# Anisotropic solutions by gravitational decoupling

## Abstract

We investigate the extension of isotropic interior solutions for static self-gravitating systems to include the effects of anisotropic spherically symmetric gravitational sources by means of the gravitational decoupling realised via the minimal geometric deformation approach. In particular, the matching conditions at the surface of the star with the outer Schwarzschild space-time are studied in great detail, and we describe how to generate, from a single physically acceptable isotropic solution, new families of anisotropic solutions whose physical acceptability is also inherited from their isotropic parent.

## 1 Introduction

In a recent paper [1], the first simple, systematic and direct approach to decoupling gravitational sources in general relativity (GR) was developed from the so-called Minimal Geometric Deformation (MGD) approach. The MGD was originally proposed [2, 3] in the context of the Randall–Sundrum brane-world [4, 5] and extended to investigate new black hole solutions [6, 7] (for some earlier work on MGD, see for instance Refs. [8, 9, 10, 11], and for some recent applications Refs. [12, 13, 14, 15, 16, 17]). The decoupling of gravitational sources by MGD (henceforth MGD-decoupling) is not only a novel idea, but also has a number of ingredients that make it particularly attractive in the search for new spherically symmetric solutions of Einstein’s field equations, as discussed below.

The converse also works. In order to find a solution to Einstein’s equations with a complex spherically symmetric energy-momentum tensor \(T_{\mu \nu }\), we can split it into simpler components, say \(T^{(i)}_{\mu \nu }\), and solve Einstein’s equations for each one of these parts. Hence, we will have as many solutions as are the contributions \(T^{(i)}_{\mu \nu }\) in the original energy-momentum tensor. Finally, by a straightforward combination of all these solutions, we will obtain the solution to the Einstein equations associated with the original energy-momentum tensor \(T_{\mu \nu }\).

To summarise, the MGD-decoupling amounts to the following procedure: given two gravitational sources A and B, standard Einstein’s equations are first solved for A, and then a simpler set of quasi-Einstein equations are solved for B. Finally, the two solutions can be combined in order to derive the complete solution for the total system A\(\cup \)B. Since Einstein’s field equations are non-linear, the above procedure represents a breakthrough in the search and analysis of solutions, especially when the involved situations are beyond trivial cases, such as the interior of self-gravitating systems dominated by gravitational sources more realistic than the ideal perfect fluid [18, 19]. Of course, we remark that this decoupling occurs because of the spherical symmetry and time-independence of the systems under investigation.

Although decoupling gravitational sources in GR by a systematic way represents in itself a fact of significant theoretical importance, its practical relevance is no less. Indeed, this simple and systematic method could be conveniently exploited in a large number of relevant cases, such as the Einstein–Maxwell [20] and Einstein–Klein–Gordon system [21, 22, 23, 24], for higher derivative gravity [25, 26, 27], *f*(*R*)-theories of gravity [28, 29, 30, 31, 32, 33, 34], Ho\(\check{\mathrm{r}}\)ava-aether gravity [35, 36], polytropic spheres [37, 38, 39], among many others. In this respect, the simplest practical application of the MGD-decoupling consists in extending known isotropic and physically acceptable interior solutions for spherically symmetric self-gravitating systems into the anisotropic domain, at the same time preserving physical acceptability, which represents a highly non-trivial problem [40] (for obtaining anisotropic solutions in a generic way, see for instance Refs. [41, 42, 43]).

This paper is organised as follows: in Sect. 2, we briefly review the effective Einstein field equations for a spherically symmetric and static distribution of matter with effective density \(\tilde{\rho }\), effective radial pressure \(\tilde{p}_r\) and effective tangential pressure \(\tilde{p}_t\); Sect. 3 is devoted to the fundamentals of the MGD-decoupling; in Sect. 4, we provide detail on the matching conditions under the MGD-decoupling; in Sect. 5, we implement the MGD-decoupling to extend perfect fluid solutions in the anisotropic domain. In particular, three new exact and physically acceptable anisotropic solutions, generated from a single perfect fluid solution, are developed in order to emphasise the robustness of the approach; finally, we summarise our conclusions in Sect. 6.

## 2 Einstein equations for multiple sources

*p*. The term \(\theta _{\mu \nu }\) in Eq. (2.2) describes any additional source whose coupling to gravity is further proportional to the constant \(\alpha \) [44]. This source may contain new fields, like scalar, vector and tensor fields, and it will in general produce anisotropies in self-gravitating systems. We just recall that, since the Einstein tensor is divergence free, the total energy-momentum tensor (2.2) must satisfy the conservation equation

*r*only, ranging from \(r=0\) (the star center) to some \(r=R\) (the surface of the star), and the fluid 4-velocity is given by \(u^{\mu }=e^{-\nu /2}\,\delta _{0}^{\mu }\) for \(0\le r\le R\). The metric (2.5) must satisfy the Einstein equations (2.1), which explicitly read

The system (2.6)–(2.8) contains seven unknown functions, namely: two physical variables, the density \(\rho (r)\) and pressure *p*(*r*); two geometric functions, the temporal metric function \(\nu (r)\) and the radial metric function \(\lambda (r)\); and three independent components of \(\theta _{\mu \nu }\). These equations therefore form an indefinite system. In the particular case where \(\theta _{\mu \nu }\) depends only on the density and the pressure, we need to prescribe only an additional equation to close the systemEqs. (2.6)–(2.8), just as we do for the perfect fluid in standard GR. However, at this stage we want to emphasise that it is not enough to know the space-time geometry to determine the gravitational source \(\{\rho , p, \theta _{\mu \nu }\}\) in general.

## 3 Gravitational decoupling by MGD

We shall implement the MGD in order to solve the system of Eqs. (2.6)–(2.9). In this approach, the system will be transformed in such a way that the field equations associated with the source \(\theta _{\mu \nu }\) will take the form of the “effective quasi-Einstein” Eqs. (3.12)–(3.14).

*m*. Now let us turn on the parameter \(\alpha \) to consider the effects of the source \(\theta _{\mu \nu }\) on the perfect fluid solution \(\{\xi ,\mu \,\rho ,p\}\). These effects can be encoded in the geometric deformation undergone by the perfect fluid geometry \(\{\xi ,\mu \}\) in Eq. (3.1), namely

*g*and

*f*are the deformations undergone by the temporal and radial metric component, respectively. Among all possible deformations (3.3) and (3.4), the so-called minimal geometric deformation is given by

- (i)The first one is given by the standard Einstein equations for a perfect fluid (the one with \(\alpha = 0\) we started from), whose metric is given by Eq. (3.1) with \(\xi (r) = \nu (r)\):$$\begin{aligned} k^2\rho= & {} \frac{1}{r^2} -\frac{\mu }{r^2} -\frac{\mu '}{r}, \end{aligned}$$(3.8)$$\begin{aligned} k^2\,p= & {} -\frac{1}{r^2}+\mu \left( \frac{1}{r^2}+\frac{\nu '}{r}\right) , \end{aligned}$$(3.9)along with the conservation equation (2.4) with \(\alpha = 0\), namely \(\nabla _\nu \,T^{\mathrm{(m)}{\mu \nu }}=0\), yielding$$\begin{aligned} k^2\,p= & {} \frac{\mu }{4}\left( 2\nu ''+\nu '^2+\frac{2\nu '}{r}\right) +\frac{\mu '}{4}\left( \nu '+\frac{2}{r}\right) , \end{aligned}$$(3.10)which is a linear combination of Eqs. (3.8)–(3.10).$$\begin{aligned} p'+\frac{\nu '}{2}\left( \rho +p\right) = 0 , \end{aligned}$$(3.11)
- (ii)The second set contains the source \(\theta _{\mu \nu }\) and reads$$\begin{aligned} k^2\,\theta _0^{\,0}= & {} -\frac{f^{*}}{r^2} -\frac{f^{*'}}{r} , \end{aligned}$$(3.12)$$\begin{aligned} k^2\,\theta _1^{\,1}= & {} -f^{*}\left( \frac{1}{r^2}+\frac{\nu '}{r}\right) , \end{aligned}$$(3.13)The conservation equation (2.4) then yields \(\nabla _\nu \,\theta ^{\mu \nu }=0\), which explicitly reads$$\begin{aligned} k^2\,\theta _2^{\,2}= & {} -\frac{f^{*}}{4}\left( 2\nu ''+\nu '^2+2\frac{\nu '}{r}\right) \nonumber \\&\quad -\frac{f^{*'}}{4}\left( \nu '+\frac{2}{r}\right) . \end{aligned}$$(3.14)Equations (3.11) and (3.15) simply mean that there is no exchange of energy-momentum between the perfect fluid and the source \(\theta _{\mu \nu }\), so that their interaction is purely gravitational.$$\begin{aligned} \left( \theta _1^{\,\,1}\right) ' -\frac{\nu '}{2}\left( \theta _0^{\,\,0}-\theta _1^{\,\,1}\right) -\frac{2}{r}\left( \theta _2^{\,\,2}-\theta _1^{\,\,1}\right) = 0. \end{aligned}$$(3.15)

As was pointed out in Ref. [1], since Eqs. (3.12) and (3.13) do not contain the standard Einstein tensor components, we should expect that the conservation equation (3.15) for the source \(\theta _{\mu \nu }\) is no longer a linear combination of Eqs. (3.12)–(3.14). However, Eq. (3.15) still remains a linear combination of the system (3.12)–(3.14). The MGD approach therefore turns the indefinite system (2.6)–(2.8) into the set of equations for a perfect fluid \(\{\rho ,p,\nu ,\mu \}\) plus a simpler system of four unknown functions \(\{f^{*},\,\theta _0^{\,0},\,\theta _1^{\,1},\,\theta _2^{\,2}\}\) satisfying the three equations (3.12)–(3.14) [at this stage we suppose that we have already found a perfect fluid solution, thus \(\nu \) is determined], or the equivalent anisotropic system of equations (3.17)–(3.19). Either way, the system (2.6)–(2.8) has been successfully decoupled.

*n*other gravitational sources \(T_{\mu \nu }^{(i)}\), namely

*n*quasi-Einstein equations for the sources \(T_{\mu \nu }^{(i)}\), namely

## 4 Matching condition for stellar distributions

*m*given by the standard GR expression in Eq. (3.2) and \(f^{*}\) the yet to be determined MGD in Eq. (3.7).

## 5 Interior solutions

*A*,

*B*and

*C*in Eqs. (5.1)–(5.4) are determined from the matching conditions in Eqs. (4.5) and (4.8) between the above interior solution and the exterior metric which we choose to be the Schwarzschild space-time. This yields

### 5.1 Solution I: mimic constraint for the pressure

Now let us match our interior metric in Eq. (2.5) with metric functions (5.1) and (5.9) with the exterior Schwarzschild solution (4.14) with \(g^{*}(r)=0\). We can see that, for a given distribution of mass \(M_0\) and radius *R*, we have four unknown parameters, namely \(\{A, B, C\}\) from the interior solution in Eqs. (5.1) and (5.9), and the mass \(\mathcal{M}\) in Eq. (4.14). Since we have only the three matching conditions (4.6), (4.7) and (4.11) at the surface of the star, the problem is not closed. An obvious solution would be to set \(B=1\) in Eq. (5.1), corresponding to the time rescaling \(t\rightarrow \tilde{t}=B\,t\). However, we want to keep *B* near its expression in the Tolman IV solution of Eqs. (5.1)–(5.4) in order to see clearly the effect of the anisotropic source \(\theta _{\mu \nu }\) on the perfect fluid. We will therefore solve for \(\{A, C, \mathcal{M}\}\) with respect to *B* as shown further below.

*p*(

*r*) in Eq. (5.4). On the other hand, the effective density and effective tangential pressure are given, respectively, by

*B*as a free parameter around the value in Eq. (5.5), in order to obtain the effective radial pressure \(\tilde{p}_r(r,\alpha )\) in Eq. (5.17) shown in Fig. 2 for two values of \(\alpha \) and \(B^2=2/5\). It appears that the anisotropy produced by \(\theta _{\mu \nu }\) decreases the effective radial pressure more and more for increasing \(\alpha \).

### 5.2 Solution II: mimic constraint for density

*B*as a free parameter with values around the expression in Eq. (5.5). Figure 3 shows the radial pressure \(\tilde{p}_r(r,\alpha )\) and tangential pressure \(\tilde{p}_t(r,\alpha )\) inside the stellar distribution, showing how the anisotropy \(\Pi (r,\alpha )\) in Eq. (5.31) increases towards the surface.

### 5.3 Solution III: extending anisotropic solutions

So far we have seen how to generate exact and physically acceptable anisotropic solutions starting from a known isotropic solution. In order to emphasise the full potential of the MGD-decoupling, schematically represented by Eqs. (3.22)–(3.28), we will use the anisotropic solution \(\{\nu ,\lambda ,\tilde{\rho },\tilde{p}_r,\tilde{p}_t\}\) from Sect. 5.2, and explicitly given by Eqs. (5.1), (5.24), (5.28)–(5.30), to generate a third anisotropic and physically acceptable solution \(\{\nu ,\lambda ,\bar{\rho },\bar{p}_r,\bar{p}_t\}\).

## 6 Conclusions

By using the MGD-decoupling approach, we presented in detail how to extend interior isotropic solutions for self-gravitating systems in order to include anisotropic (but still spherically symmetric) gravitational sources. For this purpose, we showed that the Einstein field equations for a static and spherically symmetric self-gravitating system in Eqs. (2.6)–(2.8) can be decoupled in two sectors, namely: the isotropic sector corresponding to a perfect fluid \({\hat{T}}_{\mu \nu }\) shown in Eqs. (3.8)–(3.10), and the sector described by quasi-Einstein field equations associated with an anisotropic source \(\theta _{\mu \nu }\) shown in Eqs. (3.12)–(3.14). These two sectors must interact only gravitationally, without direct exchange of energy-momentum.

The matching conditions at the stellar surface were then studied in detail for an outer Schwarzschild space-time. In particular, the continuity of the second fundamental form in Eq. (4.15) was shown to yield the important result that the effective radial pressure \(\tilde{p}_R=0\). The effective pressure (2.11) contains both the isotropic pressure of the undeformed matter source \({\hat{T}}_{\mu \nu }\) and the inner geometric deformation \(f^*(r)\) induced by the energy-momentum \(\theta _{\mu \nu }\). We recall the latter could also represent a specific matter source, like a Klein–Gordon scalar field or any other form of matter-energy, but also the induced effects of extra-dimensions in the brane-world. If the geometric deformation \(f^*(r)\) is positive and therefore weakens the gravitational field [see Eq. (4.2)], an outer Schwarzschild vacuum can only be supported if the isotropic \(p_{R}<0\) at the surface of the star. This can in fact be interpreted as regular matter with a solid crust [12] as long as the region with negative pressure does not extend too deep into the star.

In order to show the robustness of our approach, three new exact and physically acceptable interior anisotropic solutions to the Einstein field equations were generated from a single perfect fluid solution. All these new solutions inherit their physical acceptability from the original isotropic solution. In particular, it was shown that the anisotropic source \(\theta _{\mu \nu }\) always reduces the isotropic radial pressure \(\tilde{p}(r)\) inside the self-gravitating system.

We would like to remark that the MGD-decoupling is not just a technique for developing physically acceptable anisotropic solutions in GR, but represents a powerful and efficient way to exploit the gravitational decoupling in relevant physical problems. The extension of GR solutions into the domain of more complex gravitational sources is a highly non-trivial theoretical problem. For instance, it is well known that a Klein–Gordon scalar field induces anisotropic effects when it is coupled with the gravitational field through the Einstein equations. Hence the MGD-decoupling represents a useful tool for extending GR isotropic solutions for self-gravitating systems into solutions of the Einstein–Klein–Gordon system. It could be implemented, for instance, to investigate the role played by a Klein–Gordon scalar field during the gravitational collapse. In this respect, it is worth mentioning that the MGD-decoupling can be generalised for time-dependent scenarios, as long as the spherical symmetry is preserved under slowly evolving situations, which means the stellar system is always in hydrostatic equilibrium [45].

## Notes

### Acknowledgements

J.O. is supported by Institute of Physics and Research Centre of Theoretical Physics and Astrophysics, Silesian University in Opava. R.C. is partially supported by the INFN grant FLAG. R.dR. is grateful to CNPq (Grant No. 303293/2015-2), and to FAPESP (Grant No. 2017/18897-8), for partial financial support. A.S. is partially supported by Project Fondecyt 1161192, Chile.

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