# Isotropization and change of complexity by gravitational decoupling

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## Abstract

We employ the gravitational decoupling approach for static and spherically symmetric systems to develop a simple and powerful method in order to (a) continuously isotropize any anisotropic solution of the Einstein field equations, and (b) generate new solutions for self-gravitating distributions with the same or vanishing complexity factor. A few working examples are given for illustrative purposes.

## 1 Introduction

^{1}

In fact, the GD is a generalization of the minimal geometric deformation which was developed in Refs. [3, 4] in the context of the Randall-Sundrum brane-world [5, 6], where the geometric deformation is induced by the existence of extra spatial dimensions and \(\alpha \) is naturally proportional to the inverse of the brane tension [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] (for some resent applications see also [20, 21, 22, 23]). The main applications of this approach so far [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48] were to build new solutions of Eq. (1) with \(\alpha =1\) starting from known solutions generated by \(T_{\mu \nu }\) alone (that is, with \(\alpha =0\)). In order to complete this construction, one needs to make assumptions about the second source, for instance by fixing the equation of state for the tensor \(\theta _{\mu \nu }\) (for the application of the GD beyond general relativity, see for instance Refs. [49, 50]).

In this paper we are instead interested in the different purpose of showing that the GD can be used to directly control specific physical properties of a self-gravitating system. For the sake of simplicity, we shall employ the minimal geometric deformation (MGD) in which only the radial component of the metric is modified and there is no direct exchange of energy between the two energy–momentum tensors in Eq. (2). We shall then require that the complete system (for \(\alpha =1\)) enjoys specific properties, equal or different from those of the case \(\alpha =0\). In particular, we shall first require that the anisotropic pressure for \(\alpha =0\) becomes isotropic for \(\alpha =1\) in Sect. 3 and impose conditions on the complexity factor which was recently introduced in Ref. [51] in Sect. 4. It is important to remark that the MGD does not involve any perturbative expansion and all results will be exact for all values of \(\alpha \). Finally we summarise our conclusions in Sect. 5.

## 2 Gravitational decoupling of Einstein’s equations

*r*only and \(d\Omega \) denotes the usual solid angle measure. The metric (5) must satisfy the Einstein equations (3) which, in terms of the two sources in (2), explicitly read

## 3 Isotropization of compact sources

*anisotropic*system (18)–(21) generated by \(T_{\mu \nu }\) with \(\Pi \not =0\) which is transformed into the

*isotropic*system (6)–(9) with \(\tilde{\Pi }=0\) as a consequence of adding the source \(\theta _{\mu \nu }\). This change can be formally controlled by varying the parameter \(\alpha \), with \(\alpha =0\) representing the anisotropic system (18)–(21), and \(\alpha =1\) representing the isotropic system (6)–(9), for which \(\tilde{\Pi }=0\), or

*A*and

*B*can be determined from the matching conditions between this interior solution and the exterior metric for \(r>R\). If we assume the exterior is the Schwarzschild vacuum,

*A*and

*B*remain the same as shown in (36). The deformation (41) generates an effective density

## 4 Complexity of compact sources

The notion of complexity for static and spherically symmetric self-gravitating systems we are interested in here was introduced recently in Ref. [51], and further extended to the dynamical case in Ref. [54] (for some applications, see e.g. Refs. [55, 56]). The main characteristic of this notion is that it assigns a zero value of the complexity factor to uniform and isotropic distributions (the least complex system).

*r*, and it has a clear physical interpretation. In fact, we recall that we can write the Tolman mass as a function of the metric components in Eq. (5) as

*R*. Hence, we see that \(Y_{\mathrm{TF}}\) can quantify the departure of the Tolman mass \(m_{\mathrm{T}}\) of a given system from the Tolman mass \(M_{\mathrm{T}}\) of a uniform isotropic fluid when the anisotropy and density gradient do not vanish. It is in fact clear from Eq. (47) that a uniform isotropic fluid will have zero complexity factor, but this does not mean that a stellar configuration with vanishing complexity factor is uniform and isotropic.

### 4.1 Two systems with the same complexity factor

*A*,

*B*and

*C*are again determined from the matching conditions (33)–(35), which yield the same values (36) and

*A*,

*B*and

*C*in the solution (57) and (64)–(67), we find that

*A*and

*B*have the same expressions as shown in Eq. (36), but

*C*is promoted to a function of the anisotropic parameter \(\alpha \) (and the length \(\ell \)), namely

### 4.2 Generating solutions with zero complexity

^{2}Using Eqs. (22)–(24) in the condition (71), we obtain the first order differential equation for the geometric deformation

*C*via the matching conditions like in the previous case.

*A*and

*B*shown in Eq. (36), while

*C*is now given by

*M*and

*R*, and therefore for any values of \(\ell \): we have mapped the Tolman IV fluid of given mass

*M*, radius

*R*and complexity (62) into a whole family of systems with the same mass

*M*and radius

*R*but vanishing complexity parametrized by the length scale \(\ell \).

## 5 Conclusions

The GD approach is a very effective way to investigate self-gravitating systems with sources described by more than one (spherically symmetric) energy–momentum tensor. Given an exact solution generated by one of such sources, it will allow one to obtain exact solutions with more sources. In most of the previous papers, new solutions were obtained by assuming particular equations of state for the added energy–momentum tensors, or field equations for the added matter sources. In this work we have instead considered the different task of employing the GD in order to impose specific physical properties satisfied by the whole system.

In order to keep the presentation simpler, we just considered two energy–momentum tensors and the MGD in which only the radial component of the metric is modified, although the approach can be straightforwardly generalised to more sources and to the GD in which the time component of the metric is deformed as well. The specific properties we required were isotropic pressure starting from the anisotropic solution (28)–(32) and control over the complexity factor starting from the Tolman IV solution (57)–(60). The examples we provided are mostly meant to illustrate the flexibility and effectiveness of our procedure and different physical requirements could indeed be demanded.

## Footnotes

## Notes

### Acknowledgements

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 em Cantata. J. O. thanks Luis Herrera for useful discussion and comments. J. O. acknowledges the support of the Institute of Physics and Research Centre of Theoretical Physics and Astrophysics, at the Silesian University in Opava. A. S. is partially supported by Project Fondecyt 1161192 and MINEDUC-UA project, code ANT 1855. S. Z. has been supported by the Czech Science Agency Grant no. 19-03950S.

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