^2{\mathfrak{F}}^{\prime }{S}_1{S}^{\prime }+6S^2{r}^2{\mathfrak{F}}^{″}{S}_1+3S^2{r}^2{\mathfrak{F}}^{\prime }{S}_1^{\prime }+12S^{2}r{\mathfrak{F}}^{\prime }{S}_1-\mathfrak{F}{S}^{\prime }{S}_1^{\prime }S{r}^2-2\mathfrak{F}{S}^{″}{S}_{1}S{r}^2+\mathfrak{F}{S}^{\prime 2}{S}_1{r}^2\right.\hfill \\ & \left.-4\mathfrak{F}r{S}_1{S}^{\prime }S-4\mathfrak{F}r{S}_1^{\prime }{S}^2+4\mathfrak{F}{S}^2-4\mathfrak{F}{S}^2{S}_1-4f{S}^2{r}^2\right)=0,\hfill \end{array}$$

(11)

where $\mathfrak{F}=f_R$.

In the following section, we employ an algebraic method to solve these equations and derive a new BH solution.

3. Study of Black Hole Solutions in f(R) Gravity

It is not easy to derive a solution of Equations (8)–(10) in spite of the system having a closed form, i.e., we have three nonlinear differential equations in three unknowns S, $S_1$, and $\mathfrak{F}$. Therefore, we will assume a specific form for one of these unknowns to be able to derive the other two unknowns1. The function that we will assume in this study is $\mathfrak{F}$, which we assume to take the following form:

$$\begin{array}{c}\hfill \mathfrak{F}=1+{\displaystyle \frac{\alpha r^3}{r^4+g^4}},\end{array}$$

(12)

Here, the parameter $\alpha$ is a dimensional quantity with units of length, and g is another dimensional parameter, also possessing units of length.

It is important to note that if $g=0$, we obtain

$$\begin{array}{c}\hfill \mathfrak{F}=1+{\displaystyle \frac{\alpha }{r}}.\end{array}$$

(13)

Equation (13), previously studied in [52], leads to a black hole solution that exhibits problematic behavior $r\to 0$, which is in contrast to the results presented in this study.

The form $\mathfrak{F}=f_R$ given by Equation (12) reduces to Einstein GR when the parameter $\alpha =0$. Using Equation (12) in Equations (8)–(10), we obtain the explicit form of the two functions S and $S_1$, which we wrote in Appendix A.

Using Equations (12) and (A2) presented in Appendix A in Equation (3), we obtain the form of $f\left(R\right)\equiv f\left(r\right)$, which we write in Appendix A due to its length.

Using Equation (A2) presented in Appendix A, in Equation (7), we obtain the form of the Ricci scalar, which we also write in Appendix A. In the case of $\alpha =0$ we obtain

$$\begin{array}{c}\hfill \mathfrak{F}=1,S\left(r\right)=S_1\left(r\right)=1+c_2{r}^2+{\displaystyle \frac{c_1}{r}},\end{array}$$

(14)

which is the Ads/ds Schwarzschild spacetime. In the next section, we are going to discuss the physical properties of the BH solution derived in this study.

Characteristics of the BH (A2)

In this subsection, we explore the physical properties of the solution given by Equation (A2). Specifically, we analyze the asymptotic behavior of $S\left(r\right)$ and $S_1\left(r\right)$, as defined in Equation (A2) in Appendix A, in the limits of both small and large r. The behaviors of the two functions $S\left(r\right)$ and $S_1\left(r\right)$ are given as

$$\begin{array}{cc}\hfill S{\left(r\right)}_{\mid r\to 0}\approx & c_1+c_2{c}_1{r}^2-{\displaystyle \frac{\alpha c_1{r}^3}{g^4}}+{\displaystyle \frac{5{\alpha }^2{c}_1{r}^6}{14g^8}}+{\displaystyle \frac{\alpha c_1{r}^7}{g^8}}\cdots ,\hfill \\ \hfill S{\left(r\right)}_{\mid r\to \infty }\approx & c_1+c_2{c}_1{r}^2-{\displaystyle \frac{4\alpha c_1}{3r}}+{\displaystyle \frac{3{\alpha }^2{c}_1}{2r^2}}-{\displaystyle \frac{31{\alpha }^3{c}_1}{20r^3}}+{\displaystyle \frac{73{\alpha }^4{c}_1}{48r^4}}\cdots ,\hfill \\ \hfill S_1{\left(r\right)}_{\mid r\to 0}\approx & 1+c_2{r}^2-{\displaystyle \frac{3\alpha r^3}{g^4}}-{\displaystyle \frac{2\alpha c_2{r}^5}{g^4}}+{\displaystyle \frac{48{\alpha }^2{r}^6}{7g^8}}\cdots ,\hfill \\ \hfill S_1{\left(r\right)}_{\mid r\to \infty }\approx & 1+2\alpha c_{2}r+c_2{r}^2+{\displaystyle \frac{3}{2}}{\alpha }^2{c}_2+{\displaystyle \frac{\alpha (4+3{\alpha }^2{c}_2)}{6r}}+{\displaystyle \frac{{\alpha }^2(16+3{\alpha }^2{c}_2)}{48r^2}}-{\displaystyle \frac{\alpha ({\alpha }^2+120g^4{c}_2)}{208r^3}}\cdots .\hfill \end{array}$$

(15)

Using Equation (15) in Equation (6), we obtain

$$\begin{array}{cc}\hfill d{s}_{\mid r\to 0}^2\approx & -\left(1+{\Lambda }_{eff.}{r}^2-{\displaystyle \frac{3M{r}^3}{4g^4}}+{\displaystyle \frac{45M^2{r}^6}{224g^8}}+{\displaystyle \frac{3M{r}^7}{5g^8}}\right)d{t}^2\hfill \\ \hfill & +{\displaystyle \frac{d{r}^2}{1+{\Lambda }_{eff.}{r}^2-\frac{9M{r}^3}{4g^4}-\frac{3M{\Lambda }_{eff.}{r}^5}{2g^4}+\frac{27M^2{r}^6}{7g^8}}}+r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right),\hfill \\ \hfill d{s}_{\mid r\to \infty }^2\approx & -\left(1+{\Lambda }_{eff.}{r}^2-{\displaystyle \frac{M}{r}}+{\displaystyle \frac{27M^2}{32r^2}}-{\displaystyle \frac{837M^3}{1280r^3}}+{\displaystyle \frac{1971M^4}{4096r^4}}\right)d{t}^2\hfill \\ \hfill & +{\displaystyle \frac{d{r}^2}{1+{\Lambda }_{eff.}{r}^2+3/2{\Lambda }_{eff.}Mr+27/32{\Lambda }_{eff.}{M}^2+\frac{M(64+27M^2{\Lambda }_{eff.})}{128r}+\frac{M^2(768+81M^2{\Lambda }_{eff.})}{4096r^2}-\frac{M(5760g^4{\Lambda }_{eff.}+27M^2)}{1280r^3}}}\hfill \\ \hfill & +r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right),\hfill \end{array}$$

(16)

where we have put $c_1=1$, $c_2={\Lambda }_{eff.}$, and $M={\displaystyle \frac{4\alpha }{3}}$. Equation (16) shows that as $r\to 0$ and $r\to \infty$, the spacetime will approach AdS/dS spacetime according to the sign of ${\Lambda }_{eff.}$.

Next, we are going to calculate the invariants of the line element (16) and obtain the Kretschmann scalar as

$$\begin{array}{cc}\hfill & K=R_{\mu \nu \rho \sigma }{R^{\mu \nu \rho \sigma }}_{\mid r\to 0}\approx 24{\Lambda }_{eff.}^2-{\displaystyle \frac{108{\Lambda }_{eff.}M}{g^4}}r+{\displaystyle \frac{567M^2}{4g^8}}{r}^2-{\displaystyle \frac{126{\Lambda }_{eff.}^{2}M}{g^4}}{r}^3+{\displaystyle \frac{648M^2{\Lambda }_{eff.}}{4g^8}}{r}^4,\hfill \\ \hfill & K=R_{\mu \nu \rho \sigma }{R^{\mu \nu \rho \sigma }}_{\mid r\to \infty }\approx 24{\Lambda }_{eff.}^2-{\displaystyle \frac{54{\Lambda }_{eff.}^{2}M}{r}}+{\displaystyle \frac{54{\Lambda }_{eff.}^2{M}^2}{r^2}}+{\displaystyle \frac{1053{\Lambda }_{eff.}^2{M}^3}{32r^3}}+{\displaystyle \frac{45{\Lambda }_{eff.}{M}^2(64+81M^2{\Lambda }_{eff.})}{r^4}},\hfill \end{array}$$

(17)

and the squared Ricci tensor has the form

$$\begin{array}{cc}\hfill & R_{\mu \nu }{R^{\mu \nu }}_{\mid r\to 0}\approx 36{\Lambda }_{eff.}^2-{\displaystyle \frac{162{\Lambda }_{eff.}M}{g^4}}r+{\displaystyle \frac{1539M^2}{8g^8}}{r}^2-{\displaystyle \frac{189{\Lambda }_{eff.}^{2}M}{g^4}}{r}^3+{\displaystyle \frac{1863M^2{\Lambda }_{eff.}}{2g^8}}{r}^4,\hfill \\ \hfill & R_{\mu \nu }{R^{\mu \nu }}_{\mid r\to \infty }\approx 36{\Lambda }_{eff.}^2+{\displaystyle \frac{81{\Lambda }_{eff.}^{2}M}{r}}+{\displaystyle \frac{621{\Lambda }_{eff.}^2{M}^2}{8r^2}}+{\displaystyle \frac{2673{\Lambda }_{eff.}^2{M}^3}{64r^3}}+{\displaystyle \frac{9{\Lambda }_{eff.}{M}^2(224+405M^2{\Lambda }_{eff.})}{256r^4}},\hfill \end{array}$$

(18)

and finally, the Ricci scalar has the form

$$\begin{array}{cc}\hfill & R_{\mid r\to 0}\approx -12{\Lambda }_{eff.}+{\displaystyle \frac{27M}{g^4}}r+{\displaystyle \frac{63M{\Lambda }_{eff.}}{2g^4}}{r}^3-{\displaystyle \frac{81M^2}{g^8}}{r}^4-{\displaystyle \frac{126M}{g^8}}{r}^5,\hfill \\ \hfill & R_{\mid r\to \infty }\approx -12{\Lambda }_{eff.}^2-{\displaystyle \frac{27{\Lambda }_{eff.}M}{2r}}-{\displaystyle \frac{81{\Lambda }_{eff.}{M}^2}{16r^2}}-{\displaystyle \frac{81{\Lambda }_{eff.}{M}^3}{128r^3}}-{\displaystyle \frac{9M^2}{16r^4}}.\hfill \end{array}$$

(19)

Now, we are going to calculate the asymptote of the form of $f\left(r\right)$ at small and large r, which is given as

$$\begin{array}{cc}\hfill & {f\left(r\right)}_{\mid r\to 0}=-6{\Lambda }_{eff.}+{\displaystyle \frac{27Mr}{g^4}}+{\displaystyle \frac{63M{r}^3}{2g^4}}-{\displaystyle \frac{1215M^2{r}^4}{16g^8}}-{\displaystyle \frac{126M{r}^5}{g^8}},\hfill \\ \hfill & {f\left(r\right)}_{\mid r\to \infty }\approx -6{\Lambda }_{eff.}+{\displaystyle \frac{27M{\Lambda }_{eff.}}{2r}}-{\displaystyle \frac{81M^2{\Lambda }_{eff.}}{8r^2}}-{\displaystyle \frac{405M^3{\Lambda }_{eff.}}{128r^3}}-{\displaystyle \frac{9M^2(128+81M^2{\Lambda }_{eff.})}{2048r^4}}.\hfill \end{array}$$

(20)

Using the form of the Ricci scalar $R\left(r\right)$ and solving it asymptotically to obtain $r\left(R\right)$, we then substitute it in the form of Equation (20) and obtain

$$\begin{array}{cc}\hfill & f{\left(R\right)}_{\mid R\to 0}\approx {\gamma }_0+{\gamma }_{1}R+{\gamma }_2{R}^2,\hfill \\ \hfill & f{\left(R\right)}_{\mid r\to \infty }\approx -6{\Lambda }_{eff.}-{\displaystyle \frac{43{\Lambda }_{eff.}^{4/3}{M}^{4/3}}{g^{4/3}{R}^{1/3}}}-{\displaystyle \frac{101{\Lambda }_{eff.}^{5/3}{M}^{8/3}}{g^{8/3}{R}^{2/3}}}-{\displaystyle \frac{M^2{\Lambda }_{eff.}({\Lambda }_{eff.}{M}^2+5)}{g^{4}R}}.\hfill \end{array}$$

(21)

where ${\gamma }_0\cdots {\gamma }_2$ are constant depending on ${\Lambda }_{eff.}$ and M.

4. The Thermodynamic Properties of the BH Described by Equation (36)

To examine the thermodynamic properties of the newly identified BH solution, as indicated by Equation (A2), we present the idea of the Hawking temperature [88,89,90,91]2 as follows:

$$T_h={\displaystyle \frac{S^{\prime }\left(r_h\right)}{4\pi \sqrt{-S\left(r_h\right)S_1\left(r_h\right)}}},$$

(22)

where ′ is the derivative w.r.t. the horizon positioned at $r=r_h$. Here, $r_h$ corresponds to the largest positive root of the equation $S_1\left(r_h\right)=0$. The entropy in the framework of $f\left(\mathcal{R}\right)$ gravity is given by [92,93]

$$\delta \left(r_h\right)={\displaystyle \frac{1}{4}}A\mathfrak{F}\left(r_h\right),$$

(23)

where A is considered as the horizon area; the quasilocal energy in the $f\left(\mathcal{R}\right)$ gravitational theory is defined as follows [92]:

$${\displaystyle E\left(r_h\right)=\frac{1}{4}\int \left[2\mathfrak{F}\left(r_h\right)+{r_h}^2\left\{f\left(R\left(r_h\right)\right)-R\left(r_h\right)\mathfrak{F}\left(r_h\right)\right\}\right]d{r}_h.}$$

(24)

The thermodynamic stability of BHs is linked to the sign of their heat capacity, denoted as $C_h$, which is defined as [94]

$$C_h={\displaystyle \frac{d{E}_h}{d{T}_h}}={\displaystyle \frac{\partial M}{\partial r_h}}{\left({\displaystyle \frac{\partial T}{\partial r_h}}\right)}^{-1},$$

(25)

where $E_h$ represents the quasilocal energy. If $C_h$ is positive/negative, then the BH is considered stable/unstable. Lastly, the Gibbs free energy is defined as follows [93]:

$$G_h=E_h-T_h{\delta }_h.$$

(26)

In the context of the solution given by Equation (A2), the examination of all the aforementioned thermodynamic quantities commences with the constraint $S_1\left(r_h\right)=0$, leading to the following result:

$$\begin{array}{cc}& r_h={\displaystyle \frac{1}{24{\Lambda }_{eff}}}\sqrt[3]{\left(-270M^3{\Lambda }_{eff}+6\sqrt{3}\sqrt{{\displaystyle \frac{729M^6{{\Lambda }_{eff}}^3+1728M^4{{\Lambda }_{eff}}^2+18432M^2{\Lambda }_{eff}+65536}{{\Lambda }_{eff}}}}\right){{\Lambda }_{eff}}^2}\hfill \\ & -1/4\left(3M^2{\Lambda }_{eff}+32\right){\displaystyle \frac{1}{\sqrt[3]{\left(-270M^3{\Lambda }_{eff}+6\sqrt{3}\sqrt{\frac{729M^6{{\Lambda }_{eff}}^3+1728M^4{{\Lambda }_{eff}}^2+18432M^2{\Lambda }_{eff}+65536}{{\Lambda }_{eff}}}\right){{\Lambda }_{eff}}^2}}}-1/2M,\hfill \\ & M_h={\displaystyle \frac{4}{9{\Lambda }_{eff}}}\sqrt[3]{\left(\sqrt{{\displaystyle \frac{27{{\Lambda }_{eff}}^3{r_h}^6+108{{\Lambda }_{eff}}^2{r_h}^4+225{\Lambda }_{eff}{r_h}^2+64}{{\Lambda }_{eff}}}}-9r_h\right){{\Lambda }_{eff}}^2}\hfill \\ & -4/9\left(3{\Lambda }_{eff}{r_h}^2+4\right){\displaystyle \frac{1}{\sqrt[3]{\left(\sqrt{\frac{27{{\Lambda }_{eff}}^3{r_h}^6+108{{\Lambda }_{eff}}^2{r_h}^4+225{\Lambda }_{eff}{r_h}^2+64}{{\Lambda }_{eff}}}-9r_h\right){{\Lambda }_{eff}}^2}}}-4{\displaystyle \frac{4r_h}{3}}.\hfill \end{array}$$

(27)

Equation (27) demonstrates that the BH’s total mass is dependent on the horizon radius as well as on the ${\Lambda }_{eff}$. We depict $S_1\left(r\right)$ via $r_h$ in Figure 1, demonstrating the dashed-dots curve’s potential horizons for this solution. Furthermore, the dashed curve in Figure 1 illustrates the degenerate horizon, which may be found by equating ${\displaystyle \frac{\partial M_h}{\partial r_h}}$ to zero. Lastly, the dotted curve of Figure 1a illustrates the naked singularity of this solution.

Using Equation (23), we can calculate the entropy of Equation (A2) as follows:

$$\begin{array}{c}\hfill {\delta }_h={\displaystyle \frac{\pi {r_h}^2\left(4g^4+3{r_h}^{3}M+4{r_h}^4\right)}{4({r_h}^4+g^4)}}.\end{array}$$

(28)

If the dimensional quantity $g=0$, we obtain the standard entropy of Einstein’s GR.

• The pattern of $S_h$ is shown in Figure 1b, indicating a well-behaved entropy profile.

The temperature associated with Equation (A2) is computed as

$$\begin{array}{cc}& T_h={\displaystyle \frac{4\left(2{\Lambda }_{eff}{r_h}^3+M\right)\sqrt{5}}{\pi }}\left\{\left({\Lambda }_{eff}{r_h}^3+r_h-M\right)\left(1280\Lambda {r_h}^5+1920M\Lambda {r_h}^4+1080M^2{\Lambda }_{eff}{r_h}^3+1280{r_h}^3\right.\right.\hfill \\ & {\left.\left.+270M^3{r_h}^2{\Lambda }_{eff}+640M{r_h}^2-27M^3-5760M{g}^4{\Lambda }_{eff}\right)\right\}}^{-1/2},\hfill \end{array}$$

(29)

where $T_h$ is the Hawking temperature at the event horizon. We plot the Hawking temperature in Figure 1c, which shows that we always have a positive Hawking temperature.

From Equation (24), the quasilocal energy corresponding to the BH solution given by Equation (A2) is calculated as

$$\begin{array}{cc}& E_h={\displaystyle \frac{1}{16384g}}\left(243{\Lambda }_{eff}{M}^4\sqrt{2}ln\left({\displaystyle \frac{r^2-gr\sqrt{2}+g^2}{r^2+gr\sqrt{2}+g^2}}\right)-5184{\Lambda }_{eff}{M}^2{g}^2\sqrt{2}ln\left({\displaystyle \frac{r^2+gr\sqrt{2}+g^2}{r^2-gr\sqrt{2}+g^2}}\right)\right.\hfill \\ & +486\sqrt{2}{M}^2{\Lambda }_{eff}\left(-{\displaystyle \frac{64}{3}}{g}^2+M^2\right)arctan\left({\displaystyle \frac{\sqrt{2}r-g}{g}}\right)+486\sqrt{2}{M}^2{\Lambda }_{eff}\left(-{\displaystyle \frac{64}{3}}{g}^2+M^2\right)arctan\left({\displaystyle \frac{\sqrt{2}r+g}{g}}\right)\hfill \\ & -10368g\left(\left(-{\displaystyle \frac{4}{27}}M-3/8M^3{\Lambda }_{eff}\right)ln\left(r^4+g^4\right)+{\displaystyle \frac{16}{9}}{\Lambda }_{eff}M{g}^{2}arctan\left({\displaystyle \frac{r^2}{g^2}}\right)+{\Lambda }_{eff}{M}^{3}ln\left(r\right)\right.\hfill \\ & \left.\left.-2r\left({\displaystyle \frac{32}{81}}+\left({\displaystyle \frac{32}{81}}{r}^2+{\displaystyle \frac{8}{9}}Mr+M^2\right){\Lambda }_{eff}\right)\right)\right).\hfill \end{array}$$

(30)

To guarantee a positive value for the quasilocal energy, the following condition must be met:

$${\displaystyle \frac{r^2-gr\sqrt{2}+g^2}{r^2+gr\sqrt{2}+g^2}}>0,\mathrm{which}\mathrm{yields}g>0.$$

We plot the quasilocal energy in Figure 1d, which demonstrates that the Hawking temperature is always positive, indicating a consistently positive value.

Now we are going to calculate $C\left(r_h\right)$ of Equation (A2). Using Equation (25), we obtain

$$\begin{array}{cc}& C\left(r_h\right)=-{\displaystyle \frac{1}{4050}}\left({\displaystyle \frac{1}{{r_h}^4}}\left({\Lambda }_{eff}{r_h}^3+r_h-M\right)\left(1280{\Lambda }_{eff}{r_h}^5+1920M{\Lambda }_{eff}{r_h}^4+1080M^2{\Lambda }_{eff}{r_h}^3+1280{r_h}^3-27M^3\right.\right.\hfill \\ & {\left.\left.+270M^3{r_h}^2{\Lambda }_{eff}+640M{r_h}^2-5760M{g}^4{\Lambda }_{eff}\right)\right)}^{3/2}\sqrt{5}{r_h}^6\left(2{{\Lambda }_{eff}}^2{r}_h\left(27{{\Lambda }_{eff}}^3{r_h}^6+108{{\Lambda }_{eff}}^2{r_h}^4+225{\Lambda }_{eff}{r_h}^2+64\right)\right.\hfill \\ & +\left({\Lambda }_{eff}{\left(\left(-9r_h+\sqrt{{\displaystyle \frac{27{{\Lambda }_{eff}}^3{r_h}^6+108{{\Lambda }_{eff}}^2{r_h}^4+225{\Lambda }_{eff}{r_h}^2+64}{{\Lambda }_{eff}}}}\right){{\Lambda }_{eff}}^2\right)}^{2/3}+4{{\Lambda }_{eff}}^2-15{{\Lambda }_{eff}}^3{r_h}^2\right.\hfill \\ & \left.+{\left(\left(-9r_h+\sqrt{{\displaystyle \frac{27{{\Lambda }_{eff}}^3{r_h}^6+108{{\Lambda }_{eff}}^2{r_h}^4+225{\Lambda }_{eff}{r_h}^2+64}{{\Lambda }_{eff}}}}\right){{\Lambda }_{eff}}^2\right)}^{4/3}\right)\left\{{\displaystyle \frac{1}{{\Lambda }_{eff}}}\left(27{{\Lambda }_{eff}}^3{r_h}^6+108{{\Lambda }_{eff}}^2{r_h}^4+64\right.\right.\hfill \\ & {\left.\left.+225{\Lambda }_{eff}{r_h}^2\right)\right\}}^{1/2}-{\Lambda }_{eff}{r}_h\left(9{{\Lambda }_{eff}}^2{r_h}^4+24{\Lambda }_{eff}{r_h}^2+25\right)\left(3{{\Lambda }_{eff}}^2{r_h}^2+4{\Lambda }_{eff}+\left(\left(-9r_h+\left\{{\displaystyle \frac{1}{{\Lambda }_{eff}}}\left(27{{\Lambda }_{eff}}^3{r_h}^6+64\right.\right.\right.\right.\right.\hfill \\ & \left.\left.{\left.\left.\left.\left.+108{{\Lambda }_{eff}}^2{r_h}^4+225{\Lambda }_{eff}{r_h}^2\right)\right\}\right){{\Lambda }_{eff}}^2\right)}^{2/3}\right)\right)\pi {\left(\left(\sqrt{{\displaystyle \frac{27{{\Lambda }_{eff}}^3{r_h}^6+108{{\Lambda }_{eff}}^2{r_h}^4+225{\Lambda }_{eff}{r_h}^2+64}{{\Lambda }_{eff}}}}-9r_h\right){{\Lambda }_{eff}}^2\right)}^{-4/3}\hfill \\ & \times {\displaystyle \frac{1}{\sqrt{\frac{27{{\Lambda }_{eff}}^3{r_h}^6+108{{\Lambda }_{eff}}^2{r_h}^4+225{\Lambda }_{eff}{r_h}^2+64}{{\Lambda }_{eff}}}}}\left(-{\displaystyle \frac{256}{27}}{{\Lambda }_{eff}}^3{r_h}^{10}-{\displaystyle \frac{64}{9}}{{\Lambda }_{eff}}^3{r_h}^{9}M+\left(-{\displaystyle \frac{128}{9}}{{\Lambda }_{eff}}^{2}M+{{\Lambda }_{eff}}^3{M}^3\right){r_h}^7\right.\hfill \\ & +\left(-{\displaystyle \frac{280}{9}}{{\Lambda }_{eff}}^2{M}^2+{\displaystyle \frac{256}{27}}{\Lambda }_{eff}\right){r_h}^6+\left(-{\displaystyle \frac{320}{9}}M{\Lambda }_{eff}-64{{\Lambda }_{eff}}^{3}M{g}^4-{\displaystyle \frac{213}{10}}{{\Lambda }_{eff}}^2{M}^3\right){r_h}^5+\left(-13/2M^4{{\Lambda }_{eff}}^2\right.\hfill \\ & \left.-{\displaystyle \frac{64}{3}}{M}^2{\Lambda }_{eff}\right){r_h}^4+\left(-{\displaystyle \frac{320}{3}}{{\Lambda }_{eff}}^{2}M{g}^4+{\displaystyle \frac{103}{18}}{M}^3{\Lambda }_{eff}-{\displaystyle \frac{256}{27}}M\right){r_h}^3+\left({\displaystyle \frac{21}{4}}{M}^4{\Lambda }_{eff}+160{{\Lambda }_{eff}}^2{M}^2{g}^4+{\displaystyle \frac{32}{9}}{M}^2\right){r_h}^2\hfill \\ & {\left.+\left(M^5{\Lambda }_{eff}+{\displaystyle \frac{64}{27}}{M}^3\right)r_h+1/20M^4+{\displaystyle \frac{32}{3}}{M}^2{g}^4{\Lambda }_{eff}\right)}^{-1}.\hfill \end{array}$$

(31)

The above equation indicates that for a positive value of the heat capacity, we must have ${\Lambda }_{eff}>0$. We plot the heat capacity in Figure 1e, which shows that the heat capacity is always positive.

Gibbs free energy is described as [93]

$$G\left(r_h\right)=E\left(r_h\right)-T\left(r_h\right)S\left(r_h\right).$$

(32)

Here, the expressions $T\left(r_h\right)$, $S\left(r_h\right)$, and $E\left(r_h\right)$ are defined in Equations (28)–(30). Using Equations (23) and (27)–(29) in (32), we obtain

$$\begin{array}{cc}& G\left(r_h\right)={\displaystyle \frac{1}{16384\mathbb{B}g\left({r_h}^4+g^4\right)}}\left(-10368\mathbb{B}{M}^2\sqrt{2}{\Lambda }_{eff}arctan\left({\displaystyle \frac{\sqrt{2}{r}_h+g}{g}}\right)g^6+486\mathbb{B}{\Lambda }_{eff}{M}^4\sqrt{2}arctan\left({\displaystyle \frac{\sqrt{2}{r}_h+g}{g}}\right){r_h}^4\right.\hfill \\ & +486\mathbb{B}{\Lambda }_{eff}{M}^4\sqrt{2}arctan\left({\displaystyle \frac{\sqrt{2}{r}_h+g}{g}}\right)g^4+3888\mathbb{B}gln\left({r_h}^4+g^4\right)M^3{\Lambda }_{eff}{r_h}^4-18432\mathbb{B}M{\Lambda }_{eff}{g}^{3}arctan\left({\displaystyle \frac{{r_h}^2}{g^2}}\right){r_h}^4\hfill \\ & +{\Lambda }_{eff}\left[243\mathbb{B}{M}^4\sqrt{2}ln\left({\displaystyle \frac{{r_h}^2-gr\sqrt{2}+g^2}{{r_h}^2+gr\sqrt{2}+g^2}}\right)({r_h}^4+g^4)-10368\mathbb{B}g{M}^{3}ln\left(r\right){r_h}^4-5184\mathbb{B}{M}^2{g}^6\sqrt{2}ln\left({\displaystyle \frac{{r_h}^2-gr\sqrt{2}-g^2}{gr\sqrt{2}-{r_h}^2-g^2}}\right)\right]\hfill \\ & +arctan\left({\displaystyle \frac{\sqrt{2}{r}_h-g}{g}}\right)\left[486\mathbb{B}{\Lambda }_{eff}{M}^4\sqrt{2}{r_h}^4-10368\mathbb{B}{M}^2\sqrt{2}{\Lambda }_{eff}{g}^6+486\mathbb{B}{\Lambda }_{eff}{M}^4\sqrt{2}{g}^4-131072\sqrt{5}{g}^5{\Lambda }_{eff}{r_h}^3\right]\hfill \\ & -131072\sqrt{5}g{\Lambda }_{eff}{r_h}^7-49152\sqrt{5}g{M}^2{r_h}^3-65536\sqrt{5}gM{r_h}^4+1536\mathbb{B}{g}^{5}Mln\left({r_h}^4+g^4\right)+8192\mathbb{B}g{\Lambda }_{eff}{r_h}^7+8192\mathbb{B}{g}^5{\Lambda }_{eff}{r_h}^3\hfill \\ & -65536\sqrt{5}{g}^{5}M-98304\sqrt{5}g{\Lambda }_{eff}{r_h}^{6}M+8192\mathbb{B}g{r_h}^5+8192\mathbb{B}{g}^5{r}_h+ln\left({r_h}^4+g^4\right)\left[1536\mathbb{B}gM{r_h}^4+3888\mathbb{B}{g}^5{M}^3{\Lambda }_{eff}\right]\hfill \\ & +{\Lambda }_{eff}\mathbb{B}\left[18432gM{r_h}^6-18432M{g}^{7}arctan\left({\displaystyle \frac{r^2}{g^2}}\right)-10368g^5{M}^{3}ln\left(r\right)+18432g^{5}M{r_h}^2+20736g{M}^2{r_h}^5+20736g^5{M}^2{r}_h\right]\hfill \\ & \left.-5184\mathbb{B}{\Lambda }_{eff}{M}^2{g}^2\sqrt{2}\left\{ln\left({\displaystyle \frac{{r_h}^2+gr\sqrt{2}+g^2}{{r_h}^2-gr\sqrt{2}+g^2}}\right)+2\left[arctan\left({\displaystyle \frac{\sqrt{2}{r}_h-g}{g}}\right)+arctan\left({\displaystyle \frac{\sqrt{2}{r}_h+g}{g}}\right)\right]\right\}\right),\hfill \end{array}$$

(33)

where

$$\begin{array}{cc}& \mathbb{B}={\displaystyle \frac{1}{{r_h}^2}}\left\{\left({\Lambda }_{eff}{r_h}^3+r_h-M\right)\left(1280{\Lambda }_{eff}{r_h}^5+1920M{\Lambda }_{eff}{r_h}^4+1080M^2{\Lambda }_{eff}{r_h}^3+1280{r_h}^3+270M^3{r_h}^2{\Lambda }_{eff}\right.\right.\hfill \\ & {\left.\left.+640M{r_h}^2-27M^3-5760M{g}^4{\Lambda }_{eff}\right)\right\}}^{1/2}.\hfill \end{array}$$

(34)

The figure (Figure 1f) illustrates the behavior of the Gibbs energy for the BH solution outlined in Equation (A2). This visualization is based on a specific set of model parameters and demonstrates that for this solution, the Gibbs free energy is consistently positive. A positive Gibbs free energy signifies that the BH configuration is globally more stable.

5. Conclusions

In this study, we explore the concept of new BHs within the framework of $f\left(R\right)$ gravity. To achieve this, we reformulate the field equations of $f\left(R\right)$ gravity, reducing them from fourth order to third order, as shown in Equation (5). We then apply these third-order field equations to a spherically symmetric spacetime with two distinct metric potentials and derive the corresponding differential equations. Although the resulting system consists of three differential equations with three unknowns, obtaining an exact solution proved challenging. As a result, we were compelled to assume a specific form for one of the unknowns.

In this work, we assume the form of $\mathfrak{F}=f_R\left(R\right)=1+{\displaystyle \frac{\alpha r^3}{r^4+g^4}}$, which enables us to determine the other two unknowns (the metric potentials). Notably, this form of $f\left(R\right)$ reduces to the cases discussed in Refs. [45,47,49,51,52,66] when the dimensional parameter $g=0$. The nonvanishing nature of g introduces invariants that are defined as $r\to 0$ and $r\to \infty$, which is in contrast to the previous studies presented in [45,47,49,51,52,66]. Additionally, we derive the corresponding form of $f\left(R\right)$ for this solution, demonstrating that as $R\to 0$, it behaves as a polynomial function Equation (20), while as $R\to \infty$, it exhibits asymptotic behavior such as $R^{-1/3}$, $R^{-2/3}$⋯, and so on, as shown in Equation (20).

In this study, we examine the physical properties of the derived solution by analyzing the asymptotic behavior of the metric potentials at both small and large values of r. We demonstrate that the metric potentials remain finite in both limits. Additionally, we derive the asymptotic forms of the curvature invariants associated with the solution and show that they are free from singularities, unlike those in general relativity. We also show that the solution of this study depends mainly on the dimensional parameters g and $\alpha$; when $\alpha =0$, the solution reduces to AdS/dS spacetime.

Finally, we investigated the physical properties of this BH solution by calculating its thermodynamic quantities. We demonstrated that all of these values behave in a manner consistent with physical expectations, as illustrated in Figure 1. Furthermore, we established the stability of the solution, as evidenced by the positive trends in the heat capacity and Gibbs free energy plots.

In summary, we have successfully derived a new black hole solution within $f\left(R\right)$ gravitational theory. This raises several pertinent questions for future investigation: What is the geodesic completeness of this solution? What would occur if we introduced a specific form of nonlinear electrodynamics? How would the asymptotic behavior of the metric manifest, and how would the curvature invariants behave? These questions will be explored in future work.