#### Gravitational decoupled anisotropies in compact stars

Cr
Gravitational decoupled anisotropies in compact stars
Luciano Gabbanelli 1 2 3
0 Pontificia Universidad Católica de Chile , Av. Vicuña Mackenna 4860, Santiago , Chile
1 , Carlos Rubio
2 Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona , Martí i Franquès 1, 08028 Barcelona , Spain
3 , Ángel Rincón
Simple generic extensions of isotropic DurgapalFuloria stars to the anisotropic domain are presented. These anisotropic solutions are obtained by guided minimal deformations over the isotropic system. When the anisotropic sector interacts in a purely gravitational manner, the conditions to decouple both sectors by means of the minimal geometric deformation approach are satisfied. Hence the anisotropic field equations are isolated resulting a more treatable set. The simplicity of the equations allows one to manipulate the anisotropies that can be implemented in a systematic way to obtain different realistic models for anisotropic configurations. Later on, observational effects of such anisotropies when measuring the surface redshift are discussed. To conclude, the consistency of the application of the method over the obtained anisotropic configurations is shown. In this manner, different anisotropic sectors can be isolated of each other and modeled in a simple and systematic way.
1 Introduction
The study of analytical solutions of Einstein field equations
plays a crucial role in the discovery and understanding of
relativistic phenomena. Theoretical grounds provide many
exact and not exact isotropic solutions in general relativity;
however most of them have no physical relevance and do
not pass elementary tests of astrophysical observations [1–
4]. Theoretical grounds gives very few isotropic solutions
under static and spherically symmetric assumptions. Worse
yet, even fewer of these solutions have physical relevance
passing elementary tests of astrophysical observations [1–4].
Furthermore, no astronomical object is constituted of perfect
fluids only; hence isotropic approximation is likely to fail.
Anisotropic configurations have continuously attracted
interest and are still an active field of research. Strong
evidence suggests that a variety of very interesting
physical phenomena gives rise to a quite large number of local
anisotropies, either for low or high density regimes (see
[5] and references therein). For instance, among high
density regimes, there are highly compact astrophysical objects
with core densities ever higher than nuclear density (∼
3 × 1017 kg/m3) that may exhibit an anisotropic behaviour
[6]. Certainly, the nuclear interactions of these objects must
be treated relativistically. The anisotropic behaviour is
produced when the standard pressure is split in two different
contributions: the radial pressure pr and the transverse
pressure pt , which are not likely to coincide.
Anisotropies in fluid pressure usually arise due to the
presence of a mixture of fluids of different types, rotation,
viscosity, the existence of a solid core, the presence of a
superfluid or a magnetic field [7]. Even are produced by some
kind of phase transitions or pion condensation among others
[8, 9]. The sources of anisotropies have been widely
studied in the literature, particularly for different highly compact
astrophysical objects such as compact stars or black holes,
either in four dimensions [10, 11] as well as in the context of
braneworld solution in higher dimensions [12–14].
The main purpose of the present article is to
generalize anisotropic analogous solutions of a particular kind of
isotropic compact object by means of the so-called
minimal geometric deformation approach [15–17] (MGD
hereinafter). This method was originally proposed in the
context of the Randall–Sundrum braneworld [18, 19] and was
designed to deform the standard Schwarzschild solution
[20, 21]. It describes the effective 4D geometry of a
spherically symmetric stellar distribution with a physically
admissible anisotropic behaviour produced by bulk corrections over
the braneworld. The details of this method will be shown
later, however the main lines goes as follows: let us start with
a well known spherically symmetric gravitational source Tμ(0ν).
This source can be as simple as one would desire; one can
start with any known perfect fluid or even with vacuum itself.
Any classical solution works as a seed for this method. After
this, one switch on a new source of anisotropy
(
1
)
Tμν = Tμ(0ν) + α Tμ(1ν).
When the sources are couple via gravity only, i.e. they do not
exchange energy–momentum among each other, the set of
equations is split into two. On one side, the field equations for
the chosen sector which are well known. In order to contribute
to different proposals about anisotropic superdense compact
stars [22,23], we studied a deformation over the Durgapal–
Fuloria solution [24]. On the other side, one is left with a
simpler set of ‘pseudo-Einstein’ equations for the sources
of the anisotropy to be solved. Combining both sectors a
full anisotropic and physically consistent solution of Einstein
field equations is obtained. Of course one can switch on as
many arbitrary sources of anisotropies Tμ(iν) as desired, as long
as a strategy to solve the new sector can be found. Thus, the
decoupling of many gravitational sources can be done in a
simple and systematic way establishing a new window to
search for new families of anisotropic solutions of Einstein
field equations.
This method for decoupling non-linear differential
equations can be applied in a systematic way and has a vast
unexplored territory where it could give different novel
perspectives. MGD does not only give consistent interior solutions
for different perfect fluids in GR; it could also be
conveniently exploited in relevant theories such as f (R)–gravity
[25–27], intrinsically anisotropic theories as Horˇava–aether
gravity [28] or to study the stability of novel proposals of
Black Holes, described by Bose Einstein gravitational
condensate systems of gravitons [29–31]. This is a robust method
to extend physical solutions into an anisotropic domain
preserving the physical acceptability.
The paper is organized as follows: after this introduction,
we present the Einstein field equations for an anisotropic
fluid. In Sect. 3 we explain how the MGD approach is
implemented to generate arbitrary anisotropic solutions. Section
4 is dedicated to apply this method to a particular seed, the
Durgapal–Fuloria model for compact stars. We present some
physical anisotropic solutions and discus possible
observational effects. In Sect. 5 we extend the method to seeds which
are already anisotropic. The last two sections are dedicated
to discuss the main results and summarize our conclusions.
2 Anisotropic effective field equations
The simplest approach to describe compact distributions
modeling stellar structures, is to restrict the metric to be static
and spherically symmetric. In the Schwarzschild-like
coordinates the line element takes the standard form
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
(
7
)
ds2 = eν dt 2 − eλ dr 2 − r 2 (dθ 2 + sin2 θ dφ2);
where the functions ν ≡ ν(r ) and λ ≡ λ(r ) depend on
the radial coordinate only. The encoded metric is a generic
solution of the Einstein field equations
1
Rμν − 2 R gμν = κ Tμν ,
describing an anisotropic fluid sphere. The coupling constant
between matter is given by κ = 8π G/c4. Along these lines
we will work in relativistic geometrized units, G = c = 1.
The observable features of the object will be determined by
the exterior metric that will describe the geometry of the
outer part. In the present article to maintain the treatment as
simpler as possible, we will suppose a Schwarzschild vacuum
outside.
The corresponding anisotropic effective stress–energy
tensor Tμν is characterized by its diagonal components ρ,
pr and pt , that are related to the geometric functions μ, ν
through Eq. (
3
). Explicitly,
1 λ
r 2 − r
,
2 ν + ν 2 − λ ν + 2
ν − λ
r
.
The prime stand for derivatives w.r.t. r . There is another
equation consequence of the Bianchi identities: the covariant
conservation of the stress–energy tensor
Since the discovery of the first interior stellar solution by
Schwarzschild [32] and for several years, stars interior were
supposed to be constituted by perfect fluids. It was not until
1933 when Lemaître [33] develop that spherically
symmetry do not require the isotropic condition pr = pt , but only
the equality of the two tangential pressures pθ = pφ = pt .
The system of Eqs. (
4
)–(
7
) governs the matter distribution
within the star, which is assumed to be locally anisotropic
(the radial and tangential pressure do not coincide). It is
necessary to solve for five unknowns functions: two geometric
functions, ν(r ) and λ(r ); and three effective scalar functions,
ρ(r ), pr (r ) and pt (r ). However there are more unknowns
than equations, hence the system is undetermined and
constrains must be imposed. Some of them must be chosen by
consistency of regularity, stability and (or) energy conditions
of relativistic models; see for instance [5,7,34].
Throughout this article we will make use of the following
representation for the effective energy–momentum tensor
Tμν = Tμ(PνF) + α θμν .
The first term encodes a perfect fluid with isotropic pressure
p = pr = pt ,
Tμ(PνF) = (ρ + p) uμuν − p gμν .
uμ is the normalized four-velocity field that accomplish
uμuν gμν = 1. In our case, the perfect fluid will invariably
be given by the Durgapal–Fuloria interior solution. Under
this representation, the anisotropic sector is described by the
θ -term. It describes additional gravitational sources
responsible of the anisotropies. These source may contain new fields,
whether scalar, vector or (and) tensor fields, coupled to
gravity by means of a free dimensionless and constant parameter
α. One of the simplest and most treated examples in the
literature are the anisotropies that may arise due to extra
interactions resulting from the presence of charge [35]; besides there
are plenty of complex treatments of anisotropies generated
by other sophisticated physical fields [36].
The effective stress–energy tensor (
8
) contributes at the
level of Einstein equations with an effective energy density
ρ, an effective radial pressure pr and an effective tangential
pressure pt defined as
ρ = ρ + α θt t ,
pr = p − α θr r ,
pt = p − α θϕ ϕ .
Thus, each magnitude is written as a deviation from the GR
solution due to the presence of the θ -term. The additive
structure for the anisotropies allows the theory to have a
straightforward limit to GR; setting α = 0 the standard Einstein
equations for the perfect fluid are recovered.
Since the Einstein tensor is divergence free, under the
representation taken in Eq. (
8
) the covariant conservation
equation (
7
) yields
1
p + 2 ν (ρ + p)
− α (θr r ) + 21 ν (θr r − θt t ) + r2
θr r − θϕ ϕ
= 0.
This equation is a linear combination of Eqs. (
4
) and (
6
),
as commonly happens in perfect fluid solutions of Einstein
equations.
As this point let us remark the appearance of the
anisotropy: there is not an a priori restriction for the
components of θμν ; however, if θr r = θϕ ϕ when solving the
equation system (
4
)–(
6
), we will be in the presence of the
pressure anisotropy
Π ≡ pt − pr = α θr r − θϕ ϕ .
(
8
)
(
9
)
(
10
)
(11)
(12)
(13)
(14)
Therefore, an isotropic stellar distribution (perfect fluid)
becomes anisotropic when the θ -term is turned on. In these
lines we will follow a different approach to tackle the
equation system (
4
)–(
6
); we will address this system by means of
the MGD method. This theory decouples the Einstein field
equations when deforming the metric of the corresponding
GR solution [15,16,37,38].
3 Minimal geometric deformation approach
With the aim of approaching the system of equations (
4
)–(
6
)
in an alternative manner, a briefly review on the MGD
procedure will be presented. This method produces anisotropic
corrections to standard GR solutions providing physically
admissible non-uniform and spherically symmetric stellar
distributions. The input (seed) is a known solution of
Einstein equations: for instance the thermodynamic parameters
satisfying Eq. (
9
), and the corresponding geometric
functions λ(r ) and ν(r ). When a perfect fluid solution is taken
as a seed, the isotropic condition pr = pt = p is
automatically accomplished. The method will produce a drift in
the effective pressures such that pr = pt . For doing so, one
implements the most generic minimal geometric deformation
over the metric without breaking the spherical symmetry of
the initial solution; this is
e+ν(r)
with e∗ and f ∗ generic functions parametrizing the metric
deformation. In Fig. 1 a schematic picture exemplifies how
this method extends GR solutions to anisotropic domains
when releasing α. Even though the theory does not impose
(15)
(16)
limits for the coupling strength, the physical acceptability of
the new solution does so; if α is increased, the anisotropies
become at some point unstable.
Although nothing prevent us from deforming the temporal
component of the metric, it is general enough to start setting
e∗ = 0; hence the effects of the θμν source undergo in a
deformation over the radial coordinate only. The peculiarity of the
MGD method is that it entails in its formulation a decoupling
of the equations of motion. As a consequence of taking the
θ -sector as responsible of the minimal distortion of the
metric, the system of Eqs. (
4
)–(
6
) results quasi-decoupled: we
obtain the Einstein equations for the chosen perfect fluid; and
an effective ‘pseudo-Einstein’ system of equations
governing the θ -sector. The only parameter that connects the two
sectors is the temporal geometric function ν(r ). At the same
order as before, the temporal, radial and angular equations
of motion relating the geometry of the spacetime to the
thermodynamic characteristic of the perfect fluid sector reduce
to
1 μ μ
κ ρ = r 2 − r 2 − r ,
1
−κ p = r 2 − μ
1
−κ p = − 4 μ
,
The definition of a perfect fluid entails in itself the
covariant conservation of the stress–energy tensor, i.e.
∇ν Tμ(PνF) = 0.
1
p + 2 ν (ρ + p) = 0.
The resulting equation is again a linear combination of the
temporal and angular equations, (17) and (19), and yields
It is worth noting that this system of equations is equivalent
to Eqs. (
4
)–(
6
) if the coupling between the two sectors is set
to zero; this is if the anisotropic sector vanishes.
The temporal component of the metric must satisfy
binding conditions in the anisotropic sector: these are the
remaining ‘pseudo-Einstein’ field equations for the θ -sector
t f ∗
κ θt = − r 2 −
f ∗
r
,
κ θr r = − f ∗
1
κ θϕ ϕ = − 4
,
f ∗
Once again, one has the corresponding conservation
equation that is a consequence of Eqs. (
7
) and (20) satisfying
separately. This equation is
∇ν θμν = 0,
and it is explicitly written as
1 2
(θt t ) − 2 ν (θt t − θr r ) + r (θr r − θϕ ϕ ) = 0.
This time the latter equation is not necessary linearly
dependent of the ‘pseudo-Einstein’ equations, and there is no
reason why it should be. At this point it makes explicit that the
interaction between the two sectors is purely gravitational;
this is, from Eqs. (20) and (25) is clear that each sector is
separately conserved and there is no exchange of energy–
momentum between them.
To conclude this section, let us summarize. First we started
with an indeterminate system of Eqs. (
4
)–(
6
). Then, we
performed a linear mapping of the radial geometric function of
the metric (16) that results in a ‘decoupling’ of the Einstein
field equations. We ended with two sets of equations: a
perfect fluid sector {ρ; p; ν; μ}, given by Eqs. (17)–(21) where
everything is known once a perfect solution of GR is chosen;
and a simpler sector of three linearly independent equations
that can be chosen from (22) to (26), for determining four
unknown functions { f ∗; θt t ; θr r ; θϕ ϕ }. Once the second
sector is solved, we can identify directly the effective
physical quantities introduced in (
10
), (11) and (12). At this point,
is mandatory to recall that the underlying anisotropic effect
which appears as a consequence of breaking the isotropic
condition over the effective pressures, pt = pr , causes the
appearance of the anisotropy Π (α; r ) defined in Eq. (14).
4 Anisotropic Durgapal–Fuloria compact star
Let us proceed now to apply the MGD method with the aim
of solving the Einstein field equations for the interior of
anisotropic superdense stars. In the present work we will
take as a seed the well-known Durgapal–Fuloria solution
{ν; μ; ρ; p} modeling compact stars. As explained before,
once the MGD method is applied the system of equations (
4
)–
(
6
) is decoupled. Half of the decoupled equations (17)–(19)
are already solved once the relativistic perfect fluid is chosen.
For instance, the thermodynamic functions that characterize
the Durgapal–Fuloria solution are
ρ(r ) =
p(r ) =
C (9 + 2 Cr 2 + C 2r 4)
7 π(1 + Cr 2)3
2 C (2 − 7 Cr 2 − C 2r 4)
,
(25)
(26)
(
27
)
(
28
)
with C an integration constant. The gravitational mass of a
sphere of radius r is obtained integrating the density inside
the corresponding volume; in spherical coordinates is
m(r ) =
ρ dV =
V
This mass function has a well defined behaviour, vanishing
at the center of the compact object, i.e. m(r = 0) = 0. It also
determines the total mass evaluating the mass function at the
surface, MDF ≡ m(r = R).
A massive object deforms the surrounding spacetime; the
Durgapal–Fuloria solution is defined by the following metric
components
eν(r) = A 1 + Cr 2 4 ,
μ(r ) = 1 −
2 m(r )
r
It is a custom in GR to write the radial component of the
metric such that if it is evaluated at the surface r = R, we
obtain the so-called compactness parameter,
ξ =
2 MDF .
R
The spacetime results regular everywhere, even at the
center where eλ(r=0) ≡ μ(r = 0) = 1; m vanishes faster than
r as one can easily check from Eq. (
29
) inside Eq. (
31
).
A is the second (and last) integration constant to be
determined, both A and C , using boundary conditions over the
surface r = R. In the present article the outer metric will be
chosen to satisfy the Schwarzschild form—for simplicity, an
uncharged compact star. Extensions to more complex outer
spacetimes is straightforward. We can choose a Kerr metric
for obtaining a more realistic rotating object, or even switch
on a charge. Both effects are more complicated and
interesting sources of anisotropies that give rise to more realistic
scenarios. Nonetheless we will maintain the distribution in
vacuum to get the treatment as simple as possible. Both
constants A and C are positive; however they are expected to
change as far as anisotropies begin to be considered.
The remaining equations after the decoupling, Eqs. (22)–
(24), have to be solved if a generic anisotropic self-gravitating
system is desired. The system of equation is as explain before
underdetermined. A reasonable constrain is needed to close
the system, but it is mandatory not to lose the physical
acceptability of the solution. These issues will be discussed in what
follows when three different anisotropic solutions (of many)
are presented.
(
29
)
(
30
)
(
31
)
(
32
)
4.1 Pressure-like constraint for the anisotropy
In order to close the system of Eqs. (22)–(24), additional
information is needed. For instance, an equation of state for
the source θμν or some physically motivated constrain on
f ∗(r ). A first acceptable interior solution is deduced when
forcing the associated radial pressure θr r to mimic a
physically acceptable pressure
θr r (r ) = p(r ).
This means that one simple choice is to require that the
stress–energy tensor for the perfect fluid coincides with the
anisotropy in that direction. As a consequence of Eq. (
33
),
the radial Einstein equations for the GR solution (18) and the
radial ‘pseudo-Einstein’ equation (23) are equal. This gives
immediately an expression for the radial component metric
deformation
1
f ∗(r ) = −μ + 1 + r ν .
The temporal component of the metric (
30
) remains non–
deformed, so ν is computed directly. The resulting deformed
component, the radial one in Eq. (16), then becomes
1 + Cr 2
1 + 9 Cr 2
.
It is explicit that when the α → 0 limit is taken, one gets
the non-perturbed Durgapal–Fuloria solution; particularly
for the radial component of the metric, e−λ(r) = μ(r ).
With the above considerations, the metric can be written in
terms of an effective mass function of the anisotropic sphere
given by
r f ∗
.
m(r ) = m − α
2
Expressed in this form, it is obtain one branch of MGD
metrics that govern anisotropic interiors of GR solutions,
whatever the GR solution is chosen. This branch corresponds to
the pressure constrain imposed over the radial anisotropy.
Therefore, the metric (
2
) is deformed to
(
36
)
ds2 = eν dt 2 −
2 m
1 − r
−1
dr 2 − r 2 dΩ2.
(
37
)
As we have closed the system of equations with the
constrain (
33
), we can compute all the effective magnitudes that
characterized the fluid; but first, the values of the integration
constants A and C are needed to be fixed. This will be done
by means of consistent matching conditions.
4.1.1 Matching conditions
A crucial aspect in the study of stellar distributions is the
matching conditions at the star surface between the interior
and the exterior spacetime geometries [39]. In our case, the
(
33
)
(
34
)
(
35
)
cyan line) represent two anisotropic solutions. The second graph shows
a comparison between the radial and tangential pressure for α = 0.2.
The anisotropy causes the pressures values to drift apart
interior stellar geometry is given by the MGD metric (
37
).
The outer part is assumed to be empty; hence for r ≥ R
the solution is given by the Schwarzschild vacuum solution.
Between both, at the star surface Σ defined by r = R, the
Israel–Darmois junction conditions imply the metric and the
extrinsic curvature of Σ have to be continuous. The former
is the so-called first fundamental form
where the effective pressure comes from Eq. (11). On the
r.h.s. we are in vacuum, hence the pressure must nullify. The
equation system has been closed with the constrain (
33
),
therefore p(R) = 0 is equivalent to request p(R) = 0 in
Eq. (
28
). This equivalence makes the constant C not to vary
from the perfect fluid solution once the anisotropies are
considered. The value is
gμν Σ = 0
gtt r=R− = 1 −
grr
1
r=R− = 1 − 2 MRS+chw .
2 MSchw
R+
;
that yields the following two equations for each of the
relevant components
pr
r=R− =
The superindices stand for the region from where we
approach the surface, either from inside with a minus sign, or
from outside using the plus sign. The latter is the continuity
of the second fundamental form
[Gμν x ν ]Σ = 0;
x ν is a unit vector. If we make use of the field equations
(
3
), the continuity reads as [Tμν x ν ]Σ = 0. Using the full
stress–energy tensor (
8
) and projecting in the radial direction
xr = r , we have [(Tr(rP F) + α θrr ) r ]Σ = 0. This leads to
p − α θr r r=R− = 0;
(
38
)
(
39
)
(
40
)
(41)
(42)
With the constant fixed, we have fully determined the
effective radial pressure of the anisotropic Durgapal–Fuloria
solution
A natural bound is obtained, α < 1. In Fig. 2, it is shown
the dependence of the pressure with a dimensionless radial
coordinate r/R for different values of α. At first sight one can
observe that the higher α is, the smaller the radial pressure
becomes.
The decreasing of the radial pressure is needed to produce
the pressure anisotropy that is reflected in a change over the
tangential pressure along the surface. The expression for the
later pressure is written as
pt (r ; α) = p˜r + α
π 1 + Cr 2
.
(43)
(44)
(45)
The pressure (in both directions) must be a decreasing
function along the radial coordinate. This condition restricts even
more the values of α; higher values immediately triggers
instabilities. In light of what was written in Eq. (42), the
tangential pressure (45) determines another physical constrain
for α: this pressure is meaningful as long as it remains
positive everywhere pt (r ) > 0; hence, so must be α > 0 to not
contradict this statement in the surface where pr (R) = 0.
From the latter equation, the anisotropy is directly computed;
comparing with Eq. (14), we obtain
Π (r ; α) ≡ α
One can go on computing the remaining thermodynamic
parameters. For instance the density can be expressed
following Eq. (
10
) with the temporal component of the anisotropy
given by Eq. (22)
ρ(r ; α) = ρ + α
2 C
7π 1 + Cr 2 3 1 + 9 Cr 2 2
Some comments are pertinent. The Durgapal–Fuloria
solution is a fluid sphere with a solid crust. In Fig. 2 the
density shows a discontinuity in the surface. The anisotropy
smoothes this jump; the bigger the parameter α is, the lower
the value of the density on the surface of the star. This
behaviour immediately triggers the question on the profile of
the effective mass function. This parameter has been defined
in Eq. (
36
) and together with Eq. (
34
), is written as
m(r ; α) =
1 + α
2 (2 − 7 Cr 2 + C 2r 4)
(3 + Cr 2)(1 + 9 Cr 2)
An observer outside the star, would see a resulting mass
MSchw surrounded by vacuum. The continuity of the radial
component of the metric when crossing the star surface Σ ,
Eq. (
40
), is direct: Eq. (
37
) identifies the Schwarzschild mass
seen from outside with the effective mass of our solution; i.e.
MSchw ≡ m(R). Even more, a closer look at the mass
function shows that the correction (αr f ∗)/2 in Eq. (
36
) vanishes
at the surface r = R. This means that the effective total mass
of the star is the same as the isotropic total standard mass
(
29
); MDF = m(R) ≡ m(R).
This issue is not surprising at all. The anisotropy mimics
the radial pressure, hence the radial and tangential pressures
start to drift apart in the region close to the solid surface. For
this anisotropic behaviour to happen, both pressures must
decrease in magnitude at the inner region. Of course this
pressure discrepancy with respect to the isotropic solution makes
the density to be disturbed. The equilibrium between
gravm(r ).
(48)
r (1 − μ + f ∗) = K
⇒
f ∗(r ) = μ − 1
(51)
itational collapse and pressure repulsion is modified; hence
the mass function is redistributed to the center of the star.
Despite this, the total mass of the anisotropic object remains
unmodified and so as the compactness parameter ξ .
To conclude this section, let us determine the value of the
remaining constant A from Eq. (
39
). The temporal
component of the MGD metric (
30
) should match smoothly with
the outer Schwarzschild region
gtt r=R− = A(1 − C r 2) r=R− = 1 −
2 MSchw .
R+
The constant A remains unchanged. This constant close one
branch (α-dependent) of anisotropic solutions analogous to
Durgapal–Fuloria; namely {ν; λ; ρ; pr ; pt }. Of course, this
solution is not unique. Different anisotropic solutions can
be obtained starting from the Durgapal–Fuloria solution by
means of requiring different constrains when closing the
indeterminate system of equations. In next section we will
consider a different constrain and we will see that a different
anisotropic solution is obtained.
(49)
4.2 Density-like constrain for the anisotropy
Another useful constrain that gives an acceptable physical
solution, is to impose that the anisotropy ‘mimics’ a density.
This requirement is written as
θt t (r ) ≡ ρ(r )
and closes the system of Eqs. (22)–(24). The consequence
of this ansatz is direct, the temporal Einstein equation for
the perfect fluid (17) is identical to the temporal equation
of motion for the θμν -tensor (22). Equaling both equations,
one notes immediately that the resulting equation has a total
derivative structure. The integration is straightforward
(50)
where K = 0 must be imposted for the invariants R, Rμν Rμν
and Rμνγ σ Rμνγ σ to remain smooth and finite all over the
inner region. Note that with this constraint the radial
deformation is again totally determined by the solution of the
perfect fluid. Eventually, one computes the relevant component
of the metric; the minimally deformed component is written
as in Eq. (16) (naming β to the coupling between sectors) as
→ (1 + β) μ − β ≡ μ − β
We can write the metric with the structure used in Eq. (
37
).
The effective mass is written as a minimal deviation from the
GR mass m(r ) presented in Eq. (
29
)
8 Cr 2(3 + Cr 2)
7 (1 + Cr 2)2 . (52)
κ
m(r ) = m + β 2
ρ r 2dr = (1 + β) m.
(53)
In the latter equality, if we make use of spherical coordinates
and the corresponding relations, we have ρ r 2dr = m/Ω4;
(dashed red line) and β = − 0.3 (dotted cyan line) represent two
anisotropic solutions. The second set of curves shows the anisotropy
over the pressure, pr = pt for β = − 0.2
where the 4–dimensional solid angle is Ω4 = dΩ and
2 Ω4/κ = 1. Of course, this is not surprising at all, the
constrain for the anisotropy is to mimic the density, therefore the
effective mass mimics the mass being proportional one to the
other (unlike the previous case where the mass is exactly the
same with respect to the standard GR solution).
Once the system is closed and the minimal deformation
obtained, the remaining magnitudes are easily derived. As
before this will be done by means of the smooth matching
between the inner and outer region of the star.
4.2.1 Matching conditions
Here we will reproduce the same steps that we have done
before in order to find the constants A and C ; this time for
the density ansatz (50). It is already known that the constant
C is determined by the second fundamental form (42). Its
value is
In this case, the anisotropic sector has an influence on the
integration constant. It is explicitly seen that in the limit of
no coupling β → 0 the constant from Durgapal–Fuloria is
recovered.
Because of the ansatz where it has been required for the
anisotropy to mimic the density, the effective value of the
density is modified
(54)
(55)
ρ(r ; β) = (1 + β) ρ .
This is in complete accordance with the changes experienced
by the mass. In what follows we will show that this
solution only admits a minimal geometric deformation over the
metric in only one ‘direction’; the ‘direction’ to where the
density and the mass is increase. The anisotropies restricted
to the present constrain change the integration constants;
for instance C is β dependent. An analysis over Eq. (54)
shows that C increases when β becomes more negative.
This behaviour makes Eqs. (53) and (55) to increase when
β increase in modulus. Of course, theses both parameters
can not increase without a limit; as in the previous case,
anisotropies develop instabilities. In the first curve of Fig.
3 it is seen how the density function increases in the inner
region, while it slightly decreases its value over the surface’s
surroundings softening the crust. The mass function rises
throughout the interior and the total effective mass is also
increased.
The remaining thermodynamic parameters are the
effective radial pressure
pr (r ; β) = p − β
pressures are enhanced for the anisotropy to take place. When
the mass increases, the enhancement of the density requires
higher pressures for stability reasons.
In order to get some insight in the underlying sign for β in
the new solution, we will focus ourselves in the anisotropy,
given by
(58)
Π (r ; β) = −β π(1 + Cr 2)2
.
It is worth to note that this magnitude can not be negative (let
us remind that C > 0) because if this were so, over the surface
of the star where pr ( R) = 0, we would have a negative
tangential pressure pt ( R) < 0 which is not physically
acceptable. Therefore, positive pressures implies negative values
for β. Now we have a physical domain for β. The constant A
is found in an analogous manner than in the previous section,
i.e. by means of Eq. (49). The value of this constant changes
with β. The usual constant of Durgapal–Fuloria is recovered
in the limit of β → 0 as it should be.
4.3 Detectability and observational differences in
anisotropic distributions
One of the many remarkable predictions of the theory of
general relativity is the time dilation within a gravitational
well. This results in footprints in the lines of the spectrum
shifting towards the red. Although it is a useful quantity,
particularly for astronomers, which allows to get some insight
into compact stars physics, this effect is extremely difficult
to deal with because of its complexity to be disentangle from
the displacements and alterations due to various causes such
as the Doppler, Zeeman and pressure effects among others.
Theoretical derivations states that the redshift factor
associated to a star comes when relating the proper time τ of the
object with the observer clock t . This relation is given by the
standard formula dτ 2 = gtt dt 2 that yields the following for
the surface redshift
1
νe
1 + z = νo = √gtt ( R)
Therefore the relation between the emitted and observed
frequency makes the redshift manifest [40].
Over the years, the study of anisotropies in compact
objects has received considerable attention and this
parameter is a simple way to contrast theory with observations.
Formula (59) relates the measured redshift with the
compactness parameter of the star given by Eq. (
32
); in this case
written as
2 m( R)
ξ = R . (60)
Here m(r = R) is the effective total mass of the star that
could depend or not on the anisotropic coupling parameter α
or β. An observer from outside would see the Schwarzschild
metric, hence the redshift, that depends on ξ through the
total mass of the star. For simplicity, for anisotropic
contributions over fixed radius stars, we want to investigate how
this parameter evolve in our particular solutions.
Let us start with the first solution derived in Sect. 4.1.
The interest should focus in the mass function (48) evaluated
over the surface. As we have explained after this equation the
Schwarzschild mass remains unmodified with respect to the
Durgapal–Fuloria mass, this is MSchw ≡ MDF ( R) = m( R).
Hence, there is no observational evidence to differentiate an
isotropic star to these anisotropic counterpart.
On the contrary, things change in the second case. The
solution obtained in Sect. 4.1 is based on an increment of
the total mass. Then a shift occurs when observing this
anisotropic configuration
z(β) =
1 −
2MSchw(β) −1/2
R
− 1.
The Schwarzschild mass is the mass function over the
surface; i.e. MSchw = m( R) in Eq. (53). As it has been explained,
the mass increases when β increases in modulus.
Therefore, the compactness parameter is also increased; the star
becomes more and more denser with β. Then, the
parenthesis in Eq. (61) decrease and z(β) grows when β grows. This
means that this anisotropic contributions increases the
gravitational redshift as it is expected when the stars are more
dense.
(61)
5 Anisotropizing an anisotropic Durgapal–Fuloria star
In Sect. 3, we present a method to generate different
anisotropic solutions of Einstein field equations using any
well known perfect fluid as a seed. After this, we apply this
prescription to the Durgapal–Fuloria perfect sphere. In Sect.
4, with some reasonable constrains we found two novel
physical anisotropic solutions analogous to the Durgapal–Fuloria
compact star.
The decomposition of Einstein equations (
4
)–(
6
) stem on
the minimal geometric deformation (16); the anisotropic
sector (22)–(24) is decoupled with respect to any perfect fluid
sector (17)–(19). However, there is no need for the known
sector to be a perfect fluid solution exclusively. Whatever
solution of Einstein field equations, either a perfect or an
anisotropic fluid, work as a seed for implementing the MGD
decomposition. For instance, we can take any of the two
previous founded solutions; e.g. the one obtained in Sect. 4.1
given by {ν; μ; pr ; pt }. So as not to obscure how the method
works, we will minimally deform the anisotropic solution
along the radial component of the metric. While the
temporal geometric function in Eq. (
30
) remains unchanged, the
minimal distortion takes place only over the radial
component
e−λ(r)
→
e−λ¯(r) = μ(r ) + β g∗(r ).
of an anisotropic metric solution of Einstein equation; in
this case (
37
). This deformation is caused by new generic
sources of anisotropies (called ψμν in order to avoid
confusion with the deformed seed by θμν ) that acts over the
anisotropic energy momentum tensor (
8
)
T μν = Tμν + β ψμν .
The Einstein field equations connecting the latter effective
stress–energy tensor to the spacetime curvature are
1
κ ρ¯ = r 2 − e−λ¯
1
−κ p¯r = r 2 − e−λ¯
1
−κ p¯t = − 4 e−λ¯
1 λ¯
r 2 − r
,
,
2 ν + ν 2 − λ¯ ν + 2
ν − λ¯
r
Subsequently, a constrain over the solution must be imposed;
the system is indeterminate. Until now we used a constrain
that mimics the pressure to obtain the seed solution; a density
constrain will be applied now to combine both previously
found solutions. The ansatz then is to require
ψt t ≡ ρ(r ).
The minimal geometric deformation (62) decouples the
two anisotropic sectors. On the one hand, the seed sector
characterized by the density given by Eq. (47), the radial
pressure pr (44) and the tangential anisotropic pressure pt
obtained in Eq. (45). This parameters solve the already known
equation system
1 μ μ
κρ = r 2 − r 2 − r ,
1 1 ν
−κ pr = r 2 − μ r 2 + r
,
1
−κ pt = − 4 μ
and on the other hand, we are left with the new anisotropic
sector for ψμν completely decoupled. In this sector we have
the following ‘pseudo-Einstein’ equations
κ ψt t = −
κ ψr r = −g∗
1
κ ψϕ ϕ = − 4
g∗(r )
r 2
g∗ (r )
− r
1 ν
r 2 + r
,
,
g∗
Now the steps that follow are known. Equations (67) and
(70) equals and give a total derivative equivalent to Eq. (51).
The solution for the second minimal deformation function is
straightforward because the constant of integration is again
null
g∗ = μ − 1.
The radial metric component from Eq. (62) then promotes to
e−λ(r) = (1 − α)(1 + β)μ + α1(1++r νβ) − β; (75)
where the anisotropic radial component of the metric (
35
)
has been used.
The effective radial pressure p¯r = pr −βψr r is computed
from Eq. (65)
p¯r (r ; α, β) = [1 − α(1 + β)] p
− β
The first integration constant C is obtain by means of the
continuity of the second fundamental form, analogous condition
to Eq. (42). Imposing the annulment of the latter effective
radial pressure at the surface Σ , we get
C R2
−1
= 2 (1 − α) + β(9 − 2α)
7 [1 − α + β(2 − α)]
− [57(1 − α) + β(169 − 57α)] (1 − α)(1 + β) .
(74)
(76)
(77)
Both the integration constant C as well as the effective
pressure p¯r recover the corresponding values: Eqs. (43) and
(44) in the limit of β → 0 or Eqs. (54) and (56) when α → 0.
Besides, this constant is required to plot the thermodynamic
parameters.
In Fig. 4 we present the corresponding evolution of the
parameters of the theory: for a comparison, we include also
the Durgapal–Fuloria isotropic solution (α = β = 0 in solid
line). If for instance, one of the couplings move away from
zero but the other remains null, the thermodynamic
quantities behave as in Figs. 2 (if α = 0) or 3 (if β = 0), as it is
expected. After this, we plot the anisotropic seed to be
minimally deformed by fixing the coupling α (red dashed-line).
Finally, β drifts away the parameters again. We choose one
smaller order or magnitude for the second deformation to
make notorious the effect over the seed solution. An
important statement is that as the seed is anisotropic and the
corresponding tangential pressure is nonnull over the surface,
then there is no restriction for β to be negative. β is allowed
to be positive until either it decrease the tangential pressure
until it becomes null, or the anisotropy becomes unstable.
Lest we forget, we include the expression of the two
remaining parameters: first the effective density ρ¯ = ρ +
with ρ the seed density (47). And secondly, the anisotropic
tangential pressure p¯t = pt − βψϕ ϕ that can be written as
p¯t (r ; α, β) = p¯r (r ; α, β) + Π (r ; α, β).
The anisotropy is now written as
.
It is important to remark that in all expressions we recover
both previous limits when either α or β are set to zero, and
the standard Durgapal–Fuloria solution for α = β = 0. Let
us conclude presenting the profile of the anisotropy over the
surface Σ of the sphere. In Fig. 5, we plot the function Π
from Eq. (80) versus the couplings α and β. Nearby α closer
to zero and β positive, the anisotropy becomes unphysical
(Π < 0); thus deformations with this parameters are
prohibited and excluded from the physical surface. In particular this
procedure extends the range of physical values for β. Each
time α grows, new possible values of β > 0 are released.
(78)
(79)
(80)
Fig. 5 The figure illustrate the anisotropy Π as a function of the
couplings {α, β} evaluated at the star’s surface Σ. The region where α < 0.1
and β > 0, the anisotropy is unphysical
tion is minimally deformed by a ψ–sector. Two different solutions are
presented: dotted green line for {α = 0.3, β = 0.03} and the
dotteddashed cyan line for {α = 0.3, β = − 0.03}
Besides should be noted that the anisotropy can be
interpreted as a lineal combination of the first two studied cases
Π (r ; α, β) = Π (r ; β) + (1 + β)Π (r ; α);
Each single parameter anisotropy have been computed in
Eqs. (46) and (58). In this case, the resulting Π (α, β) no
longer coincides with β(ψr r − ψt t ) as mentioned in Eq. (14),
because the seed is no more an isotropic fluid. Immediately,
the linearity is translated to the stress–energy tensor; the
components of the new ψ -sector can be written as a combination
of the single minimal geometric deformations computed in
Sect. 4
ψμν = θ μ(dνensity) + α θ μ(pνressure).
making the ‘additive’ character of the method manifest. If
one starts with any perfect solution of GR, Tμ(PνF), a minimal
deformation induced by an anisotropy subjected to a pressure
structure, makes the stress energy tensor to become Tμν =
Tμ(PνF) + α θ μ(pνressure). After this, a second minimal deformation
acts over the already anisotropic solution, but now subjected
to a density constrain. The new contribution is given by Eq.
(82), therefore the effective energy–momentum tensor (63)
is decomposed as
T μν = Tμ(PνF) + α θ μ(pνressure)
+β θ μ(dνensity) + α θ μ(pνressure) .
(81)
(82)
(83)
This expression states the noncommutative structure of the
MGD-decoupling method; the order in which the
deformations take place matters. For instance, if we take as a seed
the solution found in Sect. 4.2 where the deformation obeys
a density–like constrain, and then we deform the anisotropic
solution with a different constrain analogous to Eq. (
33
), the
noncommutative character of the theory becomes manifest.
The Eqs. (81) and (82) change their form and become
Π (r ; β, α) = Π (r ; α) + (1 − α)Π (r ; β),
ψμν = θ μ(pνressure) − β θ μ(dνensity);
(84)
respectively. Likewise, the effective energy–momentum
tensor (83) becomes
T μν = Tμ(PνF) + β θ μ(dνensity) + α θ μ(pνressure) − β θ μ(dνensity)
(85)
when the order of the two anistropizations is reversed.
The reason of the noncommutativity is that the coefficients
involved in the linear combinations of the components of
the anisotropic tensor θμν depends explicitly on the coupling
constants. This is not surprising at all; we are dealing with
nonlinear differential equations. The commutativity is likely
to be lost in this kind of systems.
From a perturbation theory point of view, one can think
that the deformations over the metric (zero order) is due to
the existence of the anisotropic term which acts at O(α);
being α the coupling strength to the anisotropies. We must
emphasize that, although the MGD approach seems like a
perturbation technique, the method, in fact, is not, and this
is easily visualized by noticing that the couplings do not
necessarily have to be small, which is a crucial ingredient
in perturbation theories. The deformation being a
perturbation is just a well behaved limit of the theory, and means
that we can softly deform the seed configuration. Being the
theory noncommutative, successive and mixed perturbative
deformations give different configurations depending on the
order in which each of them are implemented. This provides
infinite manners of deforming realistic configurations
controlling rigorously the physical acceptability of the resulting
anisotropic distribution.
6 Conclusions
In this paper we have presented different branches of
solutions that models non-rotating and uncharged anisotropic
superdense stars. Each anisotropic branch opens a
possibility for new physically acceptable configuration obtained
by guided deformations over the isotropic Durgapal–Fuloria
stars, and exemplify some possible anisotropic distributions
among the many that the MGD method generates. This
prescription has been design to decouple the field equations of
static and spherically symmetric self-gravitating systems. It
associate the anisotropic sector with a deformation over the
geometric potentials. In this work we have reported radial
deformations only, but extensions to the temporal
deformation may bring intriguing results. After the decoupling, one
obtains a sector which solution is already known (seed
sector) and the anisotropic sector which obeys a set of
simpler ‘pseudo-Einstein’ equations associated to the metric
deformation. It is worth to note that the minimal
geometric deformations stem in an exclusively gravitational
interaction between sectors; i.e. there is no exchange of energy–
momentum among them.
When the equations are decoupled, no new information
is introduced. Then we have an underdetermined system of
equations; consistent constrains are needed. We have shown
how intuitive constrain leads to new physical anisotropic
solutions. Variations in the couplings between the seed and
the anisotropic sector reveals consistent evolution of the
thermodynamical parameter giving to the MGD method a new
prove of validity. We also have discussed the observational
features of the anisotropic sectors. When the anisotropy
changes the compactness of the star, the observed redshift
increases as it is expected. Not all anisotropic
contributions have observational effects because some anisotropies
only readjust the thermodynamical parameters in the
interior. However, when the anisotropy tweaks the compactness
parameter, the star suffer a redshift. Therefore, observational
data would bound the parameters of the model.
After presenting two branches of solutions that provides an
infinite number of physical compact stars, we have proceed
to generalize the method to deform anisotro-pic solutions.
Any solution of Einstein equation admits a minimal
deformation. Different anisotropic sources have additive effect,
however these effects are noncommutative. The path to the
final configuration matters, and normally deformations in
reversed order produces different resulting configurations.
Hence, the method provides a ‘fine tunning structure’ that
generates an enormous amount of different physically
acceptable anisotropic stars.
Acknowledgements The author AR was supported by the
CONICYTPCHA /Doctorado Nacional/2015-21151658. LG acknowledges the FPI
Grant BES-2014-067939 from MINECO (Spain). The author CR was
supported by Conicyt PhD Fellowship No. 21150314.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.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.
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11. I. Cho, H.C. Kim, Black holes with anisotropic fluid,
arXiv:1703.01103
12. C. Germani, R. Maartens, Phys. Rev. D 64, 124010 (2001)
13. J. Ovalle, Mod. Phys. Lett. A 23, 3247 (2008)
14. J. Ovalle, L.A. Gergely, R. Casadio, Class. Quantum Grav. 32,
045015 (2015)
15. J. Ovalle, Phys. Rev. D 95, 104019 (2017)
16. J. Ovalle, R. Casadio, R. da Rocha, A. Sotomayor, Eur. Phys. J. C
78, 122 (2018)
17. M. Estrada, F. Tello-Ortiz, arXiv:1803.02344 [gr-qc]
18. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999)
19. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999)
20. R. Casadio, J. Ovalle, R. da Rocha, Class. Quantum Grav. 32,
215020 (2015)
21. J. Ovalle, Int. J. Mod. Phys. Conf. Ser. 41, 1660132 (2016)
22. S.K. Maurya, Y.K. Gupta, S. Ray, B. Dayanandan, Eur. Phys. J. C
75, 225 (2015)
23. J. Ovalle, Int. J. Mod. Phys. D 18, 837 (2009)
24. M.C. Durgapal, R.S. Fuloria, Gen. Rel. Grav. 17, 671 (1985)
25. A. De Felice, S. Tsujikawa, Living Rev. Rel. 13, 3 (2010)
26. S. Nojiri, S.D. Odintsov, V.K. Oikonomou, Phys. Rept. 692, 1
(2017)
1. H. Stephani , D. Kramer , M. MacCallum, C. Hoenselaers , E. Herlt, Exact solutions of Einstein's field equations . Cambridge monographs on mathematical physics (Cambridge University Press, New York, 2003 )
2. M.S.R. Delgaty , K. Lake , Comput. Phys. Commun . 115 , 395 ( 1998 )
3. I. Semiz , Rev. Math. Phys. 23 , 865 ( 2011 )
4. P.S. Negi , Int. J. Theor. Phys . 45 , 1684 ( 2006 )
5. L. Herrera , N.O. Santos , Phys. Rep . 286 , 53 ( 1997 )
6. R. Ruderman , Ann. Rev. Astron. Astrophys. 10 , 427 ( 1972 )
7. M.K. Mak , T. Harko, Proc. R. Soc. Lond. A 459 , 393 ( 2003 )
8. A.I. Sokolov , JETP 79 , 1137 ( 1980 )
9. R.F. Sawyer , Phys. Rev. Lett . 29 , 382 ( 1972 )
10. S.K. Maurya , Y.K. Gupta , B. Dayanandan , S. Ray , Astrophys. Space Sci . 361 , 163 ( 2016 )
27. L.G. Jaime , L. Patino , M. Salgado , Phys. Rev. D 83 , 024039 ( 2011 )
28. D. Vernieri , S. Carloni, arXiv: 1706 . 06608
29. J. Alfaro , D. Espriu , L. Gabbanelli, Class. Quantum Grav . 35 , 015001 ( 2018 )
30. G. Dvali, C. Gomez , Phys. Lett . 719 , 419 ( 2013 )
31. R. Casadio , R. da Rocha, Phys. Lett. B 763 , 434 ( 2016 )
32. K. Schwarzschild , Sitz. Deut. Akad. Wiss. Berlin. Kl. Math. Phys 24 , 424 ( 1916 )
33. G. Lemaitre, Ann. Soc. Sci. Brux. A 53 , 51 ( 1933 )
34. B.V. Ivanov , Eur. Phys. J. C 77 , 738 ( 2017 )
35. Y.K. Gupta , S.K. Maurya , Astrophys. Space Sci . 331 , 135 ( 2010 )
36. P. Burikham, T. Harko , M.J. Lake , Phys. Rev. D 94 , 064070 ( 2016 )
37. J. Ovalle , R. Casadio , A. Sotomayor , Adv. High Energy Phys . 2017 , 9756914 ( 2017 )
38. J. Ovalle , R. Casadio , A. Sotomayor , J. Phys. Conf. Ser . 883 , 012004 ( 2017 )
39. W. Israel, Nuovo Cim . B 44 , ( 1966 ) 1 ; Erratum-ibid B 48 , ( 1967 ) 463
40. J. Hladík , Z. Stuchlík , J. Cosmo . Astropart. Phys. 7 , 12 ( 2011 )