Buchdahl Spacetime with Compact Body Solution of Charged Fluid and Scalar Field Theory

The structure and life span of compact bodies or strange stars are the biggest problems in astrophysics. The astrophysical bodies like nebula, galaxies, stars mainly follow the local solutions of Einstein equation in General Theory of Relativity. Hence the structure and their interior fluid stability became the biggest mystery after the application of Einstein equation in fluid mechanics and Astrophysics. The astronomical bodies mainly face two types of forces viz. gravitational force due to its field and Hydrodynamical force. The equilibrium condition between those forces may provide the mass versus radius limits on the astrophysical systems like stars. The hydrodynamical pressures from the ordinary fluids and its limits do not match the observations and hence the idea of interpreting different kind of fluid models has been brought in astrophysics. Different kind of fluid models, Dark energy models, Field models have been introduced here to discuss the interior fluid mechanism of stars and astrophysical bodies. perfect fluid models, polytropic fluid models, Chaplygin gas models, Inhomogeneous and anisotropic models and Nonlinear fluid models are some of the examples. The variation of fluid models can provide the variation of the equilibrium conditions and hence we can find different limits of the Mass-Radius ratios of the astrophysical bodies.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]

Recent observation provides the existence of a compact star with mass $2.59^{+0.08}_{-0.09}M_{\odot}$ (Abbott et al. (2020)). From the references of (Bailyn et al. 1998; Ozel et al. 2010; Belczynski et al. 2012) it can be observed that this mass remains within the mass gap of stellar bodies. Thus the ambiguity comes in the discussion of interior fluid of those compact bodies. From literatures of theoretical and Observational studies it can be found that the blackholes generated from stellar evolution must have the masses more than $5M_{\odot}$ whereas the stable neutron star should have a mass of mostly within $3M_{\odot}$. The neutron star with mass more than that can loose its individuality hypothetically. This kind of problems have been studied with the introduction of quark stars (The intermediate phase of starry evolution between neutron star and black hole). Different kinds of quark fluid function have been assumed to reconstruct the interior fluid model of a star or compact body. The estimated radius of some objects like (LMC X-4, 4U 1820-30, Her X-1, etc.) suggest that their characteristics and phenomenology are similar to those of quark stars. On the other hand the gravitational wave signal GW170817 don’t allow the large mass of that compact object instead suggests around $2.5M_{\odot}$. The heaviest neutron star possible is around $2.01\pm 0.04M_{\odot}$. Hence that large values of masses of compact objects within $M\in(2.5,5)M_{\odot}$ is theoretically impossible.[55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70]

The gamma ray and X-ray blasts are the results of the electron-proton interactions with the molecular clouds. The electrons and protons are accelerated from the supernova explosion shock waves or pulsar wind nebulae. This supernova generally happens from the non-equilibrium conditions found inside of a compact body. The hydrodynamical balance destruction is the primary cause behind those kinds of blasts. Hence, Compact body interior fluid hydrodynamics study is important in astrophysics. Recent observational an computational analysis (Kar et.al (2021), Abeysekara, A. U et.al (2020), Cao et al. (2021), Mondal et al (2021)) provides several examples of such high energy explosions. Hence the idea of modifications of interior fluid dynamics and the equilibrium conditions caused by those hydrodynamical and gravity pressures and their interactions became a necessity in astrophysics.[40, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80]

The interior structure reconstruction of a compact body like stars, pulsars, nebulae, giant molecular clouds etc. can be followed by several types of fluid models. Different fluid models provide different equation of states that again reconstruct the nature of fluid pressures and hydrodynamic properties. Several dark energy models like fluid dark energies, holographic dark energies can be used in this purpose. From cosmological point of view the dark energy models have extensive use in discussion of the origin of negative pressure that causes the accelerated expansion of universe. If we use this kind of fluids in the discussion of astrophysical bodies interior fluid mechanism, we can modify the equilibrium conditions of them. In gravitational point of view these compact bodies generally follow the local solutions of Einstein equation which is a result of generalized action. The action can be coupled with different kinds of fluids to find some new type interior solutions. Recent observations and theoretical analysis proved that the local fluids may have anisotropy and Inhomogeneity under certain conditions. The inhomogeneities mainly brought with the fluid viscosity, fluid-fluid interactions, Fluid-fields interactions. There are several literatures available on anisotropic fluid solutions (Malaver et.al (2013-2021)[42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]), Inhomogeneous fluid solutions (Kar et.al (2021) and Sadhukhan et.al (2020) [31, 32, 33, 34, 35, 36, 37, 38, 39]). The nonlinear fluid models are the one that can represent the hydrodynamical properties with non linear equation of states which also a different representations of dark energy (Kar et.al (2021) and Sadhukhan et.al (2020) [35, 37]). The scalar field theories can also represent the hydrodynamic nature of the interior fluid systems in scalar field potential variation approach. In general the scalar field theory generalizes the field interaction models inside of a compact body that can again modify the equilibrium conditions of the same. The scalar field models are mainly used to discuss the expanding nature of universe which can also discuss the mass accretion of any compact body. This mass accretion helps in the evolution of these compact bodies and unifies the cosmological expanding models with astrophysical compact body interior solutions. Hence with scalar field models we can discuss the evolution of compact body interior properties. The thermodynamics energy conditions in such problems are mainly assumed to be the stability conditions of the models. Although the violation of any of these conditions may bring some shift in equilibrium of compact bodies (Kar et.al (2021) [34, 35] and Malaver et.al (2021)[39]). The interior fluid interactions and few other conditions may bring local anisotropy in the interior fluid systems. One of the most successful solution of such anisotropic system is Buchdahl space time solutions ( Buchdahl et. al (1959)). Several applications have been done on it till date (Malaver et.al (2013 to 2021)[39, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and Kumar et.al (2021)).

In this paper we have mainly discussed the nature of interior fluid of a compact object with new kind of solutions of Buchdahl type functions. We have introduced the anisotropic solutions and perfect fluid model as interior fluid. The physical acceptability conditions, junction conditions, stability conditions and thermodynamical energy conditions have been discussion with the new model of compact star. The corresponding scalar field and its potential have been found that can help to interpret the inside interaction pictures. The equilibrium conditions with different force components have been discussed with this model. Finally, we have introduced the generalized entropy of this system and its time evolution with the pre-assumed scale factor solution of isotropic-homogeneous universe model (Friedmann model). The mass accretion of that compact object and its relation with the interior properties have been discussed here.

The manuscript is prepared as follows. In section 2 we have given the mathematical basis of compact stars. In section 3 we have discussed the acceptability and energy conditions of our newly derived model. In section 4 we have discussed the interior thermodynamics of our compact body. In section 5 we have given a scalar field theory for our model. Section 6 contains the interpretations of mass change and time dependency in scalar field theory. We conclude the paper with results analysis and discussion in section 7 and 8.

The work should start from the electromagnetism coupled gravity action which can be written as follows.

$$S=\int{d^{4}x\sqrt{-g}(f(R,T)+L_{m}+L_{e})}$$

(1)

We can apply the least action principle on the above action to get equation of motion. Here, $L_{e}=$ Lagrangian for electromagnetic fields and $L_{m}=$ the Lagrangian for matter fields. The following equation is the outcome of the variation principle from action in equation 1.

$$\frac{\partial f(R,T)}{\partial R}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f(R,T)+(g_{\mu\nu}\Box-\triangledown_{\mu}\triangledown_{\nu})\frac{\partial f(R,T)}{\partial R}=8\pi(T_{\mu\nu}+E_{\mu\nu})-\frac{\partial f(R,T)}{\partial T}(T_{\mu\nu}+\Theta_{\mu\nu})$$

(2)

where

$$\Theta_{\mu\nu}=g^{\alpha\beta}\frac{\partial T_{\alpha\beta}}{\partial g^{\mu\nu}}$$

(3)

And ,

$$\Box=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})$$

(4)

The electromagnetic Lagrangian $L_{e}$ can produce the EM component of the action and the energy tensor on EM field can be written as following equation.

$$E_{\mu\nu}=\frac{1}{4\pi}(F^{\alpha}_{\mu}F_{\nu\alpha}-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}g_{\mu\nu})$$

(5)

Now, the energy momentum tensor can be written also as follows.

$$T_{\mu\nu}=g_{\mu\nu}L_{m}-2\frac{\partial L_{m}}{\partial g_{\mu\nu}}$$

(6)

Here we discuss the validity of newly derived model. The acceptability conditions provide the physical viability and stability of our model to discuss the compact bodies. The acceptability conditions are mainly contained with the following conditions and points.

• •

  Matching Conditions

• •

  Compactness

• •

  Regularity and Maximality criterion

• •

  Thermodynamics Energy Conditions

• •

  $\rho\geq 0$ for $0\geq R\geq a$ where $R$ is the total radius

• •

  $p_{r}\geq 0$ and $p_{t}\geq 0$ for $0\geq R\geq a$

• •

  $p_{r}=0$ at $r=0$

• •

  $\frac{dp_{r}}{d\rho}=v_{r}^{2}\geq 0<c$ and $\frac{dp_{t}}{d\rho}=v_{t}^{2}\geq 0<c$

• •

  Adiabatic Index $\gamma=\frac{(c^{2}\rho+p_{r})}{p_{r}}\frac{dp_{r}}{c^{2}d\rho}=v_{r}^{2}\geq 1$ or $4/3$ and $\gamma=\frac{(c^{2}\rho+p_{t})}{p_{t}}\frac{dp_{t}}{c^{2}d\rho}=v_{t}^{2}\geq 1$ or $4/3$

• •

  Stability under forces

We proceed with our previously derived energy density and pressure of the compact body interiors.

We have discussed the acceptability conditions and stability of our solutions in terms of geometric parameters. In general a physical system is ultimately stable when its thermodynamics make it the same. Hence we need to discuss the interior thermodynamics of the compact body of our work. We need to visualize the radial variations of the thermodynamics parameters to get the conditions of thermodynamics stability. Hence we mainly discuss the internal energy, temperature and entropy of the interior system. Moreover we search the conditions behind the equilibrium of this compact star. The thermodynamics should start from the relations between energy density and pressure with thermodynamical parameters. Hence we may get as follows.

$$\rho=\frac{U}{V}$$

(41)

And,

$$p=-(\frac{\partial U}{\partial V})_{s}$$

(42)

Here, $U=$ Internal energy and $V=$ volume of the system with the relation of $V=\frac{4}{3}\pi(\frac{x}{c})^{3/2}$. As we have two pressures i.e. radial and transverse, hence we can have two internal energies. If we consider the equation of state parameter as constant i.e. $\omega=$ constant we can find the internal energies as follows using the relations of pressure, density and anisotropy. Here we are considering $\frac{\Delta}{\rho}=f(x)$.

$$U_{t}=U_{t0}\exp{(-\frac{3}{2}\int_{0}^{x}{\frac{f(x)}{x}dx}+\omega\int_{0}^{x}{\frac{1}{x}dx})}$$

(43)

And the radial internal energy will be as follows.

$$U_{r}=U_{r0}\exp{(-\frac{3}{2}\int_{0}^{x}{\frac{f(x)}{x}dx})}$$

(44)

Now we can observe that for constant EOS parameter we can not get non-singular transverse internal energy as because $\ln{x}\rightarrow\infty$ with $x\rightarrow 0$. Thus, we we must have to choose the $\omega$ as a function of $x$. Hence let us consider the following relation.

$$\omega=g(x)$$

(45)

Hence, the transverse component can be written again as follows.

$$U_{t}=U_{t0}\exp{(-\frac{3}{2}\int_{0}^{x}{\frac{(f(x)+g(x))}{x}dx})}$$

(46)

Now the spatial dependent temperature can be defined with the relation as follows.

$$\frac{dT}{T}=\frac{dm}{m}\frac{\partial p}{\partial\rho}$$

(47)

Now here we got $m=$ the number of particles in the volume of compact bodies. Hence we can get $m\frac{4}{}\pi x^{3}=$ Number density at radius $x=$ constant where $x=$ radius of a shell inside of the compact body. Hence we have,

$$\frac{dT}{T}=-\frac{dx}{x}\frac{\partial p}{\partial\rho}$$

(48)

Hence the radial and transverse temperatures can be written as follows.

$$T_{r}=T_{0r}\exp{(-\frac{9}{2}\int_{0}^{x}{\frac{1}{x}\frac{\partial p_{r}}{\partial\rho}dx})}$$

(49)

And,

$$T_{t}=T_{0t}\exp{(-\frac{9}{2}\int_{0}^{x}{\frac{1}{x}\frac{\partial p_{t}}{\partial\rho}dx})}$$

(50)

Now finally we can find the spatial dependent entropy of the interior fluid system of that compact body. From the first law of generalized thermodynamics we know that $ds=\frac{dU}{T}+\frac{pdV}{T}$. Hence from this we can write the transverse and radial entropies of that system are as follows.

$$s_{r}=\int_{0}^{x}{\frac{dU_{r}}{T_{r}}}+2\pi c^{-3/2}\int_{0}^{x}{\frac{p_{r}}{T_{r}}x^{1/2}dx}$$

(51)

And,

$$s_{t}=\int_{0}^{x}{\frac{dU_{t}}{T_{t}}}+2\pi c^{-3/2}\int_{0}^{x}{\frac{p_{t}}{T_{t}}x^{1/2}dx}$$

(52)

Hence with the choice of equation of state parameter as a function of $r$ or $x$ we can find the functional forms of all the parameters like internal energies, temperatures and entropies. Let us choose the most simplest version of EOS parameter as follows.

$$\omega(x)=g(x)=\lambda x$$

(53)

Now we establish the spatial dependent scalar field theory from this newly derived compact body. The whole interior fluid dynamics can be discussed with the space dependent scalar fields. From the usual action of scalar field theory we know that the energy density and pressure can be written as follows.

$$\rho_{\phi}=\frac{1}{2}(\dot{\phi})^{2}+V(\phi)$$

(54)

And,

$$p_{\phi}=\frac{1}{2}(\dot{\phi})^{2}-V(\phi)$$

(55)

Now substituting those energy density and pressure with radial and transverse energy density and pressure we can easily find the radial and transverse components of scalar field kinetic energy and potential which are as follows.

$$\frac{1}{2}(\nabla\phi)^{2}_{r}=\frac{1}{2}(\frac{\partial\phi}{\partial r})^{2}=\rho+p_{r}$$

(56)

And,

$$\frac{1}{2}(\nabla\phi)^{2}_{t}=\frac{1}{2}(\frac{1}{r^{2}}(\frac{\partial\phi}{\partial\theta})^{2}+\frac{1}{r^{2}\sin^{2}{\theta}}(\frac{\partial\phi}{\partial\psi})^{2})=\rho+p_{t}$$

(57)

Here the coordinate $\psi$ is the transverse coordinate of spherical polar system. Now the potential can be written as follows.

$$V(\phi)_{r}=\rho-p_{r}$$

(58)

And,

$$V(\phi)_{t}=\rho-p_{t}$$

(59)

The temporal part of the scalar field Lagrangian has been avoided as because the system in only space dependent. The graphical representations can be provided as follows.

Now we consider the mass accretion and its effect on the compact body stability. We consider expanding universe model to demonstrate the exterior fluid evolution that again insist in compact body mass accretion and its instability in structure. We can also provide the possible mass blast due to such instability. We know that the mass accretion is completely dependent upon the external cosmic fluids and their properties. Considering the universe contains the fluid with energy density and pressure as $\rho_{tot}$ and $p_{tot}$ we can introduce the mass accretion formalism as follows. Here the subscript ”tot” is used to define the multi-fluid contribution on mass accretion.

$$\dot{M}_{1}(t)=4\pi A(\rho_{tot}+p_{tot})$$

(60)

Now we assume that the mass accretion will increase the total mass of the compact body and hence we can say that the mass accretion will change the radius of the compact system. This can be written as follows.

$$M(R)=M_{1}(t)$$

(61)

Hence we can write the time varying radius as follows.

$$R(t)=M^{-1}(M_{1}(t))$$

(62)

The time variation can be found from the solution of polynomial equation of $R$ found from equation 5. The radius is time varying, hence all other parameters and variables discussed in this work became time varying. Therefore we can find time varying scalar field model from time varying energy densities and pressures. The scalar field kinetic energy and potential energy can be written as follows.

$$\frac{1}{2}(\dot{\phi})^{2}=\frac{1}{2}(\frac{1}{2}(\dot{\phi})^{2}_{r}+\frac{1}{2}(\dot{\phi})^{2}_{t})=\rho(t)+\frac{1}{2}(p_{r}(t)+p_{t}(t))$$

(63)

And,

$$V(t)=V(\phi)=\frac{1}{2}(V_{r}(t)+V_{t}(t))=\rho(t)-\frac{1}{2}(p_{r}(t)+p_{t}(t))$$

(64)

Hence we can observe that the scalar field theory for compact body is evolving due to continuous mass accretion. Even after the evolution the equilibrium conditions remain unaltered and this is because the change of mass mainly changes the radius which apparently changes everything with time. That means we can say that the mass accretion changes everything indirectly except the radius and for each instantaneous time the equilibrium conditions are keeping the model stable.

In section 2.3 we have discussed the path dependent variations of the astrophysical fluid variables $\rho$, $p_{r}$, $p_{t}$, $\omega$, $\Delta$ and $M$. We have shown their variations pictorially from figure 1 to 7. For our assumed parametric values we found positive values of energy density which is dissimilar to the exotic matter system in strange stars or dark energy stars. From the radial and transverse pressure graphical representations we can find positive pressures. Hence, our system can not be stabilized with the pressures from interior fluids and that is why we shall need high values of hydrodynamical forces for the same. The anisotropy remains positive throughout the whole region and thus we may conclude that the system stability may come with anisotropy also. The mass function and shearing scalar both provide positive values and thus validates of our model. In all those graphs (fig. 1 to 22) we have shown the values for different EOS parameter $\omega$. Basically we have used changing EOS parameter which changes only during phase transitions.

In the section 3.3 from figures 8 to 11 we can observe that our model violates the maximally and regularity conditions. The variables $\rho$, $p_{r}$, $p_{t}$ and $\Delta$ all of them increases with radial distances and hence they have higher values on surface that center. This may bring structure instabilities of the compact body discussed through our model. Although from figure 10, we can observe that for positive values of $\omega$, $\frac{dp_{r}}{dr}$ is less than zero around center and surface but providing $>0$ in the middle shell volume.

We have discussed the energy conditions in section 3.4. We have found that both radial and transverse pressures are satisfying the strong, weak and null energy conditions with energy densities (fig. 12 to 17). But The dominant energy condition has been violated with tangential pressure. The satisfaction of energy conditions again proved that the fluid system discussed here can not provide repulsive behaviour in compact body.

In section 3.5 we have incorporated four forces i.e. Hydrostatic force, Gravitational force, Anisotropic repulsion force and Electric force to discuss the stability of the compact body system. We have to make it zero on the surface of the system. From figure 18 we can observe that all the forces are neutralized with the increase in radius and thus becoming stable system. The hydrostatic force and anisotropic force neutralizes all other attractive forces discussed here.

In section 4 we have discussed the interior thermodynamics through internal energy, temperature and entropy for both radial and tangential pressures. We have discussed the internal energy variation for both constant and variable EOS parameter. We have proved that the constant EOS parameter can not stabilize the compact body interior system and this is because the internal energy diverges for constant $\omega$.

In section 5 we have discussed the compact body interior fluid corresponding to scalar field theory. The scalar field and potential are different for different pressures. We have shown the variation of both tangential and radial kinetic and potential energies for changing EOS parameter. The Potential energies are seemed to be decreasing in nature with radius.

The present work has been established with the solution of Einstein-Maxwell field equation using Buchdahl solution technique. We have considered nonlinear electromagnetic field whose nature is different from Coulomb’s inverse square law. The Buchdahl function assumption brought a new kind of transformations that also helped us to solve six variables solutions from six equations. The final results provide us a new kind of compact body where we have also tried to incorporate its inner fluid thermodynamics and stability analysis. Except the regularity conditions we have successfully satisfied all other conditions with our newly derived model. At the very end, we have introduced external cosmic fluid mass accretion of the compact body and proved that the mass accretion can increase the body radius of the system. The interior thermodynamics proved that a stable compact body can never contain constant equation of state parameter throughout its whole interior.

We have limited our work to the assumption of new type of Buchdahl Spacetime function that can produce a new type of compact star solution of Einstein-Maxwell equation. We have also studied the corresponding scalar field solution, There is an extensive application in future of this work to analyze several pulsars, Nebula Gamma Ray Blast and compact object analysis. All those areas are beyond the scope of this paper. The mass function and Electromagnetic Fields from such objects of our work can also be used in future to discuss the Gamma Ray Blast Shock wave, Supernovae Shock waves and radiations from them. In future we can also use the mass function to interpret the cosmic evolution and discuss the mass accretion of that compact system with universe evolution.