# Convection and cracking stability of spheres in general relativity

## Abstract

In the present paper we consider convection and cracking instabilities as well as their interplay. We develop a simple criterion to identify equations of state unstable to convection, and explore the influence of buoyancy on cracking (or overturning) for isotropic and anisotropic relativistic spheres. We show that a density profile \(\rho (r)\), monotonous, decreasing and concave , i.e. \(\rho ' < 0\) and \(\rho '' < 0\), will be stable against convection, if the radial sound velocity monotonically decreases outward. We also studied the cracking instability scenarios and found that isotropic models can be unstable, when the reaction of the pressure gradient is neglected, i.e. \(\delta \mathcal {R}_p = 0\); but if it is considered, the instabilities may vanish and this result is valid, for both isotropic and anisotropic matter distributions.

## 1 Introduction

The stability of general relativistic self-gravitating matter distributions has been extensively studied and reported in the literature through several techniques for many years. It is complex multivariate problem which depends on the micro-physics – bulk/shear viscosity, crust on the surface, magnetic field and so on – of the material constituents and their description through a macroscopic equation of state that characterizes the configuration (see standard texts in Relativistic Astrophysics and Neutron Stars [1, 2, 3, 4, 5] and references therein).

The studies of stability distinguish two different approaches associated to the global and local scales, where instabilities affect the structure. On the global scale it is examined through the dynamical perturbation scheme which, in the case of spherical symmetry, can be translated into how radial pulsations induce possible disruptions of a stellar body. On the other hand, local stability investigates the effects of convection and/or cracking of the material within the matter distribution. Pulsation is a global phenomenon characterized by the collective motion of the entire body while, convection/cracking occurs locally and is governed by the nearby values of the thermodynamical variables and their gradients (see an interesting review in [6]).

*P*the radial pressure, and \(v^{2}\) the radial sound speed, respectively [13, 14, 15, 16]. Within the global stability criteria we can also identify the Harrison–Zeldovich–Novikov condition, which implies that \(\mathrm {d}M(\rho _c) /\mathrm {d} \rho _c \ge 0\), where

*M*is the total mass of the configuration and \(\rho _c\) the central density, of the distribution [16].

The stability of a spherical star to convection implies the buoyancy principle which leads to that pressure and energy density must, decrease outwards in any hydrostatic matter configuration [17, 18, 19]. Finally, the cracking instability approach determines the tidal acceleration profiles generated by perturbations of the energy density and the anisotropy of pressures identifying the changes sign of the total force distribution within the system [20, 21, 22, 23]. This approach has been applied to an anisotropic fluid with barotropic equations of state [24] and, more recently extended to take into account the perturbation of the pressure gradient in both isotropic and anisotropic matter configurations [25, 26].

In the present paper we consider convection and cracking instabilities as well as their interplay. We develop a simple criterion to identify equations of state unstable to convection, and also explore the influence of buoyancy on cracking (or overturning) of isotropic and anisotropic relativistic spheres.

This paper is organized as follows: Sect. 2 describes the general equations of the theoretical framework of General Relativity. In Section 3 we formulate the concepts of adiabatic stability while cracking for self-gravitating anisotropic matter configurations is discussed in Sect. 4. Next we present in Sect. 5 the acceptability conditions which make any model physically reasonable. The models used and a discussion of our results for isotropic and anisotropic cases are presented in the Sects. 6 and 7. Finally, in Sect. 8 we wrap-up our final remarks.

## 2 The field equations

At the surface of the sphere \(r=r_b\), the interior solution should match continuously the exterior Schwarzschild solution, which implies that: \(e^{2\nu (r_b)}=e^{-2\lambda (r_b)}=1-2\mu _b\), and the compactness at the surface defined as \(\mu _b=2M/r_b\).

*P*(

*r*) and tangential pressure \(P_\perp (r)\) of the fluid being determined by the Einstein field equations as

*r*.

## 3 Adiabatic convection stability condition

The stability of a spherical star against convection can be easily understood. When a fluid element is displaced downward, if its density, \(\rho _{e}\), increases more rapidly than the surrounding density, \(\rho _{s} \), the element will sink downward and the star will be unstable. On the other hand, if the density of fluid element is less than its surroundings, it will float back and the star will be stable to convection.

- 1.
If \( \rho _{e}> \rho _{s} \), gravity will tend to push the fluid element downward further and the system will be unstable.

- 2.
If \( \rho _{e} = \rho _{s} \), the system is considered neutral or metastable.

- 3.
If \( \rho _{e} <\rho _{s} \), a restoring force will act on the fluid element and then the system will be stable because it tends to its original state.

*r*represents the current position of the fluid element, \(r_p \) its original position and \(-\delta r\) the downward shift.

It is clear that parabolic density profiles, \(\rho = \alpha r^2 + \beta \), with \(\alpha \) and \(\beta \) constants, will be stable against this type of convection, because buoyancy condition must also be fulfilled at the center \(r = 0\) of the sphere: \(\rho '< 0 \,\, \Rightarrow \, \rho '_c=0,\) and \( \rho ''_c<0 \). These profiles have been implemented for the MIT Bag model through a linear equation of state, \(P = \beta (\rho -\rho _{s})\), when densities become high enough for a phase transition to quark matter to occur [3, 16].

## 4 Convection and cracking sources of instability

In this section we shall consider convective instabilities in the framework of cracking induced by perturbation in the density profile. Just for completeness we outline here the main concepts and equations concerning cracking for isotropic and anisotropic matter configurations, for further details, we refer interested readers to [25, 26] and references therein.

As in the previous works [25, 26], we assume that density fluctuations induce variations into all other physical variables, i.e. \(m(r), P(r), P_\perp (r)\) and their derivatives, generating a non-vanishing total radial force distribution (\(\delta \mathcal {R} \ne 0\)) within the configuration. It is important to stress we are considering local perturbations of density, that can be properly described by any function of compact support, \(\delta \rho = \delta \rho (r)\), defined in a closed interval \(\Delta r \ll r_b\), where \(r_b\) is the total radius of the configuration.

Note that the present perturbation scenario contrasts the original one presented by Herrera and collaborators [20, 21, 22], where fluctuations in density and anisotropy were considered independent and simultaneous; and it is also different from a previous work [24] because there pressure gradient were not affected by the density perturbation.

*r*and the possible change of sign for \( \delta \mathcal {R}\) should emerge from \(\delta \mathcal {R}_a\) and the criterion against cracking is written as:

## 5 Physical acceptability conditions

- C1:
Metric potentials, positive, finite and free from singularities;

- C2:
Matching conditions at the surface of the star;

- C3:
Decrease of interior redshift

*Z*with the increase of*r*; - C4:
Positive density and pressures;

- C5:
Density and pressures having a maximum at the center and decreasing monotonically outwards, with \(P_{\perp } \ge P\);

- C6:
Energy conditions. Strong (SEC) \(\rho \ge P + 2P_{\perp }\) or dominant (DEC) \(\rho \ge P\) and \(\rho \ge P_{\perp }\);

- C7:
Causality Conditions. \(0 \le v^2,v_{\perp }^2 \le 1\);

- C8:
The adiabatic index \(\Gamma \) stability criterion as stated in Eq. (1), which is a consequence of the dynamical stability criterion;

- C9:
Stability against cracking as expressed by Eq. (26);

- C10:
Harrison–Zeldovich–Novikov stability condition: \(\mathrm {d}M(\rho _c)/\mathrm {d}\rho _c > 0\).

- 1.
\((m/r)' > 0\) which fulfills C1, C2 and C3 conditions;

- 2.
\(0 \ge P_{\perp }' \ge P'\) accomplishing: C4, C5, C6 (SEC), C7, C9;

- 3.
\(\mu = 2M/r_b \le 4/5\) executing C6 (DEC);

- 4.
\(v^2 \ge 1/3\), implementing C8;

- 5.
and the C10 condition: \(\mathrm {d}M(\rho _c)/\mathrm {d}\rho _c > 0\).

## 6 Isotropic and anisotropic models

In this section we select seven exact solutions – describing isotropic or anisotropic fluid spheres – which comply with the Ivanov criteria. With this selection we study the effect of convective instability and the reaction of the pressure gradient to density perturbations.

Four of seven density profiles are among most physically reasonable isotropic solutions (Tolman VII [27], Buchdahl-1 [28], Mehra [29] and Kuchowicz [30]) reported in [31]. The fifth selection corresponds to a one-parameter family of a generalized Tolman IV solution obtained in [32] and allows us to exemplify the correlation of convection stability with a decreasing profile of \((v^2)'\). Finally, we study two anisotropic solutions (Gokhroo and Mehra [33] and Sah and Chandra [34]) to illustrate the buoyancy effects in anisotropic matter configurations.

### 6.1 Isotropic solutions

Parabolic density profiles, represented by Tolman VII and Mehra, are used extensively to model stable neutron stars. In these profiles: *A*, *R* and *C* are constants to be determined by the boundary conditions. Rational profiles like Buchdahl sets limits to the compactness, \(\mu = 2M/r_b\) of relativistic spheres, and \(r_b\) is the boundary radius of the matter configuration. Kuchowicz solution models of charged spheres, in this case \(x=Ar^2\), the function \(F(r) = \mathrm {Ei}\left( 1 +\frac{Ar^2}{2} \right) /\mathrm {e}\) where \(F(0)= \mathrm {Ei}(1)/\mathrm {e}\) and \(\mathrm {Ei}\) is the exponential integral function, again, *A* and *C* are constants to be determined by the boundary conditions

Solution | Density \(\ \rho \) | Central density \(\rho _c\) | \([\rho '']_c\) |
---|---|---|---|

Tolman VII [27] | \(\dfrac{1}{8\pi A^2}\left[ \frac{3A^2}{R^2}-\frac{20\,{r}^{2}}{{A}^{2} }\right] \) | \({\dfrac{3}{8\pi {R}^{2}}}\) | \(-{\dfrac{5}{{A}^{4}\pi }}\) |

Buch1 [28] | \(\dfrac{3C}{16 \pi }{\dfrac{ 3+ C{r}^{2}}{ \left( 1+ C{r}^{2} \right) ^{2}}}\) | \({\dfrac{9\,C}{16\,\pi }}\) | \(-{\dfrac{15\,{C}^{2}}{8\,\pi }}\) |

Mehra [29] | \(\dfrac{15\mu }{16\pi r_b^2}\left[ 1 -\left( \frac{r}{r_b }\right) ^2 \right] \) | \(\dfrac{15\ \mu }{16\,\pi \,r_b^2}\) | \(-\dfrac{15\mu }{16\pi r_b^4}\) |

Kuch2 III [30] | \(\dfrac{\left[ x-x^2 +6 \right] \left[ A F(r) -2 C\right] \mathrm {e}^{-\frac{x}{2} }+ 2Ax}{ 16\pi (2+ x) }\) | \(\dfrac{3\left[ A F(1) -2 C \right] }{16 \pi } \) | \(\dfrac{5A\left[ A[1-F(1)]+2\,C \right] }{16\,\pi }\) |

*C*is an arbitrary constant and

*N*is a positive integer that produces an infinite family of analytic solutions. As we can see from the Eq. (27), different values of

*N*recover well-known solutions: \(N = 1\) corresponds to Tolman IV solution [27]; \(N = 3\) represents Heint IIa [41] solution; \(N = 4\) and \(N=5\) are Durg IV and Durg V solutions, respectively [42]. The case \(N = 2\) is considered in [43] studying the relationship between the central barionic density and the total mass of observed neutron stars.

As can be easily guessed, the main difficulty of the method lies in how to calculate the two integrals that appearing in equation (4) of [32], but fortunately it is possible in the present case, as can seen in the appendix.

*N*, as

*a*,

*b*;

*c*;

*d*) automatically evaluates to exact values with

*K*a constant.

Florides–Stewart–Gokhroo–Mehra (FSGM) and Sah–Chandra profiles satisfy Ivanov criteria. In these profiles: \(\alpha \), *K* and *a* are constants to be determined by the boundary conditions

Solution | Density \(\ \rho \) | Central density \( \ \rho _c\) | \([\rho '']_c\) |
---|---|---|---|

FSGM [33] | \(\dfrac{3\alpha }{8 \pi } \left( 1-{\dfrac{K{r}^{2}}{r_b^2 }} \right) \) | \( {\dfrac{15\,\mu }{8\pi \,r_b^2 \, \left( 5-3\,K \right) }}\) | \(-\dfrac{2 K \rho _c }{ r_b^2 }\) |

Sah and Chandra [34] | \(\dfrac{1}{8 \pi } \left[ 10\,a \left( 1-a{r}^{2}\right) ^{4}+ \dfrac{ 1- \left( 1 -a{r}^{2} \right) ^{5} }{{r}^{2}} \right] \) | \(\dfrac{15\,a}{8\,\pi }\) | \(-\dfrac{25a^2}{2\pi }\) |

### 6.2 Anisotropic solutions

Local anisotropy (unequal stresses: \(P\ne P_{\perp }\)) in compact objects can be associated to different physical scenarios –such as phase transition, density inhomogeneity and electromagnetic field, just to mention a few of them – and has been considered extensively since the work of Bowers and Liang [44]. The unknown physics in the tangential equation of state, \(P_{\perp } = P_{\perp }(\rho )\) is partially compensated by using heuristic criteria: geometric, simplicity or any other assumption relating radial and tangential pressures (see [15, 45] and references therein).

Our selection of anisotropic solutions shown in the Table 2 are: Florides–Gokhroo–Mehra [33, 46, 47] and the Sah and Chandra [34].

The Florides–Gokhroo–Mehra profile was due originally to Florides [46], but also corresponds one of the different solutions considered by Stewart [47] and, more recently, by Gokhroo and Mehra [33]. The Florides–Stewart–Gokhroo–Mehra (FSGM) solution represents densities and pressures which, under particular circumstances [48], give rise to an equation of state similar to the Bethe–Börner–Sato newtonian equation of state for nuclear matter [1, 2, 49].

## 7 Modeling performed and discussion of some results

The first effect to be analyzed is the convective instability and its relation to the sign of the gradient of the speed of sound, \((v^2)'\), which are illustrated in Figs. 1, 2 and 3. We show this effect by plotting the normalized buoyancy \(\hat{\rho }'' =\rho ''/\rho _{c}''\) vs \(\hat{r} = r/r_b\) and the corresponding gradient \((v^2)'\) vs \(\hat{r}\). The first two figures display the convective stability for isotropic models while Fig. 3 illustrates this property for anisotropic spheres.

As we have mentioned before, parabolic density profiles – represented by Tolman VII, Mehra and Florides–Stewart–Gokhroo–Mehra in Tables 1 and 2 – are stable to convection because they have constant \(\rho ''< 0\). The stability for the other isotropic models having rational density profiles is presented in Fig. 1, where the Buchdahl model becomes unstable because \(\rho ''/\rho _c\) changes sign, while Kuchowicz is stable against convective perturbations due to \(\rho ''/\rho _c > 0\) for all *r*.

The other effect to be discussed is the stability induced by the reaction of the pressure gradient to density perturbations. This is shown in Figs. 4 and 5, where we compare the perturbation on the total force distribution when the pressure gradient is perturbed and when it is not. In these figures we plot \(\delta \mathcal {R}_p \ne 0\), and \(\delta \mathcal {R}_p = 0\), respectively. The first case, \(\delta \mathcal {R}_p \ne 0\), represents the recent cracking scheme of Gonzalez–Navarro–Nunez [25, 26] while the second one corresponds to a variation of the previous work of Abreu–Hernandez–Nunez [24]. As we have stressed above, in the first case it is assumed a non-constant density perturbation \(\delta \rho = \delta \rho (r)\), which leads to the factor \( \frac{4 \pi r^2 \rho }{\rho '} \le 0 \) in the third term in the perturbation to the gravitational force distribution \(\delta \mathcal {R}_g\) in Eq. (22), which may cause cracking instabilities.

As it is evident for the models considered, if the pressure gradient is not perturbed, i.e. \(\delta \mathcal {R}_p~=~0\), \(\delta \mathcal {R}\) may change its sign and potential cracking instabilities may appear. On the other hand, if the gradient reacts to the perturbation, \(\delta \mathcal {R}_p \ne 0\), we find that \(\delta \mathcal {R}\) does not change sign and the matter configuration becomes stable against cracking. This tendency to make models cracking-stable if the the pressure gradient reacts to the density, was previously reported incidentally in reference [26].

This induced stability can be understood if we shape the perturbation of the hydrostatic equation. Clearly, for the isotropic case, \(\mathcal {R}_a =\delta \mathcal {R}_a = 0\) and when \(\delta \mathcal {R}_p = 0\), the cracking instability emerges only from the effect of the perturbation on the gravitational force distribution, \(\delta \mathcal {R}_g\) and particularly from the third term, which is always negative. If \(\delta \mathcal {R}_p \ne 0\) and \(P'' > 0\), the reaction of the pressure gradient to the density perturbation can neutralize the effect of the negative sign of the above mentioned gravitational term.

## 8 Conclusions and final remarks

Stability is a key concept when considering self-gravitating stellar models, because only those in stable equilibrium are of astrophysical interest. A we have stated above, in addition to solving the structure equations (7) and (8) for a particular set of equations of state: \(P=P(\rho )\) and \(P_{\perp }=P_{\perp }(\rho )\), the emerging physical variables have to comply with the several acceptability conditions stated in Sect. 5.

The stability of a spherical star against convection has almost been forgotten in most of the stability analysis. It is a very simple criterion which implements the Archimedes principle in any hydrostatic matter configuration [17, 18, 19]. This criterion has proven to be interesting because several reasonable models become unstable against convection and should be included in the acceptability criteria to guarantee physically interesting models of compact objects.

- 1.
A density profiles with its second derivative with respect to the radial marker less or equal than zero, \(\rho '' < 0\), will be stable against convective motions. This is a very simple criterion to identify potential convection instabilities within spherical matter configurations and it should be added to the above mentioned acceptability Ivanov criteria.

- 2.
A decreasing concave pressure profile

*P*(*r*), i.e. \(P' < 0\) and \(P'' < 0\), will be stable against convection if the radial sound velocity decreases monotonically outward in spherically matter configurations. Equivalently, if \( \frac{\partial ^2 P}{\partial \rho ^2} > 0\), then \((v^2)' < 0\) implies stability against convective motions. It should be pointed out that to illustrate this effect we have implemented a new family of exact solution by using the method propose by in Ref. [32]. - 3.
From (19) we obtained the possible sources of cracking: the reaction of the pressure gradient \(\delta \mathcal {R}_p\); the perturbation to the gravitational force distribution \(\delta \mathcal {R}_g\) and the perturbation of the anisotropy \(\delta \mathcal {R}_a\). Convection may cause cracking instability only when the perturbation of the pressure gradient is considered, i.e. \(\delta \mathcal {R}_p \ne 0\), in this case cracking density perturbations affect the pressure gradient, depending on the values of \((v^2)'\) and \(v^2\frac{\rho ''}{\rho '}\).

- 4.
Isotropic and anisotropic models considered can be unstable to cracking when the reaction of the pressure gradient is neglected, i.e. \(\delta \mathcal {R}_p = 0\), but if taken into account, the instabilities may vanish. Thus, there is a stabilizing effect against cracking, when the perturbation gradient is affected by density perturbation, \(\delta \mathcal {R}_p \ne 0\).

Finally, it should be stressed that the results we have presented of possible instabilities for local perturbation schemes (convection and/or cracking) have to be considered as tendencies that could lead potential evolution of fluids within a relativistic matter configuration, but this should emerge from the full integration of the Einstein Equations.

## Notes

### Acknowledgements

We gratefully acknowledge the financial support of the Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander and the financial support provided by COLCIENCIAS under Grant no. 8863.

## References

- 1.S.L. Shapiro, S.A. Teukolsky,
*The Physics of Compact Objects*(Wiley, New York, 1983)Google Scholar - 2.M. Demianski,
*Relativistic Astrophysics*, vol. 110, International Series in Natural Philosophy (Pergamon Press, Oxford, 1985)Google Scholar - 3.N.K. Glendenning,
*Compact Stars: Nuclear Physics, Particle Physics, and General Relativity*(Springer, Berlin, 2000)CrossRefGoogle Scholar - 4.P. Haensel, A.Y. Potekhin, D.G. Yakovlev,
*Neutron Stars 1: Equation of State and Structure*, vol. 326, Astronomy and Astrophysics Library (Springer Science & Business Media, Berlin, 2007)Google Scholar - 5.R. Kippenhahn, A. Weigert, A. Weiss,
*Stellar Structure and Evolution, Astronomy and Astrophysics Library*, 2nd edn. (Springer, Berlin, 2013)zbMATHGoogle Scholar - 6.Friedman, J.L., Stergioulas, N., Instabilities of relativistic stars, in
*General Relativity, Cosmology and Astrophysics*, eds. by J. Bičák, T. Ledvinka (Springer, Berlin, 2014), pp. 427–458Google Scholar - 7.S. Chandrasekhar, Dynamical instability of gaseous masses approaching the schwarzschild limit in general relativity. Phys. Rev. Lett.
**12**(4), 114–116 (1964)ADSMathSciNetCrossRefGoogle Scholar - 8.S. Chandrasekhar, The dynamical instability of gaseous masses approaching the Schwarzschild limit in general relativity. Astrophys. J.
**140**, 417 (1964)ADSMathSciNetCrossRefGoogle Scholar - 9.R.F. Tooper, Stability of massive stars in general relativity. Astrophys. J.
**140**, 811–814 (1964)ADSMathSciNetCrossRefGoogle Scholar - 10.R.F. Tooper, Adiabatic fluid spheres in general relativity. Astrophys. J.
**142**, 1541–1562 (1965)ADSMathSciNetCrossRefGoogle Scholar - 11.J.M. Bardeen, Stability and dynamics of spherical symmetric masses in general relativity. PhD thesis, California Institute of Technology, Pasadena (1965)Google Scholar
- 12.J.L. Friedman, B.F. Schutz, On the stability of relativistic systems. Astrophys. J.
**200**, 204–220 (1975)ADSMathSciNetCrossRefGoogle Scholar - 13.H. Heintzmann, W. Hillebrandt, Neutron stars with an anisotropic equation of state-mass, redshift and stability. Astrono. Astrophys.
**38**(1), 51–55 (1975)ADSGoogle Scholar - 14.R. Chan, L. Herrera, N.O. Santos, Dynamical instability for radiating anisotropic collapse. R. Astron. Soc. Mon. Not.
**265**, 533 (1993)ADSCrossRefGoogle Scholar - 15.L. Herrera, N.O. Santos, Local anisotropy in self-gravitating systems. Phys. Rep.
**286**(2), 53–130 (1997)ADSMathSciNetCrossRefGoogle Scholar - 16.B.V. Ivanov, Analytical study of anisotropic compact star models. Eur. Phys. J. C
**77**, 738 (2017)ADSCrossRefGoogle Scholar - 17.H. Bondi, Massive spheres in general relativity. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**282**(1390), 303–317 (1964)ADSMathSciNetzbMATHGoogle Scholar - 18.K.S. Thorne, Validity in general relativity of the Schwarzschild criterion for convection. Astrophys. J.
**144**, 201–205 (1966)ADSCrossRefGoogle Scholar - 19.A. Kovetz, Schwarzschild’s criterion for convective instability in general relativity. Zeitschrift für Astrophysik
**66**, 446 (1967)ADSGoogle Scholar - 20.L. Herrera, Cracking of self-gravitating compact objects. Phys. Lett. A
**165**, 206–210 (1992)ADSCrossRefGoogle Scholar - 21.A. Di Prisco, E. Fuenmayor, L. Herrera, V. Varela, Tidal forces and fragmentation of self-gravitating compact objects. Phys. Lett. A
**195**, 23–26 (1994)ADSMathSciNetCrossRefGoogle Scholar - 22.A. Di Prisco, L. Herrera, V. Varela, Cracking of homogeneous self-gravitating compact objects induced by fluctuations of local anisotropy. Gen. Relativ. Gravit.
**29**(10), 1239–1256 (1997)ADSCrossRefGoogle Scholar - 23.H. Abreu, H. Hernández, L.A. Núñez, Cracking of self-gravitating compact objects with a non local equation of state. J. Phys. Conf. Ser.
**66**, 012038 (2007)CrossRefGoogle Scholar - 24.H. Abreu, H. Hernández, L.A. Núñez, Sound speeds, cracking and stability of self-gravitating anisotropic compact objects. Class. Quantum Gravity
**24**, 4631–4646 (2007)ADSCrossRefGoogle Scholar - 25.G.A. González, A. Navarro, L.A. Núñez, Cracking of anisotropic spheres in general relativity revisited. J. Phys. Conf. Ser.
**600**(1), 012014 (2015)CrossRefGoogle Scholar - 26.G.A. González, A. Navarro, L.A. Núñez, Cracking isotropic and anisotropic relativistic spheres. Can. J. Phys.
**95**(11), 1089–1095 (2017)ADSCrossRefGoogle Scholar - 27.R.C. Tolman, Static solutions of Einstein’s field equations for spheres of fluid. Phys. Rev.
**55**(4), 364–373 (1939)ADSCrossRefGoogle Scholar - 28.H.A. Buchdahl, General relativistic fluid spheres. Phys. Rev.
**116**(4), 1027–1034 (1959)ADSMathSciNetCrossRefGoogle Scholar - 29.A.L. Mehra, Radially symmetric distribution of matter. J. Aust. Math. Soc.
**6**, 153–156, 5 (1966)CrossRefGoogle Scholar - 30.B. Kuchowicz, General relativistic fluid spheres. I. New solutions for spherically symmetric matter distributions. Acta Phys. Pol.
**33**, 541–563 (1968)Google Scholar - 31.M.S.R. Delgaty, K. Lake, Physical acceptability of isolated, static, spherically symmetric, perfect fluid solutions of Einstein’s equations. Comput. Phys. Commun.
**115**, 395 (1998)ADSMathSciNetCrossRefGoogle Scholar - 32.K. Lake, All static spherically symmetric perfect fluid solutions of Einstein’s equations. Phys. Rev. D
**67**, 104015 (2003)ADSMathSciNetCrossRefGoogle Scholar - 33.M.K. Gokhroo, A.L. Mehra, Anisotropic spheres with variable energy density in general relativity. Gen. Relativ. Gravit.
**26**(1), 75–84 (1994)ADSMathSciNetCrossRefGoogle Scholar - 34.A. Sah, P. Chandra, Spherical anisotropic fluid distribution in general relativity. World J. Mech.
**6**, 487–504 (2016)ADSCrossRefGoogle Scholar - 35.A.M. Raghoonundun, D.W. Hobill, Possible physical realizations of the Tolman VII solution. Phys. Rev. D
**92**(12), 124005 (2015)ADSCrossRefGoogle Scholar - 36.P. Bhar, M.H. Murad, N. Pant, Relativistic anisotropic stellar models with Tolman VII spacetime. Astrophys. Space Sci.
**359**, 13 (2015)ADSCrossRefGoogle Scholar - 37.M. Azam, S.A. Mardan, M.A. Rehman, Cracking of compact objects with electromagnetic field. Astrophys. Space Sci.
**359**(1), 14 (2015)ADSCrossRefGoogle Scholar - 38.A.M. Raghoonundun, Exact solutions for compact objects in general relativity. PhD thesis, University of Calgary, Alberta-Canada (2016)Google Scholar
- 39.H.A. Buchdahl,
*Seventeen Simple Lectures on General Relativity Theory*(Wiley, New York, 1981)zbMATHGoogle Scholar - 40.B.V. Ivanov, Static charged perfect fluid spheres in general relativity. Phys. Rev. D
**D65**, 104001 (2002)ADSMathSciNetCrossRefGoogle Scholar - 41.H. Heintzmann, New exact static solutions of Einsteins field equations. Z. Phys.
**228**, 489–493 (1969)ADSMathSciNetCrossRefGoogle Scholar - 42.M.C. Durgapal, A class of new exact solutions in general relativity. J. Phys. A Math. Gen.
**15**, 2637–2644 (1982)ADSMathSciNetCrossRefGoogle Scholar - 43.J.M. Lattimer, M. Prakash, Ultimate energy density of observable cold baryonic matter. Phys. Rev. Lett.
**94**(11), 111101 (2005)ADSCrossRefGoogle Scholar - 44.R.L. Bowers, E.P.T. Liang, Anisotropic spheres in general relativity. Astrophys. J.
**188**, 657 (1974)ADSCrossRefGoogle Scholar - 45.K. Yagi, N. Yunes, Approximate universal relations for neutron stars and quark stars. Phys. Rep.
**681**, 1–72 (2017)ADSMathSciNetCrossRefGoogle Scholar - 46.P.S. Florides, A new interior Schwarzschild solution. Proc. R. Soc. Lond.
**A337**, 529–535 (1974)ADSMathSciNetCrossRefGoogle Scholar - 47.B.W. Stewart, Conformally flat, anisotropic spheres in general relativity. J. Phys. A Math. Gen.
**15**, 2419–2427 (1982)ADSMathSciNetCrossRefGoogle Scholar - 48.J. Martínez, Transport processes in the gravitational collapse of an anisotropic fluid. Phys. Rev. D
**53**, 6921–6940 (1996)ADSCrossRefGoogle Scholar - 49.H.A. Bethe, G. Borner, K. Sato, Nuclei in neutron matter. Astron. Astrophys.
**7**, 279–288 (1970)ADSGoogle Scholar - 50.F. Özel, P. Freire, Masses, radii, and the equation of state of neutron stars. Annu. Rev. Astron. Astrophys.
**54**, 401–440 (2016)ADSCrossRefGoogle Scholar - 51.L. Herrera, J. Ospino, A. Di Prisco, All static spherically symmetric anisotropic solutions of Einstein’s equations. Phys. Rev. D
**77**, 027502 (2008)ADSMathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}