A conformally flat realistic anisotropic model for a compact star
Eur. Phys. J. C
A conformally flat realistic anisotropic model for a compact star
B. V. Ivanov 0
0 Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Science , Tzarigradsko Shausse 72, 1784 Sofia , Bulgaria
A physically realistic stellar model with a simple expression for the energy density and conformally flat interior is found. The relations between the different conditions are used without graphic proofs. It may represent a real pulsar.
The study of relativistic stellar structure is now more than
100 years old. It began with the discovery in 1916 by Karl
Schwarzschild of the universal vacuum exterior solution 
and the first interior stellar solution , which should be
matched to the exterior one. For a long time the star
interior was considered to be made of perfect fluid, which has
equal radial ( pr ) and tangential ( pt ) pressures. This leads to
the isotropic condition pr = pt , imposed on the Einstein
The first attempts to consider pressure anisotropy in
selfgravitating objects were made by Jeans within the context
of Newtonian gravity . Spherical symmetry demands only
the equality of the two tangential pressures. In general
relativity the first anisotropic model was proposed by Lemaitre
in 1933 . He discussed a model sustained solely by pt and
with constant energy density ρ. His work remained unnoticed
for a long time.
In 1972 Ruderman  argued for the first time that nuclear
matter at very high densities ρ of the order of 1015 g/cm3 may
have anisotropic features and its interactions are relativistic.
The work of Bowers and Liang  on building anisotropic
models in 1974 gave start to a number of such solutions.
Anisotropy may have a lot of sources : a mixture of fluids
of different types, presence of a superfluid, existence of a
solid core, phase transitions, presence of magnetic field,
viscosity, etc. Such models describe compact stellar objects like
neutron stars, strange stars, quark stars, boson stars,
gravastars, dark stars and others.
The Einstein equations describe the effect of matter upon
the metric of spacetime. For static, spherically symmetric
fluid solutions the metric may be written in comoving
canonical coordinates and has two components ν and λ. The
energymomentum tensor is represented by its diagonal components,
mentioned above: ρ, pr and pt . There are only three
equations for these five characteristics, so that two of them may
be chosen freely. They should satisfy, however, a lot of
regularity, stability and energy conditions for a realistic model.
The situation with this undetermined system of differential
equations is analogous to the one for charged isotropic star
models . This is not surprising since charge and other
characteristics can be looked upon as an effective anisotropy of
the model [9–11].
Different choices of the two given functions have been
made. The simplest one is to propose ansatz for the two metric
functions. One of the first was given in , where some of
the Tolman isotropic solutions  were modified to become
anisotropic. Other followed recently [14–20].
String theory has inspired embedding of branes like in the
Randall–Sundrum model . This rekindled the interest in
stellar models embedded in five-dimensional flat spacetime
(embedding class one). They must satisfy the Karmarkar
condition . It can be written as a relation between the metric
functions and one of them can generate the whole solution.
It is interesting that the isotropic condition, can be translated
into a similar relation, giving different generating functions
There are just two perfect fluid solutions of the Karmarkar
condition – the interior Schwarzschild one and a
cosmological one. When the fluid is anisotropic, a number of realistic
solutions has been found in the last 2 years [28–47].
Conformally flat anisotropic spheres have a vanishing
Weyl tensor. This leads to a differential equation for λ and
ν, similar to the Karmarkar one or the isotropic condition.
Early solutions were found in  where different ansatz
for the mass function m were proposed. There is a simple
relation between m and λ and from the condition for
conformal flatness ν may be found. Then expressions for all other
characteristics of the model are obtained. The first who
integrated the vanishing Weyl condition was Ponce de Leon in
1987 , but no details were given. A relation between ν
and λ is the outcome. Details were supplied in 2001  and
some models were discussed with pr = 0 or prescribed λ.
The authors worked in non-comoving coordinates and gave
the general solution in another, but equivalent form, used
later in . Conformally flat spherically symmetric
spacetimes were studied in different coordinate systems in . A
conformally flat model with polytropic equation of state was
discussed in . Other solutions were given too [54,55].
Closely related are solutions which admit conformal
motion. They depend on the conformal factor. The first
anisotropic solution of this kind was given in 1984 . Time
dependent solutions followed , as well as charged ones
 with constant anisotropy, or constant charge density and
a generalisation of the Lemaitre solution. Some recent
solutions are [59–66]. The work  has been generalized to
non-static solutions [54,55] and many new solutions were
The main shortcoming of the existing model building is
that the conditions for a realistic model are checked after
the ansatz for the two free functions are made. The
expressions for the different characteristics become very involved
even for polynomial seeding functions and one has to turn
to graphic proofs. Solutions usually have lots of constants in
order to satisfy the set C1–C10, introduced in Sect. 3. One
constant turns a 2-dimensional plot into a 3-dimensional one.
With two and more constants only partial plots are possible.
Recently,  we have argued that the combination of
free functions ρ, pr is the right choice to reduce the number
of graphic proofs. Another important fact is that the
conditions C1–C10 are not independent. There are many relations
between them and we have reduced the set to a couple of
inequalities. Only they need in general a graphic proof in the
concrete examples. To illustrate this formalism we have given
a solution with simple energy density and linear equation of
state (EoS) with bag constant.
In the present paper we apply the approach of  to
conformally flat solutions with simple metric function λ, which
leads to a simple ρ. We make a full analytic physical analysis
of the solution and show that no graphic proofs are
necessary. It implies that a certain constant of the model should
fall in a particular range. A real pulsar is shown to satisfy this
In Sect. 2 the Einstein field equations are given, as well
as the definitions of the main characteristics of a static
anisotropic star. The Weyl condition and its solution are also
introduced. In Sect. 3 we summarize the conditions for a
physically realistic model. In Sect. 4 we present the model,
which depends on three constants. In Sect. 5 we perform
a full physical analysis and find the range of the constants
where C1–C10 are fulfilled. In Sect. 6 a real star is shown
to satisfy these constraints and therefore is a candidate for a
neutron star with conformally flat interior. Section 7 contains
2 Field equations and definitions
The interior of static spherically symmetric stars is described
by the canonical line element
ds2 = eν c2dt 2 − eλdr 2 − r 2(dθ 2 + sin2 θ dϕ2),
where λ and ν are dimensionless and depend only on the
radial coordinate r . The Einstein equations read
kc2ρ = r 2 [r (1 − e−λ)] ,
k pr = − r12 (1 − e−λ) + νr e−λ,
k pt =
2ν + ν 2 + r
− ν λ − r
where ρ is the matter density, pr is the radial pressure, pt is
the tangential one, means a radial derivative and
Here G is the gravitational constant and c is the speed of
The gravitational mass in a sphere of radius r is given by
ρ (ω) ω2dω.
e−λ = 1 − 2rm .
Due to kc2, its dimension is length. Then Eq. (
The compactness of the star u is defined by
u = r
and is dimensionless.
On the other side, the redshift Z depends on ν:
Z = e−ν/2 − 1.
The field equations do not contain ν, but its first and second
derivative. One can express ν from Eqs. (
), and (
kr pr + 2m/r 2
1 − 2m/r
The second derivative ν may be excluded by differentiation
of Eq. (
) and combination with the other field equations.
The result is
pr = − 2 (ρc2 + pr )ν + r ,
where Δ = pt − pr is the anisotropic factor. Combining
) and (
) one gets the well-known TOV (Tolman,
Oppenheimer, Volkoff) equation [13,68] of hydrostatic equilibrium
in a relativistic star, found initially for isotropic solutions. Its
anisotropic version was given by Bowers and Liang :
pr = −(ρc2 + pr )
kr pr + 2m/r 2
2 (1 − 2m/r ) +
2 ( pt − pr ) .
The hydrostatic force on the left Fh is balanced by the
gravitational Fg and the anisotropic forces Fa on the right. This
equation is not independent from the field equations, but
is their consequence. It can replace one of them. It is also
equivalent to the Bianchi identities Tνμ;μ = 0, which in the
static spherically symmetric case have only one non-trivial
component [6,69–71]. In CGS units G = 6.674 × 10−8
cm3/g s2, c = 3 × 1010 cm/s, k = 2.071 × 10−48 s2/g cm,
kc2 = 1.864 × 10−27 cm/g. The mass in grams M is related
to m by
From now on we set G = c = 1, passing to usual relativistic
units. Then k = 8π . As a whole, we have three field equations
for five unknown functions: λ, ν, ρ , pr and pt .
The space-time is conformally flat when its Weyl tensor
vanishes. In our case this gives a relation between the two
metric coefficients 
1 −r2eλ − ν4λ − ν 2−r λ + ν2 + ν42 = 0. (
Similar relations arise in embeddings of class 1 , or in
the case of isotropic pressure . Equation (
) may be
integrated. It appears that for the first time this was done in
, but no details of the integration method were given.
These were provided later in [50,52,55]. The result can be
written as 
eν/2 = C1r cosh
r dr + C ,
where C and C1 are integration constants. This equation
should be added to the three field equations, so one may
choose freely one generating function to obtain solutions.
The model will be physically realistic if a number of
regularity, matching and stability conditions are satisfied too.
3 Conditions for a physically realistic model
A comparatively reasonable set of conditions includes
C1. The metric potentials are positive and should be finite
and free from singularities in the star’s interior and at the
centre should satisfy eλ(0) = 1 and eν(0) = const .
C2. Matching conditions. At the surface of the star r = rs
the interior solution should match continuously to the exterior
Schwarzschild solution ,
1 − rs dt 2 −
− r 2(dθ 2 + sin2 θ dϕ2),
1 − rs
where ms = m (rs ). This determines the metric at the surface
eν(rs ) = e−λ(rs ) = 1 − 2ms . (
In addition, the radial pressure there vanishes, prs = 0.
Neither the energy density nor the tangential pressure are obliged
to do so.
C3. The interior redshift Z , given by Eq. (
decrease with the increase of r . The surface redshift and
compactness are related, due to Eq. (
Zs = (1 − us )−1/2 − 1.
They should be less than the universal bounds, found when
different energy conditions hold (see C6). In the isotropic
case they are 2 and 8/9 correspondingly . In the
anisotropic case, when DEC holds, they are 5.211 and 0.974.
When TEC holds, one has the bounds 3.842 and 0.957 .
They are greater than those in the isotropic case, but not
arbitrary as asserted in .
C4. The density and the pressures should be non-negative
inside the star. At the centre they should be finite ρ (0) = ρ0,
pr (0) = pr0, pt (0) = pt0. Moreover, pr0 = pt0 .
C5. They should reach a maximum at the centre, so
ρ (0) = pr (0) = pt (0) = 0 and should decrease
monotonically outwards, ρ ≤ 0, pr ≤ 0, pt ≤ 0. The tangential
pressure should remain bigger than the radial one, except at
the centre, pt ≥ pr .
C6. Energy conditions. The solution should satisfy the
dominant energy condition (DEC) ρ ≥ pr , and ρ ≥ pt .
The strong energy condition (SEC)  should be satisfied
too, ρ + pr + 2 pt ≥ 0. Because of C4 it is trivial, as well
as the null energy condition (NEC). It is desirable that even
the trace energy condition (TEC) ρ ≥ pr + 2 pt should be
satisfied. Obviously, the latter is stronger than DEC.
C7. Causality condition. It says that the radial and
tangential speeds of sound should not surpass the speed of
light. The speeds of sound are defined as vr2 = d pr /dρ and
vt2 = d pt /dρ. Therefore this condition reads
C9. Stability against cracking. Cracking was introduced
by Herrera  as a possibility of breaking of perturbed
selfgravitating spheres. Abreu et al.  found a simple
requirement for avoiding this to happen, namely the region of
C10. The Harrison–Zeldovich–Novikov stability
condition [79,80]. It implies that d M (ρ0) /dρ0 > 0.
4 The model
We shall choose a simple ansatz for eλ as a generating
e−λ = (1 − x )2 , x = br22 ,
where b is some constant of dimension length, so that x is
dimensionless, as is the metric coefficient. Its range is from
0 to xs < 1. Then Eqs. (
) and (
m = 21 r (2x − x 2), u = 2x − x 2.
The derivative of Eq. (6) yields
and is positive too. Eq. (
), which is an expression for the
radial pressure, yields the formula
There is a general expression for the anisotropy factor
Δ in conformally flat models, which follows from the field
) and (
) and the requirement (
In the case of the simple ansatz (
) it becomes
kb2Δ = 2x , kb2Δx = 2.
It makes the expression for the tangential pressure very
similar to the one for the radial pressure
kb2 pt = x +
4α (1 − x )2
1 + αx
Thus, the characteristics of the model are given by simple
elementary functions. They depend on three constants b (or
ρ0), α and C1. They should be related to the mass ms and the
radius rs of the star.
kb2 pr = −x +
4α (1 − x )2
1 + αx
kρ = r 2
kb2ρ = 6 − 5x ,
b2 = kρ60 .
which is very simple. In more general form it was used in
the past, [67,81–92]. In the context of conformal flatness it
was used in , Example 4, and , model III but only a
partial physical analysis has been done. Eq. (
) clarifies the
meaning of b:
and using this formula or Eq. (
) we obtain for the energy
5 Physical analysis
Now we have to choose the free parameters of the model in
such a way that the conditions C1–C10 are satisfied.
C1. Eq. (
) shows that eλ is finite and positive and
increases monotonically with r from 1 to (1 − xs )−2.
) and (
) show that eν is also finite and positive and
increases monotonically. Equation (
) shows that eν(rs ) is
less than 1, hence eν(0) is also less than one.
C2. The matching condition for λ is fulfilled when ms =
m (rs ). Thus Eq. (
The zero index will be used for variables at the centre of the
star. Thus b is related to the central density ρ0, whose value
in CGS units is about 1015g for compact neutron stars.
Let us introduce now the constants B and α instead of C
C = 21 ln B2, α = B2 − 1. (
Then Eq. (
) gives an expression for the other metric
e = 4 (1 + α)
(1 + αx )2
1 − x
which is obviously positive. The redshift Z throughout the
star is obtained then from Eq. (
). The derivative of ν with
respect to x , which enters the field equations, is
) and (
) fix C1
4 (1 + α) (1 − xs )3
b2 (1 + αxs )2
in terms of α, b and rs . The boundary condition prs = 0,
combined with Eq. (
) expresses α as a function of xs
α = (2 − xs ) (2 − 3xs )
Thus α is positive, increases monotonically with xs and is
finite as long as xs < 2/3. Then B2 is positive as it should
0 ≥ pt ≥ pr .
− kb2 pr x ≥ 2.
Finally, Eqs. (41) and (47) yield
2 ≤ −kb2 pr x ≤ 5.
Then we shall finish the proof of C5 and C9. These are the
same inequalities, derived in , Eq. (
). In addition, since
) may be written as
d pt d pr
0 ≤ dρ ≤ dρ
we also prove C7b, the causality condition for d pt /dρ. Thus,
C5 about pt , C7 and C9 are reduced to Eqs. (
) and (
Moving to x -derivatives and subtracting pr x from Eq. (
− pr x ≥ Δx ≥ 0.
Utilizing Eq. (
), we transform the above two inequalities
To solve these two inequalities we use Eq. (
). Then the
left inequality becomes
5α2x 2 + 10αx + 1 − 4α (2 + α) ≤ 0,
while the right inequality transforms into
2α2x 2 + 4αx + 1 − α (2 + α) ≥ 0.
The terms containing x in Eq. (
) increase with x , hence, it
is enough to prove it for x = 0. Then it becomes an inequality
The l.h.s. increases with α starting from −1, therefore α
should be less or equal than the positive root of the
α ≤ α1 = −1 + √2 = 0.414. (
) is quadratic for xs with α as parameter. The
root less than one should be used to express xs , namely
xs1 = 8α + 1 − √166αα2 + 16α + 1 . (
As we mentioned after Eq. (
), xs decreases with α, hence
xs ≤ xs1 (α1) or
due to Eqs. (
) and (
C4. Because of Eq. (
) ρ will be positive as long as
x < 6/5. This inequality is true because xs < 2/3. The
energy-density is finite at the centre and taken to be about
1015g/cm3. This defines b according to Eq. (
). Both terms
in the expression for pr in Eq. (
) decrease monotonically
when r increases. We have arranged that prs = 0 (Eq. 36).
Hence, in the interior pr decreases monotonically to zero and
is positive. This is confirmed by its derivative
kb2 pr x = −1 −
4α (1 − x ) (2 + α + αx )
(1 + αx )2
It is obviously negative and since x = 2r/b2, pr is also
negative. Finally, due to Eq. (
), Δ = pt − pr ≥ 0 and
pt ≥ pr . Therefore pt is also positive and at r = 0 coincides
with pr . Their value at the centre is given by Eq. (
pt0 = pr0 = kb2 .
kb2ρx = −5.
C5. Equation (25) gives
) and (
) combine to deliver ρ (0) =
pr (0) = pt (0) = 0 and the monotonic decrease of ρ and
pr . It remains to prove that pt ≤ 0. Before doing that let us
turn to C7.
C7a. Causality condition for d pr /dρ. This ratio can be
written as pr x /ρx . We have just proved that the numerator
and the denominator are negative, so their ratio is positive.
Hence, the left inequality of the first part of Eq. (
) is true.
The right inequality demands
− kb2 pr x ≤ 5,
because of Eq. (
). Let us go now to C9.
C9. The anti-cracking condition may be written as
− 1 + dρ
≤ ρpr .
We suppose that inequality (
) holds. Then the left hand
side of Eq. (
) is negative. Let us multiply Eq. (
) by ρ ,
which was shown to be negative. We have
−1 + dρ
≥ pt ≥ pr .
Now the left hand side is positive. Combining this chain of
inequalities with the inequality pt ≤ 0, that we have to prove,
Replacing α with its expression from Eq. (
) we get
4 (1 − xs )4 ≤ xs (1 − xs )2 (8 − 7xs ) .
The fourth degree inequality surprisingly becomes a quadratic
one, when we divide both sides by the common multiplier,
and thus much easier to be solved. We have
11xs2 − 16xs + 4 ≤ 0.
The derivative of the l.h.s. is negative, so it decreases from
4 and becomes negative at the point, where the inequality
becomes an equality. We solve this quadratic equation and
take the root that is less than 1. The solution reads
xs ≥ 8 −121√5 = 0.321. (
Combining Eqs. (
) and (
) we obtain the range of xs
8 − 2√5
0.321 = 11 ≤ xs
8√2 − 7 − 33 − 16√2
≤ 6 √2 − 1 = 0.439.
In this range C5, C7 and C9 hold.
Let us discuss next the energy conditions C6. The left part
of the proven Eq. (
) may be written as
since ρ ≤ 0. A definite integral of the l.h.s. is also positive,
so that Eq. (
( pr − ρ) ≥ 0,
( pr − ρ) dr = − pr − ρs + ρ ≥ 0,
which proves DEC for pr , because ρs ≥ 0.
The r.h.s. of Eq. (
) may be written as
( pt − ρ) ≥ 0
ρ − pt ≥ ρs − pts .
and the same integral of this inequality gives
Hence DEC for pt holds in the interior, if it holds at the
surface of the star. In  a sufficient universal condition
was given, us ≤ 0.8. For our model Eqs. (
) and (
0.539 ≤ us ≤ 0.685,
so that the sufficient condition is satisfied in the whole range
of xs . We can also use the expressions for ρ (Eq. 25) and pt
(Eq. 33) to find
kb2 (ρs − pts ) = 2 (1 − xs ) (3 − 2α + 5αxs ) .
) and (
) show that α ∈ [0.184, 0.414]. In this
interval 3 − 2α is positive and consequently the r.h.s. of the
above equation is positive too. This proves that DEC holds
for pt as well.
Let us prove, finally, that TEC is true. Eq. (
) gives ρ ≥
pr + ρs . If
This chain is true due to Eq. (
) (the two pressures are equal
at the centre of the star) and Eq. (
) (the tangential pressure
decreases towards the stellar surface). Using Eqs. (
), Eq. (
As we know α increases with xs till 0.414. Inserting this and
the maximum of xs = 0.439 in the above inequality, yields
6 ≥ 5.567, which obviously is true. Hence, TEC holds for
the whole range of xs . Thus C6 holds in its entirety.
Condition C8. In  a sufficient condition was given for
) to hold, namely TEC and a lower limit for the radial
speed of sound
The derivative of the total stellar mass with respect to the
central density, when the star radius is kept constant, reads
and is obviously positive. Thus C10 is true and the whole set
C1-C10 is true as long as xs belongs to the range given by
), xs ∈ [0.321, 0.439].
6 Model of a real star
The astronomers collect data about the radius rs and the mass
M of real stars. Usually the ratio β to the solar mass Msol is
Msol = msol ≡ β.
us = 2msol r
It is known that in relativistic units msol = 1.474 km. Then
we can find the compactness from Eq. (
where rs is in km. The conformally flat solution is physically
realistic when xs ∈ [0.321, 0.439]. Equations (
) and (
or Eq. (
) give a relation between us and xs
xs = 1 −
1 − us .
0.183 ≤ r
) may be used too, us ∈ [0.539, 0.685]. Then
Thus, it is very easy to find whether a real star may be
described by our model. Only the surface compactness of
the star matters or equivalently, its surface redshift.
) gives the limits Zs ∈ [0.473, 0.782]. These are
somewhat higher than the redshifts of many other models,
discussed in the literature.
An important characteristic is the central density ρ0. We
suppose that it may be written as
ρ0 = a × 1015 g/cm3,
where a is some constant close to 1. Equations (
) and (
in CGS units give
ρ0 = kc2rs2
Thus a is given by
6 × 10−15xs
= 321.9 rx2s ,
where we have used kc2 = 1.864 × 10−27 cm/g and rs is
given in km.
Let us apply these formulas to some stars, described in a
recent paper . There a model with given λ and Δ was
used. The pulsar 4U1820-30 has β = 1.58 and rs = 9.1 km.
Then β/rs = 0.174 and is out of range. The pulsar Cen X-3
has β = 1.49 and rs = 9.178 km. Again β/rs = 0.162
is too small. However, the pulsar PSR J1614-2230 has β =
1.97 and rs = 9.69 km and β/rs = 0.203 which satisfies
) and may be a candidate for a compact neutron star
with a conformally flat interior. We get from Eq. (
) us =
0.598. Equation (
) yields xs = 0.366. Then we obtain from
) a = 1.258, which is realistic. The surface redshift is
comparatively high, Zs = 0.577 (see Eqs. (
) or (
)), but is
less than the known limits [72,73]. The other characteristics
of this star may be found from the formulas in the previous
We have followed in this paper the approach of .
Although, instead of ρ and pr , we chose an ansatz for λ and
the condition of conformal flatness, we have been able to
satisfy all physical conditions without using graphic proofs.
After all, C1–C10 involve inequalities, which, in principle,
may be proved by algebra and calculus. Their use is limited to
solving quadratic equations and integrating derivatives. The
relations between the different conditions, that we found in
the above reference allowed us to come to the same basic
couple of inequalities, Eq. (
). It is interesting that the
sufficient conditions for the other realistic features of the model
are contained in them and do not restrict further the range
of the main parameter xs . A remarkable fact is that only the
compactness (or the surface redshift) of the star is necessary
to determine, whether it may have a conformally flat interior.
The mass and the radius of the star are enough to determine
all of its characteristics.
In the perfect fluid case the Buchdahl bounds  on the
compactness and the redshift are 8/9 = 0.889 and 2 (see C3).
They are saturated by the Schwarzschild interior solution ,
which is an incompressible sphere with constant density. It is
unphysical, because the speed of sound is infinite. Something
more, the saturation occurs when the pressure is infinite at
the centre . This model is the unique conformally flat
one for perfect (isotropic) fluids. This is one of the reasons
to study conformally flat solutions in the anisotropic case. It
may provide an explanation for the intermediate ranges of
compactness and redshifts of realistic anisotropic solutions.
In the literature, in many papers the real strong energy
condition (SEC) is used, which, however, is trivial. In many
others, the trace energy condition (TEC) is called SEC and made
use of. It is really strong, because it requires that the energy
density (which in CGS units is multiplied by c2) should be
bigger than the sum of the radial and the two equal
tangential pressures. Thus, it is even stronger than the dominant
energy condition (DEC). We have tried to clarify this misuse
Finally, the proofs of C1–C10 were considerably
simplified by the simple expression for the anisotropy factor Δ.
In the case of embeddings of class one, the Karmarkar
condition leads to a more sophisticated form for Δ. Therefore,
the present paper may be considered also as a preparation to
attack this case.
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