1. No category

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10.1098/ rspa.2002.1014
Anisotropic stars in general relativity
By M. K. M a k1 a n d T. H a r k o2
1
Department of Physics, The Hong Kong University of Science and Technology,
Clear Water Bay, Hong Kong, People’s Republic of China
2
Department of Physics, The University of Hong Kong,
Pokfulam Road, Hong Kong, People’s Republic of China
([email protected]; [email protected])
Received 7 November 2001; revised 12 March 2002; accepted 29 April 2002;
published online 2 December 2002
We present a class of exact solutions of Einstein’s gravitational-­ eld equations
describing spherically symmetric and static anisotropic stellar-type con­ gurations.
The solutions are obtained by assuming a particular form of the anisotropy factor.
The energy density and both radial and tangential pressures are ­ nite and positive
inside the anisotropic star. Numerical results show that the basic physical parameters (mass and radius) of the model can describe realistic astrophysical objects such
as neutron stars.
Keywords: anisotropic stars; Einstein’s ¯eld equations; static interior solutions
1. Introduction
Relativistic stellar models have been studied ever since the ­ rst solution of Einstein’s
­ eld equation for the interior of a compact object in hydrostatic equilibrium was
obtained by Schwarzschild in 1916. The search for the exact solutions describing
static isotropic and anisotropic stellar-type con­ gurations has continuously attracted
the interest of physicists. The study of general relativistic compact objects is of
fundamental importance for astrophysics. After the discovery of pulsars and the
explanation of their properties by assuming them to be rotating neutron stars, the
theoretical investigation of superdense stars has been done using both numerical and
analytical methods and the parameters of neutron stars have been worked out by
general relativistic treatment.
There are very few exact interior solutions (both isotropic and anisotropic) of the
gravitational-­ eld equations satisfying the required general physical conditions inside
the star. From 127 published solutions analysed in Delgaty & Lake (1998), only 16
satisfy all the conditions. But the study of the interior of general relativistic stars by
­ nding exact solutions to the ­ eld equations is still an active ­ eld of research. Hence,
recently an algorithm to generate any number of physically realistic pressure and
density pro­ les for spherical perfect ®uid isotropic distributions without evaluating
integrals was proposed by Fodor (2000). The gravitational-­ eld equations for static
stellar models with a linear barotropic equation of state (Nilsson & Uggla 2001a)
and with a polytropic equation of state (Nilsson & Uggla 2001b), p = k» 1+ 1=n , were
recast into two complementary three-dimensional regular systems of ordinary di¬erential equations on compact state space; these systems were analysed numerically
Proc. R. Soc. Lond. A (2003) 459, 393{408
393
°
c 2002 The Royal Society
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M. K. Mak and T. Harkoon November 17, 2014
and qualitatively, using the theory of dynamical systems. Schmidt & Homann (2000)
discussed numerical solutions of Einstein’s ­ eld equation describing static spherically symmetric conglomerations of a photon star with the equation of state » = 3p.
Recently, upper limits for the mass{radius ratio have been derived for compact general relativistic objects in the presence of a cosmological constant (Mak et al . 2000)
and in the presence of a charge distribution (Mak et al. 2001).
Since the pioneering work of Bowers & Liang (1974), there has been an extensive
literature devoted to the study of anisotropic spherically symmetric static general
relativistic con­ gurations. The theoretical investigations of Ruderman (1972) about
more realistic stellar models show that the nuclear matter may be anisotropic at least
in certain very high density ranges (» > 1015 g cm¡3 ), where the nuclear interactions
must be treated relativistically. According to these views, in such massive stellar
objects the radial pressure may not be equal to the tangential pressure. No celestial
body is composed of a perfect ®uid. Anisotropy in ®uid pressure could be introduced
by the existence of a solid core or by the presence of type-3A super®uid (Kippenhahm
& Weigert 1990), di¬erent kinds of phase transitions (Sokolov 1980), pion condensation (Sawyer 1972) or by other physical phenomena. On the scale of galaxies, Binney
& Tremaine (1987) have considered anisotropies in spherical galaxies, from a purely
Newtonian point of view.
The mixture of two gases (e.g. monatomic hydrogen or ionized hydrogen and electrons) can also formally be described as an anisotropic ®uid (Letelier 1980; Bayin
1982). Bowers & Liang (1974) have investigated the possible importance of locally
anisotropic equations of state for relativistic ®uid spheres by generalizing the equations of hydrostatic equilibrium to include the e¬ects of local anisotropy. Their study
shows that anisotropy may have non-negligible e¬ects on such parameters as maximum equilibrium mass and surface redshift. Heintzmann & Hillebrandt (1975) studied fully relativistic, anisotropic neutron-star models at high densities by means of
several simple assumptions and showed that for arbitrary large anisotropy there is
no limiting mass for neutron stars, but that the maximum mass of a neutron star
still lies beyond 3{4M­ . Hillebrandt & Steinmetz (1976) considered the problem of
stability of fully relativistic anisotropic neutron-star models. They derived the differential equation for radial pulsations and showed that there exists a static stability
criterion similar to the one obtained for isotropic models. Anisotropic ®uid-sphere
con­ gurations have been analysed, using various ansatze, in Bayin (1982), Cosenza
et al . (1981) and Harko & Mak (2000). For static spheres in which the tangential
pressure di¬ers from the radial one, Bondi (1992) has studied the link between the
surface value of the potential and the highest occurring ratio of the pressure tensor to
the local density. Chan et al . (1993) studied in detail the role played by the local pressure anisotropy in the onset of instabilities and they showed that small anisotropies
might in principle drastically change the stability of the system. Herrera & Santos
(1995) have extended the Jeans instability criterion in Newtonian gravity to systems
with anisotropic pressures. Recent reviews on isotropic and anisotropic ®uid spheres
can be found in Delgaty & Lake (1998) and Herrera & Santos (1997).
Very recently, an analysis based on the Weyl tensor of the Lema^±tre{Schwarzschild
problem of ­ nding the equilibrium conditions of an anisotropically sustained spherical body under its relativistic gravitational ­ eld has been revisited by Fuzfa et al .
(2002). Hernandez & Nunez (2001) presented a general method for obtaining static
anisotropic spherically symmetric solutions satisfying a non-local equation of state
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Anisotropic
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from known density pro­ les, by assuming the condition of a vanishing Weyl tensor. This condition can be integrated in the spherically symmetric case. Then, the
resulting expression is used by Herrera et al . (2001) to ­ nd new, conformally ®at,
interior solutions to Einstein equations for locally anisotropic ®uid. Dev & Gleiser
(2000) presented several exact solutions for anisotropic stars of constant density.
Some classes of exact solutions of Einstein ­ eld equations, describing spherically and
static anisotropic stellar-type con­ gurations, were presented by Harko & Mak (2002)
and Mak et al . (2002). The surface redshift zs of anisotropic realistic stars has been
investigated by Ivanov (2002). zs cannot exceed the values 3:842 or 5:211, when the
tangential pressure satis­ es the strong or dominant energy conditions, respectively.
In this paper we consider a class of exact solutions of the gravitational-­ eld
equations for an anisotropic ®uid sphere, corresponding to a speci­ c choice of the
anisotropy parameter. The metric functions can be represented in a closed form in
terms of elementary functions. All the physical parameters such as the energy density, pressure and metric tensor components are regular inside the anisotropic star,
with the speed of sound less than the speed of light. Therefore, this solution can
give a satisfactory description of realistic astrophysical compact objects like neutron
stars. Some explicit numerical models of relativistic anisotropic stars, with a possible
astrophysical relevance, are also presented.
The paper is organized as follows. In x 2 we present an exact class of solutions
for an anisotropic ®uid sphere. In x 3 we present neutron-star models with possible
astrophysical applications. The results are summarized and discussed in x 4.
2. Non-singular models for anisotropic stars
In standard coordinates xi = (t; r; À ; ¿ ), the general line element for a static spherically symmetric space-time takes the form
ds2 = A2 (r) dt2 ¡
V ¡1 (r) dr 2 ¡
r 2 (dÀ
2
+ sin2 À d¿
2
):
(2.1)
Einstein’s gravitational-­ eld equations are (where natural units 8º G = c = 1 have
been used throughout):
Rki ¡ 12 R¯ ik = Tik :
(2.2)
For an anisotropic spherically symmetric matter distribution the components of
the energy{momentum tensor are of the form
Tik = (» + p? )ui uk ¡
p? ¯
k
i
+ (pr ¡
p? )À i À
k
;
(2.3)
where ui is the p
four-velocity Aui = ¯ 0i , À i is the unit space-like vector in the radial
i
direction, À = V ¯ 1i , » is the energy density, pr is the pressure in the direction of
À i (normal pressure) and p? is the pressure orthogonal to À i (transversal pressure).
We assume pr 6= p? . The case in which pr = p? corresponds to the isotropic ®uid
sphere. ¢ = p? ¡ pr is a measure of the anisotropy and is called the anisotropy factor
(Herrera & Ponce de Leon 1985).
i = 0 (where a semiA term 2(p? ¡ pr )=r appears in the conservation equations Tk;i
colon denotes the covariant derivative with respect to the metric), representing a
force that is due to the anisotropic nature of the ®uid. This force is directed outward
when p? > pr and inward when p? < pr . The existence of a repulsive force (in the
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M. K. Mak and T. Harkoon November 17, 2014
case in which p? > pr ) allows the construction of more compact objects when using
anisotropic ®uid than when using isotropic ®uid (Gokhroo & Mehra 1994).
For the metric (2.1), the gravitational-­ eld equations (2.2) become
V0
;
r2
r
µ 0
¶
µ 00
A
1
A
V0
+
+ 2V
¡
A
r
A
» =
1¡
V
¡
2A0 V
V ¡ 1
+
;
Ar
r2
¶
µ
¶
A0
1
1
¡
=
2
¢
¡
;
rA r 2
r2
pr =
where the prime denotes d=dr.
To simplify calculations it is convenient to introduce the substitutions
Z r
m(r)
1
V = 1 ¡ 2x² ;
x = r2;
² (r) =
;
m(r)
=
¹ 2 » (¹ ) d¹ ;
2 0
r3
(2.4)
(2.5)
(2.6)
where m(r) represents the total mass content of the distribution within the ®uid
sphere of radius r. Hence, we can express (2.5) in the equivalent form
µ
¶
µ
¶
d2 A
d²
dA
1 d²
¢
(1 ¡ 2x² ) 2 ¡
x
+²
¡
+
A = 0:
(2.7)
dx
dx
dx
2 dx 4x
For any physically acceptable stellar models, we require the condition that the
energy density is positive and ­ nite at all points inside the ®uid spheres. In order to
have a monotonic decreasing energy density
» =
2 d
(² r3 )
r 2 dr
inside the star, we chose the function ² in the form
a0
² =
;
2(1 + Á)
(2.8)
where Á = c0 x, and a0 , c0 are non-negative and constant. Equation (2.8) represents
an ansatz for the mass function which has been used previously in the study of the
isotropic ®uid spheres by Matese & Whitman (1980) and Finch & Skea (1989).
We also introduce a new variable ¶ by means of the transformation
¶ =
(a0 ¡
c0 )(1 + Á)
:
a0
(2.9)
We also chose the anisotropy parameter as
¢
where
=
c0 ¢ 0 Á
;
(Á + 1)2
(2.10)
a0
¢
¡ K
= 0
c0
1¡ K
or equivalently ¢ 0 = a0 (1 ¡ K)=c0 + K and K is a constant. In the following we
assume ¢ 0 > 0, with ¢ 0 = 0 corresponding to the isotropic limit. Hence, ¢ 0 can
be considered, in the present model, as a measure of the anisotropy of the pressure
distribution inside the ®uid sphere. At the centre of the ®uid sphere the anisotropy
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Anisotropic
stars in general relativity
397
vanishes: ¢ (0) = 0. For small r the anisotropy parameter increases, and after reaching a maximum in the interior of the star, it becomes a decreasing function of the
radial distance.
Therefore, with this choice of ¢ , (2.7) becomes a hypergeometric equation,
¶ (¶ ¡
1)
d2 A 1 dA
+
¡
d¶ 2
2 d¶
1
KA
4
= 0:
(2.11)
By introducing the substitution ¶ = 1 ¡ X 2 , (2.11) reduces to a standard secondorder di¬erential equation given by
X2)
(1 ¡
d2 A
dA
+X
+ KA = 0:
2
dX
dX
(2.12)
By means of the pair of transformations (Gupta & Jasim 2000)
dA
= G;
dX
X = sin ³ ;
(2.13)
equation (2.12) reads
d2 G
+ ! 2 G = 0;
d³ 2
(2.14)
where ! 2 = 1 + K.
We shall not present here the simple linear solution for G corresponding to !2 = 0.
For non-zero ! 2 and with the use of the general solution of (2.14), equation (2.13)
becomes
dA
= G = p1 cos(! sin¡1 X ) + q1 sin(! sin¡1 X );
(2.15)
dX
where p1 and q1 are arbitrary constants of integration.
In view of (2.12) and (2.15), and after some readjustments of the arbitrary constants, we obtain the general solution of (2.15) in the form
A=¬
1 [!
p
1¡
X 2 cos(! sin¡1 X) + X sin(! sin¡1 X )]
p
+ ­ 1 [! 1 ¡ X 2 sin(! sin¡1 X ) ¡ X cos(! sin¡1 X)];
(2.16)
where ¬ 1 and ­ 1 are arbitrary constants of integration.
For simplicity, the metric functions and the physical parameters » , p? , pr and ¢
can be expressed in terms of ³ by setting sin ³ = X, so
¶ =
µ
¢
¢
¡
0
0
¡
¶
1
(1 + Á) = 1 ¡
K
X 2 = cos2 ³ :
Therefore, the general solution of the gravitational-­ eld equations for a static
anisotropic ®uid sphere can be presented in the form expressed in the elementary
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M. K. Mak and T. Harkoon November 17, 2014
398
functions
p
1) sin[(! + 1)³ + B] + (! + 1) sin[(! ¡ 1)³ + B]g;
(2.17)
µ
¶µ
¶ µ
¶
¢ 0¡ K
Á
¢ 0¡ 1
V =1¡
=
tan2 ³ ;
(2.18)
1¡ K
1+Á
1¡ K
·µ
¶
¸
¢ 0¡ K
2
c0 ¢ 0
cos ³ ¡ 1
c0 ¢ 0 Á
¢ 0¡ 1
¢ =
=
;
(2.19)
·µ
¶
¸2
(1 + Á)2
¢ 0¡ K
2
cos ³
¢ 0¡ 1
µ
¶·
¸
·µ
¶
¸
¢ 0¡ K
3+Á
c0 (¢ 0 ¡ 1)2
¢ 0¡ K
2
4
» = c0
=
sec
³
+
2
sec
³
;
1¡ K
(1 + Á)2
(1 ¡ K)(¢ 0 ¡ K)
¢ 0¡ 1
(2.20)
µ
¶
½ µ
¶·
¸
¾
1¡ ¢ 0
1¡ ¢ 0
tan ³
pr = c0
sec2 ³ 2K
+ 1 ; (2.21)
1¡ K
K¡ ¢ 0
! tan(!³ + B) ¡ tan ³
A=
Cf(! ¡
p? =
c0 ¢ 0 Á
+ pr ;
(1 + Á)2
(2.22)
p
p
where ¬ 1 = 2 C sin B and ­ 1 = 2 C cos B and B and C are arbitrary constants.
The physical quantities dpr =d» and dp? =d» , describing the behaviour of the speed
of sound inside the static anisotropic ®uid sphere, can be computed from the resulting
line element.
In order to be physically meaningful, the interior solution for static ®uid spheres
of Einstein’s gravitational-­ eld equations must satisfy some general physical requirements. The following conditions have been generally recognized to be crucial for
anisotropic ®uid spheres (Herrera & Santos 1997).
(i) The density » and pressure pr should be positive inside the star.
(ii) The gradients d» =dr, dpr =dr and dp? =dr should be negative.
(iii) Inside the static con­ guration the speed of sound should be less than the speed
of light, i.e.
dpr
dp?
06
6 1 and 0 6
6 1:
d»
d»
(iv) A physically reasonable energy{momentum tensor has to obey the conditions
» > pr + 2p? and » + pr + 2p? > 0.
(v) The interior metric should be joined continuously with the exterior Schwarzschild metric, that is A2 (a) = 1 ¡ 2u, where u = M=a, M is the mass of the
sphere as measured by its external gravitational ­ eld and a is the boundary of
the sphere.
(vi) The radial pressure pr must vanish but the tangential pressure p? may not
vanish at the boundary r = a of the sphere. However, the radial pressure is
equal to the tangential pressure at the centre of the ®uid sphere.
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By matching (2.18) on the boundary of the anisotropic sphere and by denoting
Áa = c0 a2 , we obtain
µ
¶µ
¶
¢ 0¡ K
Áa
Van is (a) = 1 ¡
= 1 ¡ 2uan is ;
(2.23)
1¡ K
1 + Áa
where generally u denotes the mass{radius ratio of the isotropic or anisotropic star
and the subscript `a’ represents the value of the physical quantities at the vacuum
boundary of the star r = a.
For the isotropic case, that is for ¢ 0 = 0, it is easy to show that
µ
¶
K
Áa
uis o =
:
2(K ¡ 1) 1 + Áa
Therefore, the mass{radius ratios for the anisotropic and isotropic spheres are related
in the present model by
K¡ ¢ 0
uan is =
uis o :
(2.24)
K
Hence, the constants C and B appearing in the solution can be evaluated from
the boundary conditions. Thus, we obtain
Áa = c 0 a2 =
C = (1 ¡
2uan
is
)f(! ¡
2(1 ¡ K)uis o
2(1 ¡ K)uan is
=
2(K ¡ 1)uis o ¡ K
2(K ¡ 1)uan is ¡ K + ¢
1) sin[(! + 1)³
a
+ B] + (! + 1) sin[(! ¡
1)³
;
(2.25)
0
a
+ B]g¡2 :
(2.26)
Using (2.21) and the boundary condition that the radial pressure pr vanishes at
the radius r = a of the ®uid sphere, we obtain
·
¸
K + ¢ 0 (1 ¡ 2K)
tan ³ a ¡ tan(!³ a )
!(¢ 0 ¡ K)
·
¸
tan B =
:
(2.27)
K + ¢ 0 (1 ¡ 2K )
1+
tan(!³ a ) tan ³ a
!(¢ 0 ¡ K)
By de­ ning tan(!³ a ) = [tan(!³ )]r= a , it is easy to show that
s
(¢ 0 ¡ 1)(1 + Áa )
cos ³ a =
;
(¢ 0 ¡ K)
s
(¢ 0 ¡ K)
tan ³ a =
¡ 1;
(¢ 0 ¡ 1)(1 + Áa )
"
Ãs
!#
(¢ 0 ¡ K)
¡1
tan(!³ a ) = tan ! tan
¡ 1 :
(¢ 0 ¡ 1)(1 + Áa )
(2.28)
(2.29)
(2.30)
The mathematical consistency of (2.28) requires the condition (¢ 0 ¡ 1)(¢ 0 ¡ K) > 0
be satis­ ed.
In order to ­ nd general constraints for the anisotropy parameter ¢ 0 , K and uan is ,
we shall consider that the conditions » 0 = » (0) > 0, p0 = p(0) > 0 and R0 = R(0) =
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M. K. Mak and T. Harkoon November 17, 2014
5
p0, R0, r
0
4
3
2
1
0
0
0.1
0.2
0.3
D
0.4
0.5
0.6
0.7
0
Figure 1. Variations of the trace R0 = » 0 ¡ 3p 0 of the energy{momentum tensor at the centre
of the anisotropic star (solid line), of the central density » 0 (dotted line) and of the central
pressure p0 (dashed line) as functions of the anisotropy parameter ¢ 0 , for K = 2:6 and for a
given mass{radius ratio uis o = 0:24.
0 ¡ 3p0 > 0 hold at the centre of the ®uid sphere. Subsequently, the parameters ¢
K and uan is should be restricted to obey the conditions
»
0
=
3(¢
¡
K)
> 0;
K
p
(¢ 0 ¡ K) 1 + K + 4K(K ¡ 1) cot B
p
p0 =
> 0;
(K ¡ 1) 1 + K
p
6[(K ¡ ¢ 0 ) 1 + K ¡ 2K(K ¡ 1) cot B]
p
R0 =
> 0:
(K ¡ 1) 1 + K
»
0
0,
(2.31)
1¡
(2.32)
(2.33)
Clearly, the condition of the non-negativity of the energy density is satis­ ed only
if (¢ 0 ¡ K)(1 ¡ K) > 0.
The general behaviour of the functions
» 0 (K; ¢
0 );
p0 (K; ¢
0 ; uis o )
and
R0 (K; ¢
0 ; uis o )
is represented in ­ gure 1.
Since cos ³ a 6 1, it follows from (2.28) that the parameters K, ¢
satisfy the condition
(¢ 0 ¡ 1)(1 + Áa )
6 1:
(¢ 0 ¡ K)
0
and Áa must
(2.34)
That the mathematical consistency condition for equations (2.30) be well de­ ned
also requires that equation (2.34) holds. The upper limit of the radius a of the
anisotropic star can be obtained from (2.34) in the form
s
1¡ K
a6
:
(2.35)
c0 (¢ 0 ¡ 1)
Equation (2.35) requires that the parameters K and ¢
(1 ¡ K)(¢ 0 ¡ 1) > 0.
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satisfy the condition
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401
0.32
0.30
0.28
zc 0.26
0.24
0.22
0.20
0
0.1
0.2
0.3
D 0
0.4
0.5
0.6
Figure 2. Variation of the central redshift zc as a function of the anisotropy parameter ¢ 0
for the static anisotropic ° uid sphere with K = 2:2 and a mass{radius ratio u is o = 0:24.
Since, from (2.23), Áa can be de­ ned in terms of the total mass Man
Áa =
¢
0
2(1 ¡ K)Man is =a
K ¡ 2(1 ¡ K)Man
¡
is
=a
is
as
;
equation (2.34) gives the minimum allowed mass of the anisotropic star in the form
Man
is
>
¢ 0¡ 1
c0 a 3 :
2(1 ¡ K)
(2.36)
The conditions of the non-negativity of the energy density and maximum radius of
the star give the allowed ranges of the parameters ¢ 0 and K. There are two ranges
for these parameters, making the physical requirements mathematically consistent,
and they are given by (i) K 6 1 and ¢ 0 > 1 > K and (ii) K > 1 and ¢ 0 6 1 6 K.
For values of ¢ 0 and K which do not belong to these sets, the ­ eld equations do not
have a physical solution.
3. Astrophysical applications
When the thermonuclear sources of energy in its interior are exhausted, a spherical
star begins to collapse under the in®uence of gravitational interaction of its matter
content. The energy continues to increase and the star ends up as a compact relativistic cosmic object such as a neutron star, strange star or black hole. Important
observational quantities for such objects are the surface redshift, the central redshift
and the mass and radius of the star.
For a relativistic anisotropic star described by the solution presented in x 2, the
surface redshift zs is given by
zs = (1 ¡
2uan
is
)¡1=2 ¡
1:
(3.1)
The surface redshift is decreasing with increasing ¢ 0 . Hence, at least in principle,
the study of the redshift of light emitted at the surface of compact objects can lead
to the possibility of observational detection of anisotropies in the internal pressure
distribution of relativistic stars.
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M. K. Mak and T. Harkoon November 17, 2014
2
D
N
1
1.5
1.0
0
5
0.5
10
aN
15
MN
0
Figure 3. Variation of the anisotropy parameter of the neutron star ¢ N (in units of 104:7 ) as
a function of the radius of the star aN (in km) and of the mass MN of the star (in solar mass
units), for the mass{radius ratio uis o 2 (0:01; 0:18) and ¢ 0 2 (0; 0:6). We have used the value
K = 2:2.
The central redshift zc is of the form
zc = C ¡1=4 f(! ¡
1) sin[(! + 1)³
0
+ B] + (! + 1) sin[(! ¡
where
1)³
0
+ B]g¡1=2 ¡
1; (3.2)
r
¢ 0¡ 1
:
¢ 0¡ K
The variation of the central redshift of the neutron star against the anisotropy parameter is represented in ­ gure 2.
Clearly, in our model the anisotropy introduced in the pressure gives rise to a
decrease in zc . Hence, as functions of the anisotropy, the central and surface redshifts
have the same behaviour.
The stellar model presented here can be used to describe the interior of the realistic
neutron star. Taking the surface density of the star as » s = 2 £ 1014 g cm¡3 and with
the use of (2.25) we obtain
³
0
= cos
» s a2N =
(¢
¡1
0
(1 ¡
¡
K)Áa (3 + Áa )
;
K)(1 + Áa )2
(3.3)
or
23:144 p
(¢ 0 ¡ K)uis o [4(K ¡ 1)uis o ¡ 3K ];
(3.4)
K
where aN is the radius (in kilometres) of the neutron star corresponding to a speci­ c
surface density.
aN =
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0.4
dp / d r
0.3
0.2
0.1
0
0
0.1
0.2
0.3
D 0
0.4
0.5
0.6
Figure 4. Variations of the radial and tangential speeds of sound dp=d» at the vacuum boundary
and at the centre of the anisotropic ° uid sphere as a function of the anisotropy parameter ¢ 0
for uis o = 0:2 and K = 2:2: (dpr =d» )r= 0 (solid line), (dp ? =d» )r= 0 (dotted line), (dpr =d» )r= a N
(dashed line) and (dp ? =d» )r= a N (long-dashed line).
For the mass MN and anisotropy parameter ¢
MN =
15:67
(K ¡
K2
¢
N
¢
0 )uis o
p
(¢
0
= 3:595 02 £ 1035
(¢
¡
N
we ­ nd
K )uis o [4(K ¡
0
¡
1)uis
o
(K ¡ 1)2 ¢ 0 uis o
K)[4(K ¡ 1)uis o ¡
¡
3K]M­ ;
(3.5)
:
(3.6)
3K]
In the equations above we have expressed, for the sake of simplicity, all the quantities in international units, instead of natural units, by means of the transformations
MN
GM
! 8º 2 N ;
aN
c aN
» ! » c2
and
¢
N
!
8º G
¢
c4
N;
where G = 6:6732 £ 10¡8 dyn cm2 g¡2 (1 dyn = 105 N) and c = 2:997 925 £
1010 cm s¡1 .
The variation of the anisotropy parameter ¢ N of the neutron star as a function of
the radius aN and mass MN is represented in ­ gure 3.
The quantities
µ
¶
µ
¶
µ
¶
µ
¶
dpr
dp?
dpr
dp?
;
;
and
d»
d»
d»
d»
r= 0
r= 0
r= aN
r= aN
are represented against the anisotropy parameter ¢ 0 in ­ gure 4.
The plots indicate that the necessary and su¯ cient criterion for the adiabatic
speed of sound to be less than the speed of light is satis­ ed by our solution. However,
Caporaso & Brecher (1979) claimed that dp=d» does not represent the signal speed.
Therefore if this speed exceeds the speed of light, this does not necessary mean that
the ®uid is non-causal. But this argument is quite controversial and not all authors
accept it (Glass 1983).
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M. K. Mak and T. Harkoon November 17, 2014
4. Discussions and ¯nal remarks
In this paper we have presented the general solution of the gravitational-­ eld equations for an anisotropic static matter distribution. To derive the solution we have
used two basic assumptions: the functional form of the mass function ² (r) and a
speci­ c mathematical representation of the anisotropy parameter ¢ . In our model,
the anisotropy vanishes at the centre of the star, reaching its maximum near the
vacuum boundary. This behaviour is similar, from a phenomenological point of view,
to the behaviour of the anisotropy parameter of boson stars (Dev & Gleiser 2000).
Boson stars are gravitationally bound macroscopic quantum states made of scalar
bosons (Jetzer 1992). They di¬er from the fermionic stars in that they are only
prevented from collapsing gravitationally by the Heisenberg uncertainty principle.
Boson stars, described by non-interacting, massive scalar ­ elds ¿ with non-zero
spatial gradients, are anisotropic objects, with ¢ = p? ¡ pr = ¡ (d¿ =dr)2 . Hence,
our ¢ has the same qualitative properties as the anisotropy parameter of boson
stars.
Another possible source of the anisotropy as described by (2.10) could be an
anisotropic velocity distribution of the particles inside the star. Most of the models of stellar structure assume that the equation of state is due to a single species
of ideal noninteracting fermions, with an isotropic distribution of momenta. If, due
to the presence of a magnetic ­ eld, convection, turbulence, etc., the local velocity distribution is anisotropic, then the pressure distribution is also anisotropic. To
evaluate this e¬ect let us consider a spherical con­ guration of collisionless particles that exhibit no net velocity ®ow. The radial, azimuthal and polar velocities are vr , v¿ and v³ , respectively. One can then show that for a system
in Newtonian equilibrium ¢ = p? ¡ pr = » [hv³ 2 i ¡ hvr2 i], where the angle brackets
denote averages over the particle distribution (Herrera & Santos 1997). Moreover,
hv³ 2 i ¡ hvr2 i = ¡ © hvr2 i (Herrera & Santos 1997), where © is a parameter measuring the
anisotropy of velocity distributions, which for simplicity we assume to be a constant.
Therefore,
µ
¶·
¸
¢ 0¡ K
3+Á
2
2
¢ = » [hv³ i ¡ hvr i] = ¡ c0
© hvr2 i:
(4.1)
1¡ K
(1 + Á)2
But with the use of the virial theorem (the kinetic energy of particles is equal
to minus one-half of their potential energy) we obtain hvr2 i ¹ ¡ m=r, where m is
the mass of the star. Approximating the density by a constant (at least locally
this is generally a good approximation for general relativistic compact objects), it
follows that hvr2 i ¹ ¡ r 2 ¹ ¡ Á. Substituting this expression of the average of the
square of the radial velocity in (3.1), by choosing an appropriate value for the constant © , and by neglecting in the numerator the second-order term proportional
to Á 2 (Á ¹ r 2 =a2 ½ 1 for r < a), we obtain the exact form of the anisotropy
ansatz (2.10).
The scalar invariants of the Riemann tensor are important, since they allow a
manifestly coordinate-invariant characterization of certain properties of geometries,
such as curvature singularities, the Petrov type of the Weyl tensor, etc. Two scalars
which have been extensively used in the physical literature are the Kretschmann
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Anisotropic
stars in general relativity
405
scalars
RiemSq ² Rijkl R ijkl
µ
= 2¢ + » + pr ¡
2
RicciSq ² Rij Rij = »
4m
r3
¶2
µ
¶2
µ
2m
+ 2 pr + 3
+2 » ¡
r
+ 3p2r + 2¢ (¢
2m
r3
+ 2pr );
¶2
+4
µ
2m
r3
¶2
;
where Rijkl is the Riemann curvature tensor.
For space-times which are the product of two two-dimensional spaces, one
Lorentzian and one Riemannian, subject to a separability condition on the function which couples the 2-spaces, Santosuosso et al . (1998) have suggested that the
set
C = fR; r1 ; r2 ; w2 g;
(4.2)
forms an independent set of scalar polynomial invariants, satisfying the number of
degrees of freedom in the curvature. In (4.2) R = g il g jk Rijkl is the Ricci scalar and
the quantities r1 , r2 and w2 are de­ ned according to (Zakhary & Carminati 2001;
Carminati et al. 2002)
= 14 Sab Sba ;
AB A_ B_ ¿
r2 = ¿
B B_ CA C_ A_
AB A_ B_ ¿ C C_ ¿
CD
ABCD ª EF ª
w2 = ª
Sab
AB A_ B_
r1 = ¿
=¡
EF AB
(4.3)
1 b c a
S S S ;
8 a b c
=¡
(4.4)
1 ·
C
C· cd C· ef ab ;
8 abcd ef
(4.5)
¡ 14 R¯ ab is the trace-free Ricci tensor, ¿ AB A_ B_ denotes the
Sab , ª ABCD denotes the spinor equivalent of the Weyl
Rab
where
=
spinor
equivalent of
tensor
Cabcd and C·abcd denotes the complex conjugate of the self-dual Weyl tensor
+
Cabcd
= 12 (Cabcd ¡ i ¤ Cabcd ).
¤
In terms of the `electric’ Eac = Cabcd ub ud and `magnetic’ Hac = Cabcd
ub ud parts
ef
1
¤
a
of the Weyl tensor, where u is a timelike unit vector and Cabcd = 2 ² abef Ccd
is the
dual tensor, the invariant w2 is given by (Santosuosso et al . 1998)
w2 =
1
(3Eba Hcb Hac
32
Eba Ecb Eac ) +
¡
1
i(Hba Hcb Hac
32
¡
3Eba Ecb Hac ):
(4.6)
For the case of a perfect ®uid with anisotropic pressure distribution, the set of
invariants (4.2) is given by
R=» ¡
r1 =
r2 =
8(61=3 )(w2 )2=3 =
3pr ¡ 2¢ ;
1
[9(» + pr + 23 ¢ )2 + (»
64
1
¡ 512
[27(» + pr + 23 ¢ )3
µ
» + pr + 2¢
¡
1
(»
6
¡
¡
¡
3pr ¡
1
(»
8
4m
r3
¶2
2¢ )2 ¡
+ pr ¡
2¢
(4.7)
+ pr ¡
2
2¢ ) + 2(» + pr + 2¢ ) ];
+ (» + pr ¡
µ
+ 2 pr +
9
(» +
8
)2 ¡ 14 (»
2
3
(4.8)
3
2¢ ) + 2(» + pr + 2¢ ) ];
2m
r3
¶2
µ
+2 » ¡
pr + 23 ¢ )2
+ pr + 2¢ )2 :
2m
r3
¶2
(4.9)
¶2
2m
+4
r3
µ
(4.10)
The imaginary part of the invariant w2 is necessarily zero in spherical symmetry.
When ¢ = 0 and pr = p? = p, we recover the form of the Ricci invariants rn =
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M. K. Mak and T. Harkoon November 17, 2014
20
R, r1, r2, w2
10
0
- 10
- 20
0
0.2
0.4
D 0
0.6
0.8
Figure 5. Variations, at the centre r = 0 of the static anisotropic ° uid sphere, of the curvature scalar R (solid line), of the Ricci invariants r1 (dotted line), r 2 (dashed line) and of w 2
(long-dashed line) against the anisotropy parameter ¢ 0 , for uis o = 0:29 and K = 2:2.
cn (» + p)n+ 1 , cn = const:, for class B warped product space-times with isotropic
perfect ®uid matter sources (Santosuosso et al. 1998).
The variations of R, r1 , r2 and of w2 at the centre of the ®uid sphere are represented, as a function of the anisotropy parameter ¢ 0 , in ­ gure 5.
The scalars r1 , r2 and w2 are ­ nite at the centre of the ®uid sphere. r1 and w2
monotonically decrease, while r2 increases monotonically as the anisotropy parameter
¢ 0 increases.
For a particular choice of the equation of state at the centre of the star, 3pr0 =
3p?0 = » 0 and with a vanishing anisotropy parameter, ¢ 0 = 0, we obtain the result
uis o = 0:29 for a static isotropic ®uid sphere.
It is generally held that the trace T of the energy{momentum tensor must be
non-negative. It is also the case that this trace condition is everywhere ful­ lled if it
is ful­ lled at the centre of the star (Knutsen 1988).
The purpose of this paper is to present some exact models of static anisotropic ®uid
stars and to investigate their possible astrophysical relevance. All the solutions we
have obtained are non-singular inside the anisotropic sphere, with ­ nite values of the
density and pressure at the centre of the star. Variations of the physical parameters
mass, radius, redshift and adiabatic speed of sound against the anisotropy parameter
have been presented graphically. Because it satis­ es all the physical conditions and
requirements (i){(vi), the present model can be used to study the interior of the
anisotropic relativistic objects.
We thank the two referees, whose comments helped us to improve the manuscript.
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