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Eur. Phys. J. C (2019) 79:486

https://doi.org/10.1140/epjc/s10052-019-7022-y

Regular Article - Theoretical Physics

A causal Schwarzschild-de Sitter interior solution by gravitational

decoupling

L. Gabbanelli

1,a

, J. Ovalle

2,3,b

, A. Sotomayor

4,c

, Z. Stuchlik

2,d

, R. Casadio

5,6,e

Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1,

08028 Barcelona, Spain

Institute of Physics and Research Centre of Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University

in Opava, 746 01 Opava, Czech Republic

Departamento de Física, Universidad Simón Bolívar, AP 89000, Caracas 1080, Venezuela

Departamento de Matemáticas, Universidad de Antofagasta, Antofagasta, Chile

Dipartimento di Fisica e Astronomia, Alma Mater Università di Bologna, via Irnerio 46, 40126 Bologna, Italy

Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, I.S. FLAG, viale Berti Pichat 6/2, 40127 Bologna, Italy

Received: 24 April 2019 / Accepted: 3 June 2019 / Published online: 11 June 2019

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 cosmo-

logical 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 astrophys-

ical object.

1 Introduction

The Schwarzschild interior metric is one of the best known

solutions of Einstein’s ﬁeld equations [1]. This exact solu-

tion represents an isotropic self-gravitating object of uni-

form density (incompressible ﬂuid), and has been widely

studied, generally without considering the cosmological con-

stant. As far as we know, it is one of the few analytic solu-

tions 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

e-mail: gabbanelli@icc.ub.edu

e-mail: jovalle@usb.ve

e-mail: adrian.sotomayor@uantof.cl

e-mail: zdenek.stuchlik@fpf.slu.cz

e-mail: casadio@bo.infn.it

use of this solution for both positive and negative valuesofthe

cosmological constant. Such analyses can help to elucidate

some properties of the Schwarzschild-(anti-)de Sitter space-

time in presence of matter [4,5](seealso[6–8]). 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 stel-

lar 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. For-

tunately, the so-called method of gravitational decoupling by

Minimal Geometric Deformation (MGD-decoupling, hence-

forth) [15,16], which has been widely used recently [17–32],

has proved to be a powerful method to extend known solu-

tions into more complex scenarios.

The original version of the MGD approach was devel-

oped in Refs. [33,34] in the context of the brane-world sce-

nario [35,36], and it was eventually extended to study black

hole solutions in Refs. [37,38] (for some earlier works on

the MGD, see for instance Refs. [39–42], and Refs. [43–52]

for some recent applications). On the other hand, the MGD-

decoupling has three main characteristics that make it par-

ticularly useful in the search for new solutions of Einstein’s

ﬁeld equations, namely:

123

486 Page 2 of 9 Eur. Phys. J. C (2019) 79 :486

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

μν

for which the

metric is known and add the energy-momentum tensor

of a second source,

μν

→ T

μν

+ T

(1)

μν

. (1.1)

We can then repeat the process with more sources T

(i)

μν

extend the solution of the Einstein equations associated

with the gravitational source

μν

into the domain of more

intricate forms of gravitational sources T

μν

;

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

μν

by splitting it into simpler

components,

μν

→

μν

+ T

(i)

μν

, (1.2)

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

μν

. Finally, by a simple combination of all these solu-

tions, we will obtain the solution to the Einstein equations

associated with the original energy-momentum tensor

μν

. We emphasise that the MGD-decoupling works as

long as the sources do not exchange energy-momentum

directly among them, to wit

∇

μν

=∇

(1)μν

=···=∇

(n)μν

= 0, (1.3)

which further clariﬁes that their interaction is purely

gravitational;

III. it can be applied to theories beyond general relativity.

For instance, given the modiﬁed action [16]

= S

+ S





2 k

+ L



√

−gd

(1.4)

where L

contains all matter ﬁelds in the theory and L

is the Lagrangian density of a new gravitational sector

with an associated (effective) energy-momentum tensor

μν

√

−g

δ(

√

−g L

)

δg

μν

= 2

δL

δg

μν

− g

μν

, (1.5)

the method in I. allows one to extend all the known

solutions of the Einstein-Hilbert action S

into the

domain of modiﬁed gravity represented by S

. This rep-

resents a straightforward way to study the consequences

of extended gravity on general relativity.

In this paper we will apply the procedure I. to the Schwarzsc-

hild interior solution in order to build a new interior con-

ﬁguration with non-uniform matter density and anisotropic

pressure.

The paper is organised as follows: in Sect. 2,westart

from the Einstein equations with cosmological constant for a

spherically symmetric stellar distribution and we show how

to decoupling two spherically symmetric and static gravi-

tational sources {T

μν

,θ

μν

}. 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 con-

clusions in Sect. 4.

2 Spherically symmetric stellar distribution

Let us start from the standard Einstein ﬁeld equations

μν

−

μν

+  g

μν

= k

μν

, (2.1)

where  is a positive cosmological constant. The energy-

momentum tensor T

μν

in Eq. (2.1) is given by

μν

+ θ

μν

, (2.2)

where

μν

represents the energy-momentum tensor of a per-

fect ﬂuid, and θ

μν

adds anisotropic effects on T

μν

. Since the

Einstein tensor is divergence free, the total energy momen-

tum tensor T

μν

must satisfy the conservation equation

∇

μν

= 0. (2.3)

In Schwarzschild-likecoordinates, the spherically symmetric

metric reads

= e

ν(r )

−e

λ(r)

−r



dθ

+ sin

θ dφ



, (2.4)

where ν = ν(r) and λ = λ(r) are functions of the areal

radius 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

vac

=−p

vac

= /k

. Explicitly, the ﬁeld equations read



ρ + θ



+  =

− e

−λ



−





, (2.5)



−p + θ



+  =

− e

−λ







, (2.6)

We use the metric signature (+−−−) and the constant k

= 8 π G

123

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