Charged anisotropic matter with linear or nonlinear equation of state

Victor Varela*

Institute of Mathematics, Kings College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Farook Rahaman†

Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India

Saibal Ray‡

Department of Physics, Government College of Engineering and Ceramic Technology, Kolkata 700 010, West Bengal, India

Koushik Chakrabortyx

Department of Physics, Government Training College, Hooghly 712103, India

Mehedi Kalamk1Department of Physics, Netaji Nagar College for Women, Kolkata 700092, India

(Received 13 April 2010; published 31 August 2010)

Ivanov pointed out substantial analytical difficulties associated with self-gravitating, static, isotropic

fluid spheres when pressure explicitly depends on matter density. Simplifications achieved with the

introduction of electric charge were noticed as well. We deal with self-gravitating, charged, anisotropic

fluids and get even more flexibility in solving the Einstein-Maxwell equations. In order to discuss

analytical solutions we extend Krori and Barua’s method to include pressure anisotropy and linear or

nonlinear equations of state. The field equations are reduced to a system of three algebraic equations for

the anisotropic pressures as well as matter and electrostatic energy densities. Attention is paid to compact

sources characterized by positive matter density and positive radial pressure. Arising solutions satisfy the

energy conditions of general relativity. Spheres with vanishing net charge contain fluid elements with

unbounded proper charge density located at the fluid-vacuum interface. Notably the electric force acting

on these fluid elements is finite, although the acting electric field is zero. Net charges can be huge (1019C)

and maximum electric field intensities are very large (1023–1024 statvolt=cm) even in the case of zero net

charge. Inward-directed fluid forces caused by pressure anisotropy may allow equilibrium configurations

with larger net charges and electric field intensities than those found in studies of charged isotropic fluids.

Links of these results with charged strange quark stars as well as models of dark matter including massive

charged particles are highlighted. The van der Waals equation of state leading to matter densities

constrained by cubic polynomial equations is briefly considered. The fundamental question of stability

is left open.

DOI: 10.1103/PhysRevD.82.044052 PACS numbers: 04.40.Nr, 04.20.Jb, 04.40.Dg

I. INTRODUCTION

Self-gravitating fluid models are essential to many ap-plications of general relativity, ranging from the histori-cally important classical models of elementary particles tocurrent computational and observational problems in as-trophysics and cosmology. These systems are usually char-acterized by sets of physical variables which outnumberthe independent field equations. Consistent problems canbe posed if additional restrictions on the variables arespecified, which may take the form of equations of state(EOS).

Using an EOS to describe a self-gravitating fluid hasimportant consequences when it comes to solving the fieldequations. For example, Ivanov [1] has pointed out thatfinding analytical solutions in the static, spherically sym-metric, uncharged case of a perfect fluid with linear EOS isan extremely difficult problem.Interestingly, analytical difficulties may alleviate with

increasing physical complexity. This situation has beenillustrated by Sharma and Maharaj [2] in the case of astatic, spherically symmetric, uncharged anisotropic fluid.These authors chose a particular mass function to reduceand easily solve the system of field equations combinedwith a linear EOS.The surprising simplification of the field equations for a

charged perfect fluid satisfying a linear EOS was discussedby Ivanov [1], who showed how to reduce the systeminvolving the most general linear EOS to a linear differen-

*victor.varela.abdn@gmail.com†farook_rahaman@yahoo.com‡saibal@iucaa.ernet.inxkchakraborty28@yahoo.comkmehedikalam@yahoo.co.in

PHYSICAL REVIEW D 82, 044052 (2010)

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tial equation for one metric component. However, Ivanovalso pointed out that the use of a polytropic EOS leads tononintegrable equations.

Electrically charged fluids with anisotropic pressuresconstitute the next level of physical complexity.

Charged, self-gravitating anisotropic fluid spheres havebeen investigated in general relativity since the pioneeringwork of Bonnor [3]. Recently, this type of charged matterhas been considered by Horvat, Ilijic, and Marunovic instudies of gravastars [4]. Models with prescribed EOSremain relatively unexplored.

Motivated by MIT bag models of strange stars,Thirukkanesh and Maharaj [5] derived solutions forcharged anisotropic fluids with linear EOS from specificchoices of one metric function and the electrostatic energydensity. Their method—an extension of the procedurepresented in [2] for uncharged anisotropic fluids-providesmotivation for the completely different approach devel-oped in this paper.

Current dark matter and dark energy models are asso-ciated with nonlinear EOS. Density perturbations may leadto fluid nucleation characterized by heterogeneous matterdensities and pressures. Assuming specific EOS, boundedstatic fluid distributions and asymptotically flat spacetimes,we aim to study the arising self-gravitating objects. Weinsist on analytical solutions and note that prescribed EOSmay cause serious trouble. However, we find that theconvenient choice of a ‘‘frozen’’ internal metric reducesthe Einstein-charged fluid equations to a system of linearalgebraic equations for matter and electrostatic energydensities as well as anisotropic pressures. In a way, weare led to the simplest solution method available for thistype of source with prescribed EOS and asymptotically flatspacetime. Models based on linear and nonlinear EOS arecompletely solved following essentially the same proce-dure. Remarkably, our approach uncovers the effects ofdifferent EOS on hydrostatic and electrical variables with-out interfering changes of internal metric (apart from ad-justable numerical factors depending on junctionconditions). This treatment offers a fresh view of therelationships among EOS, charge distributions and pres-sure anisotropy.

In Sec. II we write the field equations and briefly reviewthe work of Thirukkanesh and Maharaj. Also, the originalKrori and Barua solution method is generalized to dealwith charged anisotropic sources with regular interiors.Our approach to linear and nonlinear EOS leading tomodels with positive definite matter density is presentedin Sec. III. In Sec. IV we discuss junction conditions andreformulate our procedure in terms of dimensionless quan-tities. Section V includes a detailed analysis of models withpositive matter density and positive radial pressure, payingspecial attention to conditions for physical acceptability aswell as equilibrium conditions and the values of key physi-cal parameters in Gaussian-cgs units. In Sec. VI we specu-

late on the origin of charge in models with linear andnonlinear EOS, and check our solution method against amore complicated EOS describing quintessence stars.Finally, we suggest avenues for further research involvingstability analysis and gravitational collapse.

II. CHARGED ANISOTROPIC MATTER

The starting point is the static, spherically symmetricline element represented in curvature coordinates. It reads

ds2 ¼ e�dt2 � e�dr2 � r2d�2 � r2sin2�d�2; (1)

where � ¼ �ðrÞ and � ¼ �ðrÞ. For the static, chargedsource with density � ¼ �ðrÞ, radial pressure pr ¼ prðrÞ,tangential pressure pt ¼ ptðrÞ, proper charge density � ¼�ðrÞ, and electric field E ¼ EðrÞ the Einstein-Maxwell(EM) equations take the form

8��þ E2 ¼ e��

��0

r� 1

r2

�þ 1

r2; (2)

8�pr � E2 ¼ e��

��0

rþ 1

r2

�� 1

r2; (3)

8�pt þ E2 ¼ e��

2

��00 þ �02

2þ �0 � �0

r� �0�0

2

�; (4)

� ¼ e�ð�=2Þ

4�r2ðr2EÞ0; (5)

where the primes denote differentiation with respect to r,and geometrized units (G ¼ c ¼ 1) are employed.Equations (2)–(5) are invariant under the transformation

E ! �E, � ! ��. In this work we exclusively deal withthe positive square root of E2.With our choice of radial coordinate, the metric function

e� and electrostatic energy density E2 assumed in [5] takethe forms

e� ¼ 1þ ar2

1þ ða� bÞr2 ; (6)

E2 ¼ kð3þ ar2Þð1þ ar2Þ2 ; (7)

where a, b, k are arbitrary constants.The combination of (6) and (7) with (2) yields

� ¼ ðb� kÞð3þ ar2Þ8�ð1þ ar2Þ2 : (8)

Hence � and E2 are proportional when k � 0 and k � b.This expression for �ðrÞ is joined to a linear EOS toprovide prðrÞ. The explicit form of prðrÞ together with(6), (7), and (3) imply a linear differential equation for�ðrÞ which is analytically solved. Finally, when (6) and (7)and the explicit form of �ðrÞ are substituted into (4) we getthe corresponding expression for ptðrÞ.

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The above procedure leads to analytical solutions whichdepend on a number of free parameters. The analysispresented in [5] considers sensible choices for these pa-rameters in the context of charged stars. However, we finda singularity in the charge distribution at r ¼ 0 when (6)and (7) are combined with (5).

From (6) we see that �ðrÞ satisfies �ð0Þ ¼ 0. Assumingk > 0, (5) implies

�ðrÞ �ffiffiffiffiffi3k

p2�

1

r(9)

for small r, which diverges at r ¼ 0. The electric fieldassociated with (7) does not vanish at r ¼ 0. This choicefor EðrÞ prevents the regularity of the charge distribution atthe center of the sphere.

In our view, the vanishing of the electric field at thecenter of a spherically symmetric charge distributionshould be a condition for physical relevance of the solu-tion. Furthermore the significance of charge density singu-larities is unclear and we aim to investigate their possibleoccurrence in models satisfying Eð0Þ ¼ 0. To this end, weextend Krori and Barua’s approach to charged isotropicfluid sources [6] and deal with charged anisotropic sourceswith prescribed EOS.

Krori and Barua (KB) constructed singularity-free mod-els of static, charged perfect fluids with metric (1) given by

� ¼ Ar2; (10)

� ¼ Br2 þ C; (11)

where A, B, and C are constants. This internal metricsatisfies the conditions for regularity at r ¼ 0 discussedby Lake and Musgrave [7]. As a consequence of thischoice, Eqs. (2)–(4) with pr ¼ pt ¼ p were reduced to asystem of three linear algebraic equations for �, p, and E2.Furthermore � was obtained combining (5) with thechosen square root of E2 and the assumed form of �.

We rewrite the field equations as

8��þ E2 ¼ fðrÞ; (12)

8�pr � E2 ¼ hðrÞ; (13)

8�pt þ E2 ¼ jðrÞ; (14)

where fðrÞ, hðrÞ, jðrÞ are determined by the right sides of(2)–(4), (10), and (11), namely

fðrÞ ¼ e�Ar2�2A� 1

r2

�þ 1

r2; (15)

hðrÞ ¼ e�Ar2�2Bþ 1

r2

�� 1

r2; (16)

jðrÞ ¼ e�Ar2½Bþ B2r2 þ ðB� AÞ � ABr2�: (17)

We impose center and boundary conditions on EðrÞ andprðrÞ, respectively:

Eð0Þ ¼ 0; (18)

prðaÞ ¼ 0; (19)

where a is a positive constant and r ¼ a defines thecharged fluid-vacuum interface.From (12)–(19), we obtain

8��ð0Þ ¼ 3A; (20)

4��ðaÞ ¼ ðAþ BÞe�a2A; (21)

8�prð0Þ ¼ 2B� A; (22)

8�ptð0Þ ¼ 2B� A; (23)

8�ptðaÞ ¼ e�a2A

�4B� Aþ a2ðB2 � ABÞ þ 1

a2

�� 1

a2;

(24)

E2ðaÞ ¼ 1

a2� e�a2A

�2Bþ 1

a2

�: (25)

From (22) and (23), we conclude that only one pressurevalue is associated with r ¼ 0. Hence prð0Þ and ptð0Þdenote the same quantity i.e. central pressure.Equations (12)–(14) yield general expressions for pt and

E2 namely

pt ¼ jðrÞ � fðrÞ8�

þ �; (26)

E2 ¼ fðrÞ � 8��; (27)

where � is still undetermined.

III. LINEAR OR NONLINEAR EQUATIONOF STATE

At this point we select an EOS with the general form

pr ¼ prð�;�1; �2Þ; (28)

where �1 and �2 are constant parameters.These parameters are constrained by the system of equa-

tions

prð0Þ ¼ pr½�ð0Þ; �1; �2�; (29)

0 ¼ pr½�ðaÞ; �1; �2�: (30)

Combining (12) and (13) we get

�þ pr ¼ fðrÞ þ hðrÞ8�

; (31)

which may be solved with the assumed EOS to generate

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specific forms of � and pr. The arising � is put into (26)and (27) to yield general expressions for pt and E2.

First, we consider the linear EOS

pr ¼ �1 þ �2�: (32)

Following the outlined procedure we obtain

� ¼18� ½fðrÞ þ hðrÞ� � �1

1þ �2

; (33)

pr ¼�1 þ �2

8� ½fðrÞ þ hðrÞ�1þ �2

; (34)

pt ¼ jðrÞ þ hðrÞ þ �2½jðrÞ � fðrÞ� � 8��1

8�ð1þ �2Þ ; (35)

E2 ¼ 8��1 þ �2fðrÞ � hðrÞ1þ �2

: (36)

Taking into account (32) we solve the system (29) and (30)and get

�1 ¼ � �ðaÞprð0Þ�ð0Þ � �ðaÞ ; �2 ¼ prð0Þ

�ð0Þ � �ðaÞ : (37)

Second, we deal with the nonlinear EOS

pr ¼ 1 þ 2

�n ; (38)

where n � �1. Combining (38) with (31) we obtain apolynomial equation for � which is quadratic only for n ¼�2 or n ¼ 1. The second choice for n yields

� ¼ kðrÞ � 1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½kðrÞ � 1�2 � 42

p2

; (39)

where

kðrÞ ¼ fðrÞ þ hðrÞ8�

: (40)

Solving (29) and (30) with (38) and n ¼ 1 we get

1 ¼ �ð0Þprð0Þ�ð0Þ � �ðaÞ ; 2 ¼ ��ð0Þ�ðaÞprð0Þ

�ð0Þ � �ðaÞ : (41)

If 2 < 0 then each root in (39) has definite sign.Particularly, the root with the positive radical determinesa positive definite matter density.

Equation (38) is a modification of the Chaplygin gasEOS used by Bertolami and Paramos to describe neutraldark stars [8]. Actually, the Chaplygin gas EOS is poly-tropic with negative polytropic index. The additionalterm 1 allows for a charged fluid-vacuum interface wherepr ¼ 0.

Third, we consider the modified Chaplygin EOS

pr ¼ 1�þ 2

�(42)

used in the study of static, neutral, phantomlike sourcespresented by Jamil, Farooq, and Rashid [9].Equations (29) and (30) combined with (42) yield

1 ¼ �ð0Þprð0Þ�ð0Þ2 � �ðaÞ2 ; 2 ¼ ��ð0Þprð0Þ�ðaÞ2

�ð0Þ2 � �ðaÞ2 : (43)

Putting (42) into (31) we get a quadratic equation for �which admits the solutions

� ¼ kðrÞ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikðrÞ2 � 4ð1þ 1Þ2

p2ð1þ 1Þ : (44)

If 1þ 1 > 0 and 2 < 0 then the sign of the radical in(44) uniquely determines the sign of �. As in the previouscase, a positive radical leads to positive definite matterdensity.Finally, the expressions for � selected from (39) and (44)

are combined with the corresponding EOS as well as (26)and (27) to yield solutions for pr, pt, and E2.It is clear that the constant parameters appearing in (28)

play a central role in our extension of the KB method.Their values determine the existence of positive and nega-tive definite matter densities �ðrÞ. Furthermore, they lead

to numerical values for quantities like dpr

d� , which is asso-

ciated with sound propagation in anisotropic charged fluids[10].

IV. JUNCTION CONDITIONS ANDADIMENSIONAL FORMULATION

Equations (37), (41), and (43) show that the constantparameters are determined by �ð0Þ, �ðaÞ and prð0Þ, which,according to (20) and (22), are functions of the KB con-stants A and B appearing in (10) and (11). Notably, thesetwo constants as well as C are fixed by suitable junctionconditions imposed on the internal and external metrics atr ¼ a.The external Reissner-Nordstrom (RN) metric is given

by (1) with

e�ðrÞ ¼ 1� 2m

rþ q2

r2; e�ðrÞ ¼

�1� 2m

rþ q2

r2

��1:

(45)

Junevicus [11] derived expressions for A, B, C from thecontinuity of the first and second fundamental forms acrossthe surface of the charged fluid sphere. In terms of the

dimensionless parameters � ¼ ma and � ¼ jqj

a his results

take the form

A ¼ � 1

a2lnð1� 2�þ �2Þ; (46)

B ¼ 1

a2

��� �2

1� 2�þ �2

�; (47)

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C ¼ �2 ��

1� 2�þ �2þ lnð1� 2�þ �2Þ: (48)

Combining (25) with (46) and (47) we obtain the elec-trostatic energy density at the surface of the sphere,namely,

EðaÞ2 ¼ q2

a4: (49)

On the other hand, using (45) in the electrovac (� ¼ 0)case of (2) we see that the same quantity evaluated in theexternal (r > 0) spacetime region is

EðrÞ2 ¼ q2

r4: (50)

These results are compatible with the continuity of theelectric field at r ¼ a.

For simplicity in numerical calculations, we reformulateour results in terms of dimensionless quantities. From (10)and (11) we see that A, B have dimension of length�2

whereas C is dimensionless. Clearly fðrÞ, hðrÞ, jðrÞ aswell as �, pr, pt, E

2, � also have dimension of length�2.We get the dimensionless version of any variable or con-stant parameter multiplying by the appropriate power of a.Here adimensionality is denoted by tildes, though quanti-ties which are originally dimensionless are denoted by theoriginal symbol. The dimensionless radial marker x ¼ r

a is

used so the interior of the fluid sphere is described withx 2 ½0; 1Þ.

For the first model we get

~p r ¼ ~�1 þ �2 ~�; (51)

~� ¼18� ½~fðxÞ þ ~hðxÞ� � ~�1

1þ �2

; (52)

~p r ¼~�1 þ �2

8� ½~fðxÞ þ ~hðxÞ�1þ �2

; (53)

~p t ¼ ~|ðxÞ þ ~hðxÞ þ �2½~|ðxÞ � ~fðxÞ� � 8�~�1

8�ð1þ �2Þ ; (54)

~E 2 ¼ 8�~�1 þ �2~fðxÞ � ~hðxÞ

1þ �2

; (55)

~� 1 ¼ � ~�ð1Þ~prð0Þ~�ð0Þ � ~�ð1Þ ; �2 ¼ ~prð0Þ

~�ð0Þ � ~�ð1Þ ; (56)

where

~fðxÞ ¼ e� ~Ax2�2 ~A� 1

x2

�þ 1

x2; (57)

~hðxÞ ¼ e� ~Ax2�2 ~Bþ 1

x2

�� 1

x2; (58)

~|ðxÞ ¼ e� ~Ax2

2½2 ~Bþ 2 ~B2x2 þ 2ð ~B� ~AÞ � 2 ~A ~Bx2�; (59)

~�ð0Þ ¼ 3 ~A

8�; (60)

~�ð1Þ ¼ ð ~Aþ ~BÞe� ~A

4�; (61)

~p rð0Þ ¼ 2 ~B� ~A

8�; (62)

~A ¼ a2A; ~B ¼ a2B: (63)

The second model is described with

~p r ¼ ~1 þ~2

~�; (64)

~� ¼~kðxÞ � ~1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½~kðxÞ � ~1�2 � 4 ~2

q2

; (65)

~ 1 ¼ ~�ð0Þ~prð0Þ~�ð0Þ � ~�ð1Þ ;

~2 ¼ � ~�ð0Þ~�ð1Þ~prð0Þ~�ð0Þ � ~�ð1Þ : (66)

For the third model we obtain

~p r ¼ 1 ~�þ ~2

~�; (67)

~� ¼~kðxÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~kðxÞ2 � 4ð1þ 1Þ~2

q2ð1þ 1Þ ; (68)

1 ¼ ~�ð0Þ~prð0Þ~�ð0Þ2 � ~�ð1Þ2 ; ~2 ¼ � ~�ð0Þ~prð0Þ~�ð1Þ2

~�ð0Þ2 � ~�ð1Þ2 : (69)

For models with nonlinear EOS, ~pt and ~E2 are calculatedwith the dimensionless versions of (26) and (27),respectively.From (49) we get

~Eð1Þ2 ¼ �2 (70)

which applies to the three models.Finally, the dimensionless proper charge density is given

by

~� ¼ e�ð ~Ax2=2Þ

4�x2d

dxðx2 ~EÞ: (71)

V. ANALYSIS OF THREE MODELS

We consider only external solutions (45) which excludehorizons. The standard analysis of the roots of g00 ¼ 0implies that the values of � are restricted by the values of�. Three cases arise:

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(1) � 2 ð0; 1Þ, 2�� 1<�2 <�2;(2) � 2 ð0; 1Þ, � ¼ �;(3) �> 0, �>�.From Eqs. (20) and (46) we see that �ð0Þ is positive if

and only if 2�� 1< �2 < 2�, which we admit as anadditional restriction on �.

The geometric mass in (45) is given by m ¼ GMc2

, where

G is the gravitational constant, M is the mass of thecharged source, and c is the speed of light—all in conven-tional units. If the mass M of the charged star equals onesolar mass and a ¼ 10 Km, then � ¼ GM

c2a� 0:147. Hence

2�� 1 is negative andffiffiffiffiffiffiffi2�

p � 0:543.Using (60) and (62) the central density ~�ð0Þ and central

pressure ~prð0Þ can be evaluated for � ¼ 0:147 and � 2½0; 0:543Þ. The monotonic decreases of these two parame-ters with increasing � are shown in Figs. 1 and 2. Weobserve that ~prð0Þ is positive only when � takes valuesin the range 0 � �< �0, where � ¼ �0 is the only zero of~prð0Þ in the interval � 2 ½0; 0:543Þ. It is approximatelygiven by �0 � 0:191.

For fixed�we see that ~�ð0Þ, ~�ð1Þ and ~prð0Þ depend onlyon �. Hence Eqs. (56), (66), and (69) provide expressionsfor the EOS parameters as functions of �. As shown in

Figs. 3–5, ~2ð�Þ< 0, 1ð�Þ> 0 and ~2ð�Þ< 0 for � ¼0:147 and � 2 ½0; �0Þ. These results allow us to select thepositive definite roots from (39) and (44), and discussmodels with ~�ðxÞ> 0.

For concreteness we carry out the numerical and graph-ical analysis of models with � ¼ 0:147 and � 2 ½0; �0Þ,characterized by positive matter density and positive pres-sure at x ¼ 0. Clearly � ¼ 0 describes sources with zero

net charge. This condition is compatible with nonzeroproper charge densities ~�ðxÞ. Higher values of � are asso-ciated with increasingly repulsive electrostatic forces thataffect pressure and matter density profiles. We aim to findout how the different forces that allow equilibrium con-figurations accommodate to varying net charge.To begin with we examine sources with linear EOS and a

selection of� values. The corresponding profiles for matter

FIG. 1 (color online). Central density ~�ð0Þ as a function of �for � ¼ 0:147 [y � ~�ð0Þ].

FIG. 2 (color online). Central pressure ~prð0Þ as a function of �for � ¼ 0:147 [y � ~prð0Þ].

FIG. 3 (color online). Coefficient ~2 as a function of � for� ¼ 0:147 (y � ~2).

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density, radial and tangential pressures, and electric fieldare displayed in Figs. 6–9. We observe decreasing values ofmatter density and pressure associated with increasingvalues of �. Figure 9 shows electric field profiles satisfying~Eð0Þ ¼ 0 and ~Eð1Þ ¼ �. Very large negative gradients of ~Edevelop near the surface of the sphere for � ¼ 0, and ~Eincreases at every x 2 ð0; 1� with increasing �. Notably

Fig. 7 indicates that increasing net charges and electricfields are associated with decreasing radial pressure gra-dients at each point.

FIG. 4 (color online). Coefficient ~1 as a function of � for� ¼ 0:147 (y � ~1).

FIG. 5 (color online). Coefficient ~2 as a function of � for� ¼ 0:147 (y � ~2).

FIG. 6 (color online). Matter density ~� as a function of x forfive different values of �. In this figure, the curves 1-5 from thetop (blue, cyan, green, tan, and red curves) correspond to � ¼ 0,0.05, 0.10, 0.15, and 0.19, respectively, (y � ~�).

FIG. 7 (color online). Radial pressure ~pr as a function of x forfive different values of �. In this figure, the curves 1-5 from thetop (blue, cyan, green, tan, and red curves) correspond to � ¼ 0,0.05, 0.10, 0.15, and 0.19, respectively, (y � ~pr).

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Figures 10 and 11 show profiles for the squares of radial

and tangential sound speeds, defined by v2sr ¼ dpr

d� and

v2st ¼ dpt

d� , respectively. We observe that v2sr is independent

of x and decreases with increasing �. For fixed �, v2st

monotonically decreases with increasing x; and increaseswith increasing � for fixed x. These parameters satisfy the

FIG. 8 (color online). Tangential pressure ~pt as a function of xfor five different values of �. In this figure, the curves 1-5 fromthe top (blue, cyan, green, tan, and red curves) correspond to� ¼ 0, 0.05, 0.10, 0.15, and 0.19, respectively, (y � ~pt).

FIG. 9 (color online). Electric field ~E as a function of x for fivedifferent values of �. In this figure, the curves 1-5 from thebottom (blue, cyan, green, tan, and red curves) correspond to� ¼ 0, 0.05, 0.10, 0.15, and 0.19, respectively, (y � ~E).

FIG. 10 (color online). Square of radial sound velocity v2sr as a

function of x for five different values of �. In this figure, thecurves 1-5 from the top (blue, cyan, green, tan, and red curves)correspond to � ¼ 0, 0.05, 0.10, 0.15, and 0.19, respectively,(y � v2

sr).

FIG. 11 (color online). Tangential sound velocity v2st as a

function of x for five different values of �. In this figure, thecurves 1-5 from the bottom (blue, cyan, green, tan, and redcurves) correspond to � ¼ 0, 0.05, 0.10, 0.15, and 0.19, respec-tively, (y � v2

st).

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inequalities 0 � v2sr � 1 and 0 � v2

st � 1 everywherewithin the charged fluid for the five values of � considered.

Based on the standard analysis of energy conditions forcharged anisotropic fluids (see, for example, [10]), we findthat the weak energy condition (WEC), the strong energycondition (SEC) and the dominant energy condition (DEC)are simultaneously satisfied if and only if the following sixinequalities hold at every point within the source:

~�þ ~pr � 0; (72)

~�þ ~E2

8�� 0; (73)

~�þ ~pt þ~E2

4�� 0; (74)

~�þ ~pr þ 2~pt þ~E2

4�� 0; (75)

~�þ ~E2

8��

��������~pr �~E2

8�

��������� 0; (76)

~�þ ~E2

8��

��������~pt þ~E2

8�

��������� 0: (77)

Inequalities (72) and (73) hold automatically for thesources considered here. Straightforward plotting of theleft sides of (74)–(77) shows that these inequalities aresatisfied as well at every x 2 ½0; 1�.

Further analysis shows that models with nonlinear EOS(64) and (67) are very similar to those satisfying (51).Particularly, matter densities and radial pressures areeverywhere positive, matter densities as well as radialand tangential pressures monotonically decrease with in-creasing x, radial and tangential sound speeds satisfy 0 �v2sr � 1 and 0 � v2

st � 1, vsr decreases with increasing �for fixed x, and WEC, SEC, and DEC are satisfied for� ¼0:147 and � ¼ 0, 0.05, 0.1, 0.15, 0.19. However, modelswith nonlinear EOS get vsr profiles which increase mono-tonically with increasing x and fixed �.

Another difference among the three models concerns the

anisotropy parameter ~� ¼ ~pt � ~pr. As shown in Figs. 12–

14, each EOS affect the dependence of ~� on x in a different

way. Particularly, the sign of ~� is the same at every x 2ð0; 1� for fixed � only in models with linear EOS. Theeffect of (64) on pressure anisotropy is notorious as three~�ðxÞ profiles develop sign changes in that case. Howeverthe predominance of radial pressure over tangential pres-sure for the highest value of � is a common feature of thethree models.

Are these three types of models physically meaningful?Delgaty and Lake found that only a relatively small num-ber of proposed perfect fluid sources for the Schwarzschildmetric satisfy a set of well established conditions forphysical acceptability [12]. These conditions include regu-larity of the origin, positive matter density and pressure,

decreasing matter density and pressure with increasing r,causal sound propagation, and smooth matching of theinternal and external metrics at the fluid-vacuum interface

FIG. 12 (color online). Anisotropic parameter ~� for the firstmodel as a function of x for five different values of �. In thisfigure, the curves 1-5 from the top (blue, cyan, green, tan, andred curves) correspond to � ¼ 0, 0.05, 0.10, 0.15, and 0.19,respectively, (y � ~�).

FIG. 13 (color online). Anisotropic parameter ~� for the secondmodel as a function of x for five different values of �. In thisfigure, the curves 1-5 from the top (blue, cyan, green, tan, andred curves) correspond to � ¼ 0, 0.05, 0.10, 0.15, and 0.19,respectively, (y � ~�).

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(r ¼ a). Some authors add the condition of monotonicallydecreasing sound speed with increasing r (see, for ex-ample, [13]). Regarding anisotropic fluid sources, the cau-sality condition 0 � v2

s � 1 has been imposed on radialand tangential sound speeds (see, for example, [14,15].Disagreement arises in connection with the sign of tangen-tial pressure which is unrestricted for some authors (see,for example, [14]) and strictly positive for others (see, forexample, [15]). The above discussion indicates that ourthree charged anisotropic sources satisfy most of the usualacceptability criteria. Conflict may arise only in connectionwith negative tangential pressures occurring for the highest� values, constant sound speeds vsr associated with thelinear EOS, and the increase of vsr with increasing xappearing in the models with nonlinear EOS.

Electric interactions and charge distributions in ourmodels deserve further analysis. We have found that ~Eprofiles are affected by the choice of EOS. Particularly,in models with � ¼ 0 the maximum ~E values are approxi-mately 0.03, 0.07, and 0.05, corresponding to (51), (64),and (67), respectively. In the case of EOS (51), ~�ð0Þ isfinite and increases with increasing �; and ~�ð1Þ is finite for� ¼ 0:05, 0.1, 0.15, 0.19 but unbounded for � ¼ 0.Moreover, only sources with � ¼ 0 contain electric chargeof both signs. We could have anticipated this behavior of ~�from Maxwell Eq. (71) and the slope changes displayed inFig. 9. Figure 15 shows the arising ~� profiles in therestricted interval [0,0.999]. The most important differenceamong ~� profiles associated with the three EOS is that

models with nonlinear EOS and � ¼ 0:05 also involvepositive and negative electric charges.We have noticed that the charge distributions considered

by Thirukkanesh and Maharaj [5] are singular at the origin,where the electric field does not vanish. Alternatively, allour sources have vanishing electric field and finite propercharge density at r ¼ 0. On the other hand, our modelswith � ¼ 0 involve charge distributions that are singular atthe fluid-vacuum interface. We remark that charged thinshells are not considered in our study and electric fields arecontinuous at r ¼ a. Also, charged sources with � ¼ 0 arecharacterized by EðaÞ ¼ 0 and unbounded limr!a�E

0ðrÞ.Arbitrarily large electric field gradients near the fluid-vacuum are puzzling and we proceed with a preliminarydiscussion of their significance.The proper charge density � appears explicitly in the

inhomogeneous Maxwell Eq. (5). It determines the netcharge inside a sphere of radius r through the formula

qðrÞ ¼ 4�Z r

0�ðrÞe�ðrÞ=2r2dr: (78)

We remark that qðaÞ is the total charge of the source,denoted by q in (45). It is qðrÞ the quantity that actuallydetermines the electric field

EðrÞ ¼ qðrÞr2

: (79)

Proper charge density �ðrÞ is not an observable quantityin standard EM theory. We have seen that EðrÞ takes part in

FIG. 14 (color online). Anisotropic parameter ~� for the thirdmodel as a function of x for five different values of �. In thisfigure, the curves 1-5 from the top (blue, cyan, green, tan, andred curves) correspond to � ¼ 0, 0.05, 0.10, 0.15, and 0.19,respectively, (y � ~�).

FIG. 15 (color online). Proper charge density ~� as a functionof x 2 ½0; 0:999� for five different values of �. In this figure, thecurves 1-5 from the bottom (blue, cyan, green, tan, and redcurves) correspond to � ¼ 0, 0.05, 0.10, 0.15, and 0.19, respec-tively, (y � ~�).

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the formulation of the energy conditions of general rela-tivity through the contribution of the Maxwell field to theenergy-momentum tensor. In contrast, the authors are notaware of any condition for physical acceptability imposeddirectly on �.

Both spacetime curvature and electric field gradientsdeviate the world lines of charged test particles. Balkin,van Holten, and Kerner [16] and van Holten [17] havederived covariant formulas for the relative acceleration ofparticles with the same charge/mass ratio. Their equationsinclude terms containing derivatives of the Maxwell tensor.The question arises as to whether these terms could lead toinfinite relative accelerations for pairs of charged testparticles passing through the fluid-vacuum interface ofmodels with � ¼ 0. Hence very large relative accelerationsof test particles could allow indirect observation of infinitecharge density located at the vacuum-fluid interface of oursources.

The fundamental invariant of the electromagnetic fieldI1 ¼ F��F

�� has been used in the analysis of genuine

singularities of static solutions for the Einstein-Maxwellequations [18]. This invariant is bounded at the fluid-vacuum interfaces of our three types of models. The in-homogeneous Maxwell equation implies that I2 ¼F��

;�F!�;! is proportional to �ðrÞ2. The fact that �ðrÞ is

unbounded at r ¼ a in models with � ¼ 0 determines thesingular behavior of a differential invariant of the theory.Questions about physical acceptability of solutions withregular I1 and singular I2 are analogous to those regardingspacetimes with regular curvature invariants and singulardifferential invariants discussed by Musgrave and Lake[19].

The discussion of physical acceptability for sources withzero net charge points at the equilibrium condition forsections with infinite charge density. Fluid elements withunbounded � are located at r ¼ a, where the electric fieldvanishes due to the choice � ¼ 0. Nevertheless, the con-clusion that no electric force acts on elements of chargedfluid with infinite � is not straightforward and deservescloser examination. Other features of our models motivatefurther analysis of equilibrium conditions. For example, aswe keep � constant and increase �, repulsive electricforces increase with decreasing matter density, decreasingpressure, and decreasing radial pressure gradients. Thequestion is, how gravitational and other fluid forces coun-teract increasing electrostatic repulsion when the chargedfluid becomes more diluted and pressure gradients tend tovanish? The situation is clarified using the generalizedTolman-Oppenheimer-Volkov (TOV) equation as pre-sented by Ponce de Leon [10]. (See also [20].) It reads

�MGð�þ prÞr2

eð���Þ=2 � dpr

drþ �

q

r2e�=2 þ 2

rðpt � prÞ¼ 0; (80)

where MG ¼ MGðrÞ is the effective gravitational mass in-

side a sphere of radius r and q ¼ qðrÞ is given by (78). Theeffective gravitational mass is given by the expression

MGðrÞ ¼ 12r

2eð���Þ=2�0; (81)

derived from the Tolman-Whittaker formula and theEinstein-Maxwell equations [10].Equation (80) expresses the equilibrium condition for

charged fluid elements subject to gravitational, hydrostaticand electric forces, plus another force due to pressureanisotropy. Combined with (10), (11), (63), and (79), ittakes the adimensional form

~F g þ ~Fh þ ~Fe þ ~Fa ¼ 0; (82)

where

~F g ¼ � ~Bxð~�þ ~prÞ; (83)

~F h ¼ � d~pr

dx; (84)

~F e ¼ ~� ~Eeð ~Ax2=2Þ; (85)

~F a ¼ 2

xð~pt � ~prÞ: (86)

The profiles of ~Fg, ~Fh, ~Fe, and ~Fa for sources with linear

EOS, � ¼ 0:147 and � ¼ 0 are shown in Fig. 16. Notably,the electric force acting on fluid elements with unbounded

FIG. 16 (color online). Four different forces acting on fluidelements in static equilibrium as functions of x for � ¼ 0:147and � ¼ 0. In this figure the curves 1–4 from the bottom (blue,green, tan, cyan curves) correspond to ~Fg, ~Fe, ~Fa, and~Fhrespectively. The dummy variable y on the vertical axisrepresents any of these forces.

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~� located at x ¼ 1 is finite although ~Eð1Þ ¼ 0 in this case.Equivalently limx!1 ~� ~E is finite. Furthermore, ~Fe is theweakest force and changes sign at x � 0:85. Both ~Fh and~Fa point outwards at every x 2 ð0; 1�. Electric forces in-crease and radial pressure gradients decrease with increas-ing �. When� ¼ 0:19 the hydrostatic force ~Fh is relativelyinsignificant and ~Fa points inwards, acting in conjunctionwith gravitational attraction to compensate the electrostaticrepulsion. This situation is described in Fig. 17. Clearly, thesign of ~Fa changes due to the predominance of pr over pt

for the largest � values. This sign inversion is essential forthe configuration of our static, charged anisotropic sourceswith linear EOS.

We have extended the analysis of these four forces tomodels with nonlinear EOS and found essentially the sameequilibrium configuration discussed above for linear EOSand � ¼ 0:19. Particularly, ~Fh is comparatively small, ~Fg

and ~Fa have nearly the same profile and jointly counteractthe electrostatic repulsion. Hence the choice of EOS has anegligible effect on the compensation of relatively largeelectrostatic repulsion by gravitational attraction and pres-sure anisotropy.

Equations (2)–(5) indicate that matter density, radialpressure, tangential pressure, and electric field strengthaffect the spacetime metric in our relativistic fluid models.Our choice � ¼ 0:147 corresponds to a compact stellarobject (M ¼ 2� 1033 g, a ¼ 106 cm), so pressure is ex-

pected to play a substantial role here. The question arises asto whether the electrostatic energy density significantlycontributes to the source of gravity. We compute dimen-sionless values of the physical variables and get maximaand minima for ~�0 (central matter density), ~p0 (centralpressure), ~pta (tangential pressure at r ¼ a), and ~E2

a (elec-trostatic energy density at r ¼ a) as well as ~E2

m (maximumelectrostatic energy density for � ¼ 0). Approximate nu-merical results for the model with linear EOS are displayedin Table I.We observe that the maximum values of ~p0 and ~pta are

just 1 order of magnitude smaller than the maximum of ~�0.These maxima correspond to the source with � ¼ 0 i.e.zero net charge and weakest electrostatic repulsion.Notably, these density and pressure values are very similarto the dimensionless density ~�s and dimensionless centralpressure ~p0s of the (uncharged) Schwarzschild internalsolution (SIS) with the same � value. Using well-knownformulas for the SIS in geometrized units [21], we derivethe dimensionless expressions

~� s ¼ 3�

4�; (87)

~p 0s ¼ 3�2

8�; (88)

leading to the approximate results ~�s ¼ 0:036 and ~p0s ¼0:003.The value of ~E2

m shown in Table I indicates a substantialcontribution of the Maxwell field to the source of gravity inthe case of zero net charge, although matter density andpressure predominate.In sources with maximum net charge (� ¼ 0:19) ~p0

decreases in 2 orders of magnitude, ~pta doubles its absolutevalue and gets the opposite sign, and ~�0 and ~E2

a have thesame approximate value. From Figs. 13 and 14 we see thatthe reduction of radial pressure with increasing net chargemakes gravitational attraction weaker in spite of the largervalues of electrostatic energy density. We have discussedabove that the sign inversion of tangential pressure leads toan extra force which complements gravitational attraction,so that the stronger electric repulsion gets compensated.Certainly static equilibrium is attainable in these chargedsources thanks to pressure anisotropy.The discussion of the above quantities in conventional

units is interesting as well. Adimensional density andpressure values are expressed in geometrized units andthen converted to cgs units using well-known conversion

FIG. 17 (color online). Four different forces acting on fluidelements in static equilibrium as functions of x for � ¼ 0:147and � ¼ 0:19. In this figure the first (green) and second (cyan)curves from the top correspond to ~Fe and ~Fh, respectively. Thethird (tan) and fourth (blue) curves intersect at x � 0:75 andrepresent ~Fa and ~Fg, respectively. The dummy variable y on the

vertical axis represents any of these forces.

TABLE I. Dimensionless values of physical variables for themodel with linear EOS.

� ~�0 ~p0 ~pta~E2a

~E2m

0 0.042 0.003 0.001 0 0.0009

0.19 0.036 0.000 05 �0:002 0.036

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factors [22]. Similarly, adimensional values of electricalvariables are expressed in geometrized units, then con-verted to Heaviside-Lorentz units and finally toGaussian-cgs units. The corresponding results are dis-played in Tables II and III.

These values of central matter density and pressure aswell as electric field strength and net charge in conven-tional units are similar to those discussed by previousauthors in the context of charged compact objects withisotropic pressure [23]. It is understood that the hugegravitational attraction determined by these values of cen-tral matter density and pressure compensates repulsiveelectrostatic forces associated with field strengths and netcharges with orders of magnitude 1024 statvolt=cm(1026 V=cm) and 1029 statcoul (1019C), respectively.However, our study of charged anisotropic sources pointsat the crucial role that forces caused by pressure anisotropycan play in the construction of equilibrium states. Thisanisotropic effect is equally important for the three modelsconsidered here, at least for the assumed value of � ¼0:147. Provided that pressure anisotropy supplies aninward-directed force that compensates electrostatic repul-sion, we expect our sources can achieve higher charges andelectric field strengths than sources with isotropic pres-sures and the same � value. Also, our results for sourceswith zero net charge suggest a possible role for pressureanisotropy and electric charge in the structure of staticsources for the external Schwarzschild metric.

VI. CONCLUDING REMARKS

The linear equation (32) links our analytical approachwith the numerical treatment of electrically chargedstrange quark stars by Negreiros et al. [24] based on theMIT bag model. They also assume (18) and impose van-ishing isotropic pressure at the fluid-vacuum interface.

We have borrowed nonlinear EOS (38) (with n ¼ 1) and(42) from current models of dark matter and dark energy.Most applications of Chaplygin and modified Chaplygingases are cosmological and describe nonstatic, neutral

gravitational fluids with isotropic pressures. Instead, ourbounded sources involve static, asymptotically flat space-times and charged anisotropic fluids. Actually we havetargeted the effects of these EOS on the interior regionsand fluid-vacuum interfaces of charged stars apart fromany cosmological framework. The question arises as tohow dark matter-dominated star interiors could get electriccharge. Specific charge transfer mechanisms hypothesizedin studies of charged neutron stars [25] could be consid-ered. Alternatively, a fraction of dark matter could be madeup by massive particles with electric charge (CHAMPs)[26], so dark stars could be charged. We remark that ourprocedures and results are totally independent of anycharge generation mechanism.Lobo initiated the study of van der Waals (VDW) quin-

tessence stars [27]. Apart from variations of � and pr in theinterior of the VDW fluid, which may be caused by gravi-tational instabilities, this author imposes a cut-off of theenergy-momentum tensor at r ¼ a, where the internalmetric matches the external Schwarzschild solution.Again, the motivation for introducing this new type ofbounded source is cosmological. Provided that the quin-tessence EOS leads to interesting descriptions of the lateuniverse, stellar objects arising from fluid nucleationthrough density perturbations are explored.The extended KB approach can be applied to a sphere of

charged quintessence fluid. Apart from the hypotheticallink of dark matter with CHAMPs, we cannot justify theintroduction of charge in this model. However, we aim totest the applicability of the present approach against themore complicated VDW EOS

pr ¼ 1�

1� 3�� 2�

2; (89)

which describes dark matter and dark energy as a singlefluid [27].Assuming that the interior metric is joined to (45), and

that the electric field and radial pressure vanish at r ¼ 0and r ¼ a respectively, we evaluate (89) at the center andboundary of the charged sphere and solve two linear equa-tions for 1 and 2 as functions of �ð0Þ, �ðaÞ, prð0Þ, and 3. Then (89) is combined with (31) to obtain

2 3�3 � ð 2 þ 3Þ�2 þ ½1þ 1 þ 3gðrÞ��� gðrÞ ¼ 0;

(90)

where gðrÞ ¼ fðrÞþhðrÞ8� .

We do not go further into the analysis of the real roots of(90) and the corresponding sources. However, we antici-

pate difficulties in the selection of �, �, and ~ 3 valuesleading to positive definite matter densities.The analysis of stability is beyond the scope of this

paper. Results presented by Andreasson [28] regardingstability properties of charged anisotropic spheres couldshed light on this fundamental issue. We highlight the

TABLE II. Density and pressures for the model with linearEOS expressed in cgs units.

� �0 (gr=cm3) p0 (dyn=cm2) pta (dyn=cm2)

0 5:6� 1014 3:6� 1034 1:2� 1034

0.19 4:8� 1014 6� 1032 �2:4� 1034

TABLE III. Electric field strengths and net charge for themodel with linear EOS expressed in Gaussian-cgs units.

� Ea (statvolt=cm) Em (statvolt=cm) q (statcoul)

0 0 3:7� 1023 0

0.19 2:3� 1024 1:9� 1029

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potentially interesting study of gravitational collapse ofsources with zero net charge, and hope our results willmotivate further research on these topics.

Finally, we point out that different EOS could bematched at specific radii to provide composite models.For example, a linear interior (core) with a nonlinearexterior (layer) matched at a location inside the chargedfluid sphere might provide a better model than a simpleEOS throughout. Using the extended KB approach devel-oped here, both the core and the layer would be describedwith the same general metric (1) and the ansatze (10) and(11) with presumably different values of A, B, and C ineach region. Less restricted choices of these constants

could lead to charged thin shells emerging at the interfacebetween the fluids.

ACKNOWLEDGMENTS

V.V. is grateful to Professor Graham Hall and theInstitute of Mathematics of Aberdeen University. Also heis indebted to Dr. Olga Savasta for valuable computingassistance. F. R., S. R., K. C., and M.K. are grateful to theauthorities of IUCAA for research facilities and hospitality.The authors thank an anonymous referee for suggesting thestudy of composite charged fluid models based on theextended KB solution method.

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