# A causal Schwarzschild-de Sitter interior solution by gravitational decoupling

## Abstract

We employ the minimal geometric deformation approach to gravitational decoupling (MGD-decoupling) in order to build an exact anisotropic version of the Schwarzschild interior solution in a space-time with cosmological constant. Contrary to the well-known Schwarzschild interior, the matter density in the new solution is not uniform and possesses subluminal sound speed. It therefore satisfies all standard physical requirements for a candidate astrophysical object.

## 1 Introduction

The Schwarzschild interior metric is one of the best known solutions of Einstein’s field equations [1]. This exact solution represents an isotropic self-gravitating object of uniform density (incompressible fluid), and has been widely studied, generally without considering the cosmological constant. As far as we know, it is one of the few analytic solutions for a bounded distribution which fits smoothly with the Schwarzschild exterior metric [2]. However, it cannot be used to represent a stellar model as its speed of sound is not subliminal (see Ref. [3] where a model of two fluids is used to circumvent the problem of causality). Despite of the above, some studies on the possible impact of the vacuum energy on perfect fluids have been carried out which made use of this solution for both positive and negative values of the cosmological constant. Such analyses can help to elucidate some properties of the Schwarzschild-(anti-)de Sitter space-time in presence of matter [4, 5] (see also [6, 7, 8]). Moreover, some alternatives to black holes, like the gravastars [9, 10], are mainly generated from this solution [11, 12], and an exact time-dependent version was recently reported in Ref. [13]. Therefore, it is not only natural, but also useful, to construct a possible extension of this solution for more realistic stellar scenarios, such as that represented by non-uniform and anisotropic self-gravitating objects. Above all, it would be very important to develop versions that do not suffer from the causal problem. However, given the complexity of Einstein’s field equations, we know that extending a known solution to more complex scenarios is an arduous and difficult task [14], even more so if we wish to keep it physically acceptable. Fortunately, the so-called method of gravitational decoupling by Minimal Geometric Deformation (MGD-decoupling, henceforth) [15, 16], which has been widely used recently [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], has proved to be a powerful method to extend known solutions into more complex scenarios.

- I.one can extend simple solutions of the Einstein equations into more complex domains. In fact, we can start from a source with energy-momentum tensor \({\hat{T}}_{\mu \nu }\) for which the metric is known and add the energy-momentum tensor of a second source,We can then repeat the process with more sources \(T^{(i)}_{\mu \nu }\) to extend the solution of the Einstein equations associated with the gravitational source \({\hat{T}}_{\mu \nu }\) into the domain of more intricate forms of gravitational sources \(T_{\mu \nu }\);$$\begin{aligned} {\hat{T}}_{\mu \nu } \rightarrow T_{\mu \nu } = {\hat{T}}_{\mu \nu } +T^{(1)}_{\mu \nu }. \end{aligned}$$(1.1)
- II.one can also reverse the previous procedure in order to find a solution to Einstein’s equations with a complex energy-momentum tensor \({T}_{\mu \nu }\) by splitting it into simpler components,and solve Einstein’s equations for each one of these components. Hence, we will have as many solutions as the components in the original energy-momentum tensor \({T}_{\mu \nu }\). Finally, by a simple combination of all these solutions, we will obtain the solution to the Einstein equations associated with the original energy-momentum tensor \({T}_{\mu \nu }\). We emphasise that the MGD-decoupling works as long as the sources do not exchange energy-momentum directly among them, to wit$$\begin{aligned} {T}_{\mu \nu } \rightarrow {\hat{T}}_{\mu \nu }+T^{(i)}_{\mu \nu }, \end{aligned}$$(1.2)which further clarifies that their interaction is purely gravitational;$$\begin{aligned} \nabla _{\mu }{\hat{T}}^{\mu \nu } = \nabla _{\mu } T^{(1)\mu \nu } = \cdots = \nabla _{\mu } T^{(n)\mu \nu } = 0, \end{aligned}$$(1.3)
- III.it can be applied to theories beyond general relativity. For instance, given the modified action [16]where \(\mathcal{L}_{\mathrm{M}}\) contains all matter fields in the theory and \(\mathcal{L}_{\mathrm{X}}\) is the Lagrangian density of a new gravitational sector with an associated (effective) energy-momentum tensor$$\begin{aligned} S_{\mathrm{G}} = S_{\mathrm{EH}}+S_{\mathrm{X}} = \int \left[ \frac{R}{2\,k^2}+\mathcal{L}_{\mathrm{M}}+\mathcal{L}_{\mathrm{X}}\right] \sqrt{-g}\,d^4\,x, \end{aligned}$$(1.4)the method in I. allows one to extend all the known solutions of the Einstein-Hilbert action \(S_{\mathrm{EH}}\) into the domain of modified gravity represented by \(S_{\mathrm{G}}\). This represents a straightforward way to study the consequences of extended gravity on general relativity.$$\begin{aligned} \theta _{\mu \nu } = \frac{2}{\sqrt{-g}}\frac{\delta (\sqrt{-g}\,\mathcal{L}_{\mathrm{X}})}{\delta g^{\mu \nu }} = 2\,\frac{\delta \mathcal{L}_{\mathrm{X}}}{\delta g^{\mu \nu }}-g_{\mu \nu }\,\mathcal{L}_{\mathrm{X}}, \end{aligned}$$(1.5)

The paper is organised as follows: in Sect. 2, we start from the Einstein equations with cosmological constant for a spherically symmetric stellar distribution and we show how to decoupling two spherically symmetric and static gravitational sources \(\{T_{\mu \nu },\,\theta _{\mu \nu }\}\). After providing details on the matching conditions at the star surface under the MGD-decoupling, in Sect. 3, we implement the MGD-decoupling following the scheme I. to generate the extended version of the Schwarzschild solution; finally, we summarise our conclusions in Sect. 4.

## 2 Spherically symmetric stellar distribution

^{1}

*r*only, ranging from \(r=0\) (the star’s centre) to some \(r=R>0\) (the star’s surface). The cosmological constant can be thought to contribute the stress-energy tensor being responsible for the expansion of the universe, with a non-zero vacuum energy density and negative pressure satisfying \(\rho _{vac}=-p_{vac}=\Lambda /k^2\). Explicitly, the field equations read

Eqs. (2.5)–(2.7) contain five unknown functions, namely, three physical variables: the density \(\tilde{\rho }(r)\), the radial pressure \(\tilde{p}_r(r)\) and the tangential pressure \(\tilde{p}_t(r)\); and two geometric functions: the temporal metric function \(\nu (r)\) and the radial metric function \(\lambda (r)\). Therefore these equations form an indefinite system [53, 54] which requires additional information to produce any specific solution.

### 2.1 Gravitational decoupling by MGD

In order to solve the Einstein Eqs. (2.5)–(2.8) we implement the MGD-decoupling. In this approach, one starts from a solution for the isotropic fluid and the field equations with the anisotropic source \(\theta _{\mu \nu }\) will take the form of effective “quasi-Einstein” equations [see Eqs. (2.22)–(2.24) below].

*g*and

*f*are the deformations undergone by the temporal and radial metric component of the perfect fluid geometry \(\{\xi ,\mu \}\), respectively. Among all possible deformations (2.15) and (2.16), the so-called minimal geometric deformation is given by \(g=0\) and \(f=f^*\), where \(f^*\) satisfies a suitable condition [33, 34] in order to minimise the departure from General Relativity. Only the radial metric component therefore changes to

### 2.2 Matching conditions at the surface

*m*given in Eq. (2.14) and \(f^{*}\) the geometric deformation in Eq. (2.17). On the other hand, the exterior (\(r>R\)) space-time will be described by the Schwarzschild-de Sitter metric

*r*, must be continuous across the sphere \(\Sigma \), which can be written in terms of the Einstein tensor as

^{2}

The expression in Eq. (2.36) in particular contains critical information about the conditions that the self-gravitating system must fulfil in order to be consistently coupled with the Schwarzschild-de Sitter geometry (2.28). First of all, the effective radial pressure \(\tilde{p}\) at the surface must vanish, which is a very well-known result. However, if the geometric deformation \(f^*(r<R)\) is positive, hence weakening the gravitational field, [see Eq. (2.27)], the exterior geometry (2.28) can only be compatible with a non-vanishing \(\theta _{\mu \nu }\) if the perfect fluid has \(p_{R}<0\), which may be interpreted as regular matter with a solid crust [43]. If we want to avoid having a solid-crust and keep the standard condition \(p_{R}=0\), we must impose that the anisostropic effects on the radial pressure pressure vanish at \(r=R\). For instance, this is achieved if we assume that \((\theta _1^{\,\,1})^{-}_{R}\sim \,p_{R}\) in Eq. (2.35), which leads to a vanishing inner deformation \(f^*_R=0\) [see further Eq. (3.13)].

## 3 Anisotropic Schwarzschild-de Sitter interior solution

*R*formed by an incompressible perfect fluid of total mass

*M*and is given by [6]

*A*,

*B*and

*C*can be expressed in terms of the physical parameters

*M*,

*R*and \(\Lambda \) by means of the matching conditions with the outer Schwarzschild-de Sitter vacuum (2.28). In fact, Eqs. (2.29), (2.30) and (2.36) with \(\alpha =0\) yield \(\mathcal{M}=M\) and

^{3}, for which we have

*M*and radius

*R*, we have four unknown parameters, namely

*A*,

*B*and

*C*from the interior solution in Eqs. (3.1) and (3.15), and the mass \(\mathcal{M}\) in Eq. (2.28). However, the mass

*M*is related to the constant

*C*and the radius

*R*by the definition (2.14). We therefore have only three unknown constants to be determined by the three conditions (2.29), (2.30) and (2.35) at the star surface. The continuity of the metric given by Eqs. (2.29) and (2.30) leads to

*A*shown in Eq. (3.5). This result is in agreement with our prescription to avoid having a solid crust and also ensures that

*C*is also related to

*M*,

*R*and \(\Lambda \) by the same expression given in Eq. (3.7). We remark that the condition (3.24) shows that, despite the anisotropic effects, the total mass of the stellar distribution remains unchanged. Finally, by using Eqs. (3.5) and (3.25) in the matching condition (3.21), we obtain the same expression for the constant

*B*in Eq. (3.6). We therefore conclude that the relations between the constants

*A*,

*B*and

*C*in terms of

*M*,

*R*and \(\Lambda \) are not affected by the anisotropy.

Given a stellar distribution of mass *M* and radius *R* in a background with cosmological constant \(\Lambda \), we can now analyse the anisotropic effects on physical variables for different values of \(\alpha \). The first thing to notice is that the effective radial pressure \({\tilde{p}}_r\) remains proportional to the isotropic expression (3.12), and we therefore find the usual Buchdahl limit for the star compactness [58] with cosmological constant in Eq. (3.9). This result is further supported by the fact that both the effective density \({\tilde{\rho }}\) and the tangential pressure \({\tilde{p}}\) do not show any singularity for \(0\le r\le R\) when Eq. (3.9) holds. Moreover, the effective density is not uniform and \({\tilde{\rho }}'\) turns out to be proportional to \(-\alpha \), which implies that the effective density decreases (increases) towards the surface for \(\alpha >0\) (respectively, \(\alpha <0\)). In the following we shall therefore only consider cases with \(\alpha >0\) so that the effective density decreases from the centre outwards.

## 4 Conclusions

By using the MGD-decoupling approach, we found the anisotropic and non-uniform version of the Schwarzschild-de Sitter interior solution with cosmological constant given by the exact and analytical expressions displayed in Eqs. (3.1), (3.15), and (3.17)–(3.19). Contrary to the well known Schwarzchild interior solution, this new system satisfies all of the physical requirement, namely, it is regular at the origin, pressure and density are positive everywhere (at least for \(\alpha >0\)), mass and radius are well defined when the Buchdahl limit (3.9) holds, density and pressure decrease monotonically from the centre outwards (for \(\alpha >0\)), the dominant energy condition is satisfied, and the sound speeds are subluminal. Regarding this last requirement, our interior anisotropic solution is causal, showing thus that the anisotropic effects produce a more realistic stellar structure.

*p*(

*r*), as shown in Eq. (3.13). This leads to a vanishing inner deformation \(f^*_R=0\) and therefore the total mass

*M*of the standard Schwarzschild interior solution is not affected by the anisotropy, as we can see in the condition (2.30). A direct consequence of this is that the surface redshift

In this paper we included a cosmological constant \(\Lambda \) for generality. However, since the present value of \(\Lambda \) would introduce corrections of order \(10^{-52}\), its effects could be significant mainly at very large scales, with no sizeable consequences on self-gravitating stellar objects. In particular, it was shown [8] that \(\Lambda \) plays a significant role for very extended polytropic spheres that could describe galactic dark matter halos (see also Ref. [62] where the effects of \(\Lambda \) in several astrophysical situations is summarised). On the other hand, a large effective cosmological constant could be related to phase transitions in the early universe, and that could influence compact objects created during this period, like primordial black holes. For example, the electroweak phase transition at \(T_{\mathrm{ew}} \sim 100\,\)GeV corresponds to \(\Lambda _{\mathrm{ew}} \sim 0.028\,\)cm\(^{-2}\), while for the quark confinement at \(T_{\mathrm{qc}} \sim 100\,\)MeV one would have \(\Lambda _{\mathrm{qc}} \sim 2.8\cdot 10^{-10}\,\)cm\(^{-2}\) [63].

We conclude by mentioning that, for the study of non-primordial compact configurations, we can safely ignore the cosmological constant without jeopardising our causal solution. This yields even simpler expressions which could be exploited more easily to investigate some interesting cases, such as the gravastar limit and the extended Kerr source [11], or even a possible generalisation including a nontrivial time dependence, as those in the exact time-dependent version found in Ref. [13], but without the space-time singularities present therein.

## Footnotes

## Notes

### Acknowledgements

J.O. and S.Z. have been supported by the Albert Einstein Centre for Gravitation and Astrophysics financed by the Czech Science Agency Grant No.14-37086G and by the Silesian University at Opava internal grant SGS/14/2016. J.O. thanks Luis Herrera for useful discussion and comments. L.G. acknowledges the FPI Grant BES-2014-067939 from MINECO (Spain). A.S. is partially supported by Project Fondecyt 1161192, Chile and MINEDUC-UA project, code ANT 1855. R.C. is partially supported by the INFN grant FLAG and his work has also been carried out in the framework of activities of the National Group of Mathematical Physics (GNFM, INdAM) and COST action *Cantata*.

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