Existence of Non-singular Stellar Solutions within the context of Electromagnetic Field: A Comparison between Minimal and Non-minimal Gravity Models

Cosmologists have recently revealed revolutionary discoveries that defy traditional beliefs regarding the spatial organization of celestial structures in our cosmos. Rather than presenting a random dispersion, these formations exhibit a discernible order, sparking significant curiosity among researchers. The meticulous study of these entities has become a central topic of exploration for scientists committed to unraveling the mystery surrounding the accelerated expansion of the universe. Empirical testimonies strongly indicate the existence of an extensive counter-force to gravitational attraction, driving the observed rapid expansion. Referred to as dark energy, this enigmatic force presents a significant puzzle for scientists. While Einstein’s general relativity (GR) provides some insights into this expansion, it encounters difficulties in fully explaining dark energy, particularly in relation to the cosmological constant $\Lambda$. Therefore, it has been necessary to introduce modifications to the existing theory to better comprehend and enhance our knowledge regarding fundamental dynamics of the cosmos.

Einstein’s GR has straightforwardly been modified to $f(\mathcal{R})$ gravity, representing a substantial advancement in experimental physics. This theory alters the action function by interchanging the curvature scalar $\mathcal{R}$ and its general functional. Notable progress has been made within this gravity theory, with implications reaching into the study of celestial structures 2 -4 . Astashenok with his collaborators 1n investigated the upper mass limit for massive objects in the current framework. Their research produced an intriguing result that as a second object in the binary GW190814, there must be either a rapidly rotating neutron star or a black hole. A significant body of literature underscores the remarkable contributions made by various researchers 1o -9 . One notable contribution comes from Bertolami and his colleagues 10 , who were instrumental in put forwarding the coupling between matter and spacetime geometry in $f(\mathcal{R})$ gravity. Their methodology involved integrating the matter Lagrangian and $\mathcal{R}$ into a unified functional form, known as $f(\mathcal{R},\mathcal{L}_{\mathrm{m}})$ theory. This novel concept prompted astronomers to focus on discussions related to the rapid universe’ expansion 11 .

Following these developments, Harko et al. 20 introduced a ground-breaking gravitational theory, called $f(\mathcal{R},\mathcal{T})$ gravity at the action level. This theory utilizes a generalized function that leads to a non-conserved phenomenon, resulting in the emergence of an extra force, causing moving particles to follow non-geodesic path 22 . Houndjo 22a employed a particular model based on the minimal interaction to explain the shift from one cosmic era to the other phase in which we are living right now. Among the various functional forms of $f(\mathcal{R},\mathcal{T})$ theory, the $\mathcal{R}+2\beta\mathcal{T}$ candidate has attracted considerable attention in scientific literature due to its ability to generate physically existing internal structures. Different researchers, including Das et al. 22b , utilized a similar model to develop a three-layer gravastar geometry. Various methodologies were implemented to explore diverse geometrical structures in this context 25af -25ae . An essential facet of the $f(\mathcal{R},\mathcal{T})$ gravity is its incorporation of some effects at quantum level, which introduces the potential for particle creation. This characteristic is of great significance in astronomical investigations as it sets out a connection between the extended theory and quantum mechanics. Notable findings in this area have been produced and can be seen in 1j ; 1k . In a recent research endeavor, Zaregonbadi et al. 1l have examined the feasibility of this modification to GR to study the impact of dark matter on clusters of galaxies.

The $f(\mathcal{R},\mathcal{T})$ theory has indeed presented an intriguing extension to GR, showing a diverse range of phenomenology in modern research. However, researchers 25ag ; 25aga delved into the challenges associated with constructing a viable and realistic cosmology within this theory. Their study demonstrated that the currently discussed models of this theory do not yield an expandable cosmic background. In response to these challenges, Haghani and Harko 25ah undertook a considerable effort by simultaneously unify two categories of gravitational theories, and call it the $f(\mathcal{R},\mathcal{L}_{m},\mathcal{T})$ gravity. This strategic approach aims to address the limitations encountered in the previously discussed gravity models and offers a more comprehensive understanding of the intricate dynamics governing the universe. They explored the Newtonian limit of the field equations and provided some terms representing an extra-acceleration, particularly focusing on scenarios involving small velocities of particles and weak fields of gravity. This exploration enlightens how different choices of Lagrangian influence the description of the cosmic expansion. Zubair et al. 25ai reconstructed some cosmological solutions such as de Sitter and $\Lambda$CDM models in this theory and found them to be cosmologically stable through suitable perturbations.

The study of celestial entities characterized by the field equations possessing high non-linearity, either in the framework of GR or extended theories, has prompted astronomers to actively seek their numerical or exact solutions. The significance of compact interiors lies in the physical interest they hold, contingent upon the satisfaction of specific conditions by the developed model. Various methodologies have been engaged in the scientific literature to derive such solutions, including the utilization of a specific ansatz or the implementation of particular equations of state, among other techniques. One approach to solving this challenge is through the implementation of the embedding class-one phenomenon, which posits that one can embed any space in another having at least one higher dimension. Bhar et al. 36 employed the same method, coupled with particular metric potentials and derived physically existing anisotropic solutions. Maurya et al. 37 ; 37a used the same approach to construct a new solution, delving into its stability and exploring the impact of anisotropic pressure on relativistic systems. Singh with his collaborators 37b devised a singularity-free solution for spherical geometry by proposing a specific metric function within the framework of this technique. Exploring this condition into a matter-geometry coupled theory, several works have yielded stable as well as viable solutions 38c -38a .

In this paper, we explore various isotropic solutions in conjunction with the Maxwell field within the framework of $f(\mathcal{R},\mathcal{L}_{m},\mathcal{T})$ theory. The paper’s structure is organized as follows. The following section establishes some basics of this extended theory and derives the generalized field equations. Section III presents the Karmarkar condition, which aids in determining the metric potentials. Additionally, we utilize the Reissner-Nordström vacuum solution and compute the constants associated with the overhead condition. We outline particular criteria that, once fulfilled, guarantee the model’s physical validity in section IV. Advancing further, section V reveals the newly formulated solutions and offers a visual representation to aid in understanding the physical relevance of the obtained results. Conclusively, in the final section, our findings are encapsulated, summarizing the main outcomes and insights acquired in this study.

The action of the modified $f(\mathcal{R},\mathcal{L}_{m},\mathcal{T})$ theory is obtained after replacing the Ricci scalar with this functional 25ah . This has the form

where the electric charge and ordinary matter have Lagrangian densities, denoted by $\mathcal{L}_{\mathcal{E}}$ and $\mathcal{L}_{m}$, respectively. Also, $g=|g_{\epsilon\omega}|$ with $g_{\epsilon\omega}$ being the metric tensor and the two lines enclosing it symbolize the determinant. Varying the action (1) w.r.t. $g_{\epsilon\omega}$, the tensorial form of the modified field equations become

where the entity $\mathcal{G}_{\epsilon\omega}$, namely the Einstein tensor, expresses the geometry of the considered fluid distribution and $\mathcal{T}_{\epsilon\omega}^{(eff)}$ refers to the matter enclosed by that geometry. This effective term is further classified into three different energy-momentum tensors as

where

• •

  $\mathcal{T}^{(m)}_{\epsilon\omega}$ correspond to the ordinary matter configuration,

• •

  $\mathcal{E}_{\epsilon\omega}$ indicates the presence of charge in the self-gravitating system,

• •

  $\mathcal{T}^{(cr)}_{\epsilon\omega}$ are modified correction terms.

We express $\mathcal{T}^{(m)}_{\epsilon\omega}$ as follows

On the other hand, the last term on the right side of Eq.(3) have the value given by

where $f_{\mathcal{T}}=\frac{\partial f(\mathcal{R},\mathcal{L}_{m},\mathcal{T})}{\partial\mathcal{T}}$, $f_{\mathcal{L}_{m}}=\frac{\partial f(\mathcal{R},\mathcal{L}_{m},\mathcal{T})}{\partial\mathcal{L}_{m}}$ and $f_{\mathcal{R}}=\frac{\partial f(\mathcal{R},\mathcal{L}_{m},\mathcal{T})}{\partial\mathcal{R}}$. The mathematical definitions of the D’Alembertian operator and covariant derivative are $\Box\equiv(-g)^{\frac{-1}{2}}\partial_{\epsilon}\big{(}\sqrt{-g}g^{\epsilon\omega}\partial_{\omega}\big{)}$ and $\nabla_{\epsilon}f_{\mathcal{R}}=f_{\mathcal{R},\epsilon}-\Gamma^{\omega}_{\epsilon\omega}f_{\mathcal{R}}$, respectively. Equations (2)-(4) provides after combining as

The energy-momentum tensor plays a pivotal role in formulating the gravitational field equations, enabling a precise representation of the interaction between matter and spacetime curvature. This tensor proves indispensable in understanding a wide array of physical phenomena, from celestial bodies’ gravitational influences to the dynamics of fluid systems. Its incorporation not only facilitates the development of accurate models for diverse astrophysical scenarios but also contributes to the exploration of fundamental principles in the broader context. The models possessing the isotropic fluid among all existing in the literature holds significance. Its application proves instrumental in various scientific disciplines, contributing to the development of accurate and tractable models for the study of diverse physical processes. Such matter distributions can be defined in the following way 1i

where $P$ being the pressure, $\rho$ symbolizes the energy density and $\mathcal{V}_{\epsilon}$ indicates the four-velocity. The stress-energy tensor expressing the electromagnetic field is defined by 38b

whereas we can write Maxwell equations in concise (or tensorial) form as

Here, $\mathcal{W}_{\epsilon\omega}=\varphi_{\omega;\epsilon}-\varphi_{\epsilon;\omega}$ is written in terms of the four potential defined by $\psi_{\omega}=\psi(r)\delta^{0}_{\omega}$. Also, the current $\varepsilon^{\epsilon}$ and charge density $\varpi$ are combined with each other through the relation $\varepsilon^{\epsilon}=\varpi\mathcal{V}^{\epsilon}$.

Determining the trace of Eq.(5), we have the following

As functional of this theory is generalized in terms of the geometry and matter terms, the divergence of the stress-energy tensor becomes non-null. As a result, a supplementary force emerges within the gravitational field of a massive object, leading to modifications in the geodesic trajectory of moving test particles. This force is mathematically expressed as follows

Considering a spherical spacetime as an interior geometry is a significant starting point as its investigation involves understanding the curvature dynamics and gravitational interactions specific to a spherical space. The following metric represents such geometry as

where radial/temporal components depend only on the radial coordinate, showing that the geometry under consideration is static. We observe the presence of the four-vector in Eq.(6) which now becomes

Equation (7) (left) along with the metric (9) yields

where ${}^{\prime}=\frac{\partial}{\partial r}$. Implementing an integration on the above second-order equation results in the following expression

where the total interior charge is defined as $s\equiv s(r)=\int_{0}^{r}\varpi e^{\frac{\varrho_{2}}{2}}\bar{r}^{2}d\bar{r}$.

The isotropic modified field equations representing spherical structure are now formulated by combining Eqs.(5), (6) and (9). The non-vanishing components are given by

Also, the terms $\mathcal{T}$ and $\mathcal{R}$ are defined as

Solving Eqs.(11)-(13) presents a complex challenge due to the intricate relationships among multiple quantities, including $(\varrho_{1},\varrho_{2},\rho,P,q)$. To address this complexity and arrive at a definitive solution, it is essential to introduce specific constraints. Without these constraints, obtaining a unique solution proves to be an insurmountable task.

In this section, we review multiple conditions that have been discussed in the literature whose satisfaction leads to the compact interior models to be physically relevant ab -af . We highlight some interesting and necessary conditions among them that must be discussed while studying the stars in the following.

• •

  A critical aspect involves the investigation of geometric quantities such as $e^{\varrho_{1}}$ and $e^{\varrho_{2}}$. It must be verified that both these components are positive to maintain physical significance. Additionally, the regularity of these functions should be confirmed within the defined physical domain, ensuring they do not exhibit singularities.

• •

  Within compact stars, the behavior of energy density and pressure, along with their first two derivatives, is critical in understanding the internal configuration of these astrophysical objects. Typically, as one moves from the stellar surface towards the center, both these parameters tend to increase, reaching their maximum values at the core. The first derivatives with respect to radial distance capture the rate of change of these quantities, highlighting the distribution of mass and the response of matter to gravitational forces.

• •

  A point of discussion among researchers is the mass function that describes the fluid content enclosed by a body. This helps in understanding the gravitational impact of that structure. We express this in the form of energy density as

  $$m(r)=\frac{1}{2}\int_{0}^{\mathrm{R}}\bar{r}^{2}\rho d\bar{r}.$$

  (28)

  The strength of a field surrounding a self-gravitating structure due to its gravity in relation with its size is measured by the compactness. It is actually a ratio between the mass of a body and its radius. Its expression is given by

  $$\lambda(r)=\frac{m(r)}{r},$$

  (29)

  which must be less than $\frac{4}{9}$ to get a physically relevant interior 42a . The redshift characterizes the extent to which photons are stretched as they climb out of the gravitational well of a compact star. We describe it as

  $$\textbf{z}(r)=\frac{1-\sqrt{1-2\lambda(r)}}{\sqrt{1-2\lambda(r)}}.$$

  (30)

  It has been found that the redshift at the surface boundary must not be higher than $2$ 42a , i.e., $\textbf{z}_{\Sigma}\leq 2$. On the other hand, when Ivanov dealt with anisotropic pressure fluid, he established this limit to be $5.211$ 42c .

• •

  The incorporation of energy conditions holds paramount significance in discussions about compact stars. These conditions play a pivotal role in constraining the matter distribution within these dense astrophysical objects. By imposing constraints on energy density and pressure, energy conditions ensure the physical viability of solutions, guiding the development of realistic models for self-gravitating structures. Upholding these conditions not only fosters mathematical consistency but also provides crucial insights into the nature of matter supporting these stellar objects. For the case of charged fluid, they have the form

  $\displaystyle\rho+P\geq 0,\quad\rho-P+\frac{s^{2}}{4\pi r^{4}}\geq 0,\quad\rho+3P+\frac{s^{2}}{4\pi r^{4}}\geq 0.$

• •

  Various approaches have been proposed to assess the stability of celestial systems, with one method involving the consideration of the causality condition derived from the sound speed, expressed as $v_{s}^{2}=\frac{dP}{d\rho}$. According to Abreu et al. 42bb , this condition ensures that information within the stellar medium propagates at speeds less than the speed of light, preventing causality violations, i.e., $0<v_{s}^{2}<1$. At the same time, one can check the stability by studying the thermodynamic behavior of the celestial object. This can be discussed through the adiabatic index, indicated by $\Gamma_{ai}$, whose formula is given as follows

  $$\Gamma_{ai}=\frac{\rho+P}{P}\left(\frac{dP}{d\rho}\right).$$

  To maintain the equilibrium of a compact model, the outward pressure must be as enough as it can counterbalance the force of gravity acting inward. This can only be achieved if the adiabatic index gain its value greater than $\frac{4}{3}$ everywhere 42ba .

In this section, we obtain different solutions and perform a comprehensive analysis on their physical properties corresponding to two distinct models of the considered modified gravitational theory. We further extend our exploration by choosing two different forms of the matter Lagrangian density, one in terms of the energy density and other in the form of an isotropic pressure. A large body of literature guarantees the formation of acceptable solutions for both these choices. Now, we discuss them one by one in the following.