Thin-Shell Gravastar Model in $f(Q,T)$ Gravity

There has been a large scientific interest in understanding the problems in both cosmology and astrophysics during the past few decades. Compact objects are a crucial source for this reason because they provide a platform to test many pertinent ideas in the high-density domain. The Gravitationally Vacuum Condense Star, or simply gravastar, is an excellent notion for an extremely compact object that addresses the singularity problems in classical black hole (CBH) theory. It was first postulated by Mazur and Mottola [1, 2]. They construct a cold, compact object with an internal de Sitter condensate phase and an exterior Schwarzschild geometry of any total mass M that is free of all known limitations on the known CBH. As a result, this hypothesis has gained popularity among researchers and it could be seen as an alternative for the CBH.

The gravastar, in particular, has three separate zones with different equations of states (EoS), according to Mazur and Mottola’s model:

1. 1.

   An internal region that is full of dark energy with an isotropic de Sitter vacuum situation.

2. 2.

   An intermediate thin shell consists of stiff fluid matter.

3. 3.

   The outer area is completely vacuum, and Schwarzschild geometry represents this situation appropriately.

Recent studies on the brightness of type Ia distant supernovae [3, 4, 5] indicate that the universe is expanding more quickly than previously thought, which suggests that the universe’s pressure $p$ and energy density $\rho$ should contradict the strong energy condition, that is, $\rho+3p<0$. ”Dark energy” is the substance that causes this requirement to be fulfilled at some point in the evolution of the universe [6, 7, 8]. There are several substances for the status of dark energy. The most well-known contender is a non-vanishing cosmological constant, which is equivalent to the fluid that satisfies the EoS $p=-\rho$. There are two interfaces (junctions) located at $R_{1}$ and $R_{2}$ apart from the center, where $R_{1}$ and $R_{2}$ stand for the thin shell’s interior and outer radii. The presence of stiff matter on the shell with thickness $R_{2}-R_{1}=\epsilon<<1$ is required to provide the system’s stability, which is achieved by exerting an inward force to counteract the repulsion from within.

Astrophysicists proposed a new solution of a compact, spherically symmetric astrophysical phenomenon, known as a gravastar, to solve the singularity problem in black hole geometry. There are several arguments for and against the theory that gravitational waves (GW), detected by LIGO, are the consequence of merging gravastars or black holes, despite the fact that no experimental observations or discoveries of gravastars have yet been made. A method for identifying gravastar was devised by Sakai et al. [9] by looking at the gravastar shadows. Since black holes don’t exhibit microlensing effects of maximal brightness, Kubo and Sakai [10] hypothesized that gravitational lensing may be used to find gravastars. The finding of GW150914 [11, 12] by interferometric LIGO detectors increased the likelihood that ringdown signals originated from sources without an event horizon. In a recent examination of the picture taken by the First M87 Event Horizon Telescope (EHT), a shadow that resembled a gravastar has been discovered [13].
One could observe that there are numerous publications on the gravastar available in the literature that focuses on various mathematical and physical problems in the framework of general relativity postulated by Albert Einstein [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Bilic et al. [16] replaced the de Sitter interior with a Chaplygin gas equation of state and saw the system as a Born-Infield phantom gravastar to examine the gravastar’s interior, whereas Lobo [17] replaces the inner vacuum with dark energy. Although it is commonly known that Einstein’s general relativity is an exceptional tool for revealing many hidden mysteries of nature, certain observable evidence of the expanding universe and the existence of dark matter has put a theoretical challenge to this theory. Hence, a number of modified theories have been put over time, like $f(R),f(Q),f(T),f(R,T),f(Q,T)$ gravity, etc. The $f(R),f(R,T)$ gravity which is based upon the Riemannian geometry in which Ricci scalar curvature plays an important role. Another way to represent the gravitational interaction between two particles in space-time is by torsion and non-metricity upon which $f(T)$ and $f(Q)$ gravity theory has been built, respectively. In the current project, our objective is to investigate the gravastar using one of the alternative theories of gravity, $f(Q,T)$ gravity, and to examine many physical characteristics and stability of the object. The $f(Q,T)$ gravity is the extension of the symmetric teleparallel gravity in which the gravitational action is determined by any function $f$ of the nonmetricity $Q$ and the trace of the matter energy-momentum tensor $T$, such that $L=f(Q,T)$. There are very few articles in which compact object has been studied under the framework of $f(Q,T)$ gravity [24]. Xu et al. have investigated the cosmological implication of this theory, and they have obtained the cosmological evolution equation for isotropy, homogeneous, flat geometry [25]. In [26], the author has investigated the different FRW models with three specific forms of $f(Q,T)$ gravity models. One could see the references to the recent work of gravastar in the framework of modified gravity [27, 28, 29, 30, 31, 32]. In the article, [33] researchers have studied the gravastar model in $f(Q)$ gravity. Ghosh et al. [34] has studied the gravastar in Rastall gravity. In the work [35], the author has studied traversable wormhole solutions in the presence of the scalar field. Wormhole solutions in $f(R,T)$ gravity have been studied in [36]. Elizalde et al. [37] discussed the cosmological dynamics in $R^{2}$ gravity with logarithmic trace term. Godani and Samanta [38] discussed the gravitational lensing effect in traversable wormholes. In [39], the researchers have investigated wormhole solutions with scalar field and electric charge in modified gravity. In [40], the authors studied the cosmologically stable $f(R)$ model and wormhole solutions. Salvatore et al. [41] studied the non-local gravity wormholes, and they obtained stable and traversable wormhole solutions. Shamir et al. [42] has explored the behavior of anisotropic compact stars in $f(R,\phi)$ gravity. The Bardeen compact stars in Modified $f(R)$ gravity have been researched in the work [43].

Our paper is organized as follows: In sec I we have given a brief introduction to the gravastar model and the recent research work regarding that. After that in sec II we provide the geometrical aspects of $f(Q,T)$ gravity. In sec III we have derived the modified field equation and the modified energy conservation equation in $f(Q,T)$ gravity. Sec IV gives the solution of the field equation for different regions using different EoS. After that in sec V we have studied the junction requirement and EoS and we have obtained the limiting range for the radius of the gravastar. The physical features of the model has been analyzed in sec VI. The most important thing is to check the stability of the model which is given in sec VII. Finally, we provide the conclusion of our analysis in sec VIII.

The $f(Q,T)$ theory of gravity which introduces an arbitrary function of scalar non-metricity $Q$ and trace $T$ of the matter energy-momentum tensor, is an intriguing modification to Einstein’s theory of gravity. The action of $f(Q,T)$ theory coupled with matter Lagrangian $\mathcal{L}_{m}$ is given by [44]

$$S=\int\sqrt{-g}\left[\frac{1}{16\pi}f(Q,T)+\mathcal{L}_{m}\right]d^{4}x,$$

(1)

where $g$ represents the determinant of $g_{\mu\nu}$. The non-metricity and disformation tensor is defined as

	$\displaystyle Q\equiv-g^{\mu\nu}\left(L^{\alpha}_{\,\,\beta\mu}L^{\beta}_{\,\,\nu\alpha}-L^{\alpha}_{\,\,\beta\alpha}L^{\beta}_{\,\,\mu\nu}\right),$		(2)
	$\displaystyle L^{\lambda}_{\,\,\,\,\mu\nu}=-\frac{1}{2}g^{\lambda\gamma}\left(\nabla_{\nu}g_{\mu\gamma}+\nabla_{\mu}g_{\gamma\nu}-\nabla_{\gamma}g_{\mu\nu}\right).$		(3)

The non-metricity tensor is defined as the covariant derivative of the metric tensor, and its explicit form is

$$Q_{\alpha\mu\nu}\equiv\nabla_{\alpha}g_{\mu\nu}.$$

(4)

with the trace of a non-metricity tensor as

$$Q_{\lambda}=Q_{\lambda\,\,\,\,\,\mu}^{\,\,\,\mu},\quad\quad\tilde{Q}_{\lambda}=Q^{\mu}_{\,\,\,\,\lambda\mu}.$$

The Superpotential $P_{\,\,\mu\nu}^{\lambda}$ is defined as

$$P_{\,\,\,\,\mu\nu}^{\lambda}=-\frac{1}{2}L^{\lambda}_{\,\,\,\,\mu\nu}+\frac{1}{4}\left(Q^{\lambda}-\tilde{Q^{\lambda}}\right)g_{\mu\nu}-\frac{1}{4}\delta^{\lambda}_{\,\,(\mu\,}Q_{\nu)},$$

(5)

giving the relation of scalar non-metricity as

$$Q=-Q_{\lambda\mu\nu}P^{\lambda\mu\nu}.$$

(6)

The field equations of $f(Q,T)$ theory by varying the action (1) with respect to the metric tensor inverse $g^{\mu\nu}$ is obtained as

$$-\frac{2}{\sqrt{-g}}\nabla_{\lambda}\left(f_{Q}\sqrt{-g}\,P^{\lambda}_{\,\,\,\,\mu\nu}\right)-\frac{1}{2}f\,g_{\mu\nu}+f_{T}\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)\\ -f_{Q}\left(P_{\mu\lambda\alpha}Q_{\nu}^{\,\,\,\lambda\alpha}-2Q^{\lambda\alpha}_{\,\,\,\,\,\,\,\,\mu}\,P_{\lambda\alpha\nu}\right)=8\pi T_{\mu\nu}.$$

(7)

The terms used in the above are defined as

	$\displaystyle\Theta_{\mu\nu}$	$\displaystyle=$	$\displaystyle g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta g^{\mu\nu}},\quad T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta g^{\mu\nu}},$		(8)
	$\displaystyle f_{T}$	$\displaystyle=$	$\displaystyle\frac{\partial f(Q,T)}{\partial T},\quad f_{Q}=\frac{\partial f(Q,T)}{\partial Q}.$		(9)

Where $T_{\mu\nu}$ is known as the energy-momentum tensor.

To derive the modified field equation let’s take the static, spherically symmetric line element given by,

$$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}).$$

(10)

To describe the fluid distribution we are going to take the energy-momentum tensor in the form :

$$T_{\mu\nu}=(\rho+p_{t})u_{\mu}u_{\nu}-p_{t}\delta_{\mu\nu}+(p_{r}-p_{t})v_{\mu}v_{\nu},$$

(11)

where $\rho$ is the density of the fluid, $p_{r}$ and $p_{t}$ are the pressures of the fluid in the direction of $u_{\mu}$(radial pressure) and orthogonal to $u_{\nu}$ (tangential pressure) respectively. $u_{\mu}$ is the time like four-velocity vector. $v_{\mu}$ is the unit space-like vector in the direction of the radial coordinate. Therefore the stress energy momentum tensor $T_{\mu\nu}$ and the components of $\Theta_{\mu\nu}$ can be expressed as,

$$\begin{gathered}T_{\mu\nu}=diag(e^{\nu}\rho,e^{\lambda}p_{r},r^{2}p_{t},r^{2}p_{t}sin^{2}\theta)\\ \Theta_{11}=-e^{\nu}(P+2\rho),\,\Theta_{22}=e^{\lambda}(P-2p_{r}),\\ \Theta_{33}=r^{2}(P-2p_{t}),\,\,\Theta_{44}=r^{2}sin^{2}\theta(P-2p_{t}).\end{gathered}$$

(12)

Where we have taken the Lagrangian matter density $\mathcal{L}_{\mathrm{m}}=-P=-\frac{p_{r}+2p_{t}}{3}$. By utilizing the aforementioned constraints the derived modified field equation for spherically symmetric metric in $f(Q,T)$ gravity is,

$$8\pi\rho=\frac{1}{2r^{2}e^{\lambda}}[2rf_{QQ}Q^{\prime}(e^{\lambda}-1)+f_{Q}[(e^{\lambda}-1)(2+r\nu^{\prime})\\ +(1+e^{\lambda})r\lambda^{\prime}]+fr^{2}e^{\lambda}]-f_{T}[P+\rho],$$

(13)

$$8\pi p_{r}=-\frac{1}{2r^{2}e^{\lambda}}[2rf_{QQ}Q^{\prime}(e^{\lambda}-1)+f_{Q}[(e^{\lambda}-1)\\ (2+r\nu^{\prime}+r\lambda^{\prime})-2r\nu^{\prime}]+fr^{2}e^{\lambda}]+f_{T}[P-p_{r}],$$

(14)

$$8\pi p_{t}=-\frac{1}{4re^{\lambda}}[-2rf_{QQ}Q^{\prime}\nu^{\prime}+f_{Q}[2\nu^{\prime}(e^{\lambda}-2)-r\nu^{\prime 2}+\lambda^{\prime}(2e^{\lambda}+r\nu^{\prime})-2r\nu^{\prime\prime}]+2fre^{\lambda}]+f_{T}[P-p_{t}].$$

(15)

Now here we are going to take a particular functional form of $f(Q,T)$ gravity as $f(Q,T)=\alpha\,Q+\beta\,T$. One can see there are many references [44, 24, 25] in which this cosmological models has been studied widely. Then we can rewrite the field equation in a as like :

$$e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}=\rho^{eff},$$

(16)

$$e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=p_{r}^{eff},$$

(17)

$$e^{-\lambda}\left(\frac{\nu"}{2}-\frac{\lambda^{\prime}\nu^{\prime}}{4}+\frac{\nu^{\prime 2}}{4}+\frac{\nu^{\prime}-\lambda^{\prime}}{2r}\right)=p_{t}^{eff},$$

(18)

Where,

$$\rho^{eff}=\frac{8\pi\rho}{\alpha}+\frac{\beta}{3\alpha}(3\rho+p_{r}+2p_{t})-\frac{\beta}{2\alpha}(\rho-p_{r}-2p_{t}),$$

(19)

$$p_{r}^{eff}=\frac{8\pi p_{r}}{\alpha}-\frac{2\beta}{3\alpha}(p_{t}-p_{r})+\frac{\beta}{2\alpha}(\rho-p_{r}-2p_{t}),$$

(20)

$$p_{t}^{eff}=\frac{8\pi p_{t}}{\alpha}-\frac{\beta}{3\alpha}(p_{r}-p_{t})+\frac{\beta}{2\alpha}(\rho-p_{r}-2p_{t}).$$

(21)

One can verify that for $\alpha=1,\beta=0$ i.e. for $f=Q$ the above field equation reduces to Einstein’s GR. However in this article, we limit ourselves to the isotropic scenario in order to establish the simplest possibility where $p_{r}=p_{t}$. Now the energy conservation equation is given by,

$$\frac{dp^{eff}}{dr}+\frac{\nu^{\prime}}{2}(p^{eff}+\rho^{eff})=0.$$

(22)

By using the equation (19) and (20) we get the modified energy conservation equation in $f(Q,T)$ gravity as:

$$\frac{dp}{dr}+\frac{\nu^{\prime}}{2}\left[(1+\frac{\beta}{8\pi})(p+\rho)\right]+\frac{\beta}{16\pi}(\rho^{\prime}-3p^{\prime})=0.$$

(23)

The above equation is different from that obtained in GR and can be retrieved in the limit $\beta\to 0$.

We are specifically interested in the geometrical interpretation and their related analytical solution in the three different zones for the gravastar under study. It is simple to imagine the idea that the inside of the star is considered to be encircled by a thin shell made of ultrarelativistic stiff fluid, but the outside area is completely vacuum. Schwarzschild’s measure is therefore assumed to be appropriate for this outer area. The shell’s structure is believed to be extremely thin, with a limited width ranging $R_{1}=R\leq r\leq R+\epsilon=R_{2}$, where $r$ is the radial coordinate and $R_{1},R_{2}$ denotes the inner and outer radius of the shell.

It is established that the gravastar is divided into three regions viz. the interior (I), the intermediate thin shell (II), and the exterior (III). This shell keeps the internal area and outer region connected. So, This region is crucially significant in the construction of the gravastar. According to the fundamental junction requirement, regions I and III must match smoothly at the junction. The derivatives of these metric coefficients may not be continuous at the junction surface, despite the fact that the metric coefficients are continuous there. In order to calculate the surface stresses at the junction, we will now employ the Darmois-Israel [54, 55, 56] condition. The Lanczos equation [57, 58, 59, 60] provides the intrinsic surface stress-energy tensor $S_{ij}$ in the following manner:

$\displaystyle S_{ij}=-\frac{1}{8\pi}(k_{ij}-\delta_{ij}k_{\gamma\gamma}).$

(45)

In the above expression, $k_{ij}=K^{+}_{ij}-K^{-}_{ij}$ denotes the discontinuity in some second fundamental expression. Where the second fundamental expression is given by,

$$K^{\pm}_{ij}=-n^{\pm}_{\sigma}\left(\frac{\partial x_{\sigma}}{\partial\phi^{i}\partial\phi^{j}}+\Gamma^{l}_{km}\frac{\partial x^{l}}{\partial\phi^{i}}\frac{\partial x^{m}}{\partial\phi^{j}}\right),$$

(46)

where $\phi^{i}$ denotes the intrinsic co-ordinate in the shell area, along with $n^{\pm}$ represents the two-sided unit normal to the surface, which can be written as,

$$n^{\pm}=\pm\left|g^{lm}\frac{\partial f}{\partial x^{l}}\frac{\partial f}{\partial x^{m}}\right|^{-1/2}\frac{\partial f}{\partial x^{\sigma}},$$

(47)

with $n^{\gamma}n_{\gamma}=1$. Utilizing the Lanczos method [57] the surface energy tensor can be written as $S_{ij}=diag(-\sum,P)$, where the surface energy density and surface pressure are denoted by $\sum$ and $P$ respectively and are defined by,

$$\sum=-\frac{1}{4\pi\mathcal{R}}\left[\sqrt{e^{-\lambda}}\right]^{+}_{-},$$

(48)

$$P=-\frac{\sum}{2}+\frac{1}{16\pi}\left[\frac{(e^{-\lambda})^{\prime}}{\sqrt{e^{-\lambda}}}\right]^{+}_{-},$$

(49)

$$\text{Also the EoS}\,(\omega)=\frac{P}{\sum}.$$

(50)

Here $-$ and $+$ represent the interior space-time and Schwarzschild space-time respectively. Calculating the Eq. (48)-(50) we get the expression of the above quantities as ,

$$\sum=\left(-\frac{1}{4\pi R}\right)\left(\sqrt{1-\frac{2M}{R}}-\sqrt{\frac{2(\beta-4\pi)\text{{\hbox{\rho}}c}R^{2}}{3\alpha}+1}\right)$$

(51)

$$P=\frac{1}{16\pi}\left(\frac{2M}{R^{2}\sqrt{1-\frac{2M}{R}}}-\frac{4(\beta-4\pi)\rho_{c}R}{3\alpha\sqrt{\frac{2(\beta-4\pi)\rho_{c}R^{2}}{3\alpha}+1}}\right)-\frac{1}{2}\left(-\frac{1}{4\pi R}\right)\left(\sqrt{1-\frac{2M}{R}}-\sqrt{\frac{2(\beta-4\pi)\rho_{c}R^{2}}{3\alpha}+1}\right),$$

(52)

$$\omega=\frac{\frac{1}{16\pi}\left(\frac{2M}{R^{2}\sqrt{1-\frac{2M}{R}}}+\frac{4(4\pi-\beta)\rho_{c}R}{\alpha\sqrt{\frac{6(\beta-4\pi)\rho_{c}R^{2}}{\alpha}+9}}\right)}{\left(-\frac{1}{4\pi R}\right)\left(\sqrt{1-\frac{2M}{R}}-\sqrt{\frac{2(\beta-4\pi)\rho_{c}R^{2}}{3\alpha}+1}\right)}-\frac{1}{2}.$$

(53)

There is some set of conditions known as energy conditions that must be applied in order for a geometric structure to be physically viable. The well-recognized energy criterion are :

1. 1.

   NEC: $\sum+P>0$,

2. 2.

   WEC: $\sum>0$, $\sum+P>0$ ,

3. 3.

   SEC: $\sum+P>0$,$\sum+3P>0$,

4. 4.

   DEC:$\sum>0,\sum\pm P>0$ .

The presented model is physically feasible if these energy criteria are established. Here, we’re investigating to see if the null energy requirement, which guarantees the presence of ordinary or exotic matter in the thin shell, is satisfied or not. In this context, it is noteworthy to say that violation of null energy conditions (NEC) leads to violation of other energy conditions. It is illustrated in Fig.(3) that the NEC is satisfied over a range of model parameter values throughout the entire region.

Besides that, we have plotted the variation of surface energy density with respect to the thickness parameter($\epsilon$) which shows that the surface energy density is monotonically decreasing towards the boundary of the shell. The mass of the thin shell now is easily determined using the equation for the surface energy density given by,

$$m_{s}=4\pi R^{2}\sum=-R\left(\sqrt{1-\frac{2M}{R}}-\sqrt{\frac{2(\beta-4\pi)\rho_{c}R^{2}}{3\alpha}+1}\right).$$

(54)

Now for determining the real value of shell mass, we have the inequality, $m_{s}>0$ from which we get the upper bound of the radius as $R<\left(\frac{3M\alpha}{\rho_{c}(4\pi-\beta)}\right)^{\frac{1}{3}}.$ Thus we get the limiting value on the radius as

$$2M<R<\left(\frac{3M\alpha}{\rho_{c}(4\pi-\beta)}\right)^{\frac{1}{3}}.$$

(55)

In this section, we are interested to investigate the stability of the thin shell gravastar model by analyzing some physical parameters.