Int J Theor Phys (2012) 51:1408–1415DOI 10.1007/s10773-011-1016-3

Geometry of Quark and Strange Quark Matter in HigherDimensional General Relativity

G.S. Khadekar · Rupali Wanjari

Received: 6 September 2011 / Accepted: 3 November 2011 / Published online: 16 November 2011© Springer Science+Business Media, LLC 2011

Abstract In this paper we study quark matter and strange quark matter in higher-dimensional spherical symmetric space-times. We analyze strange quark matter for thedifferent equations of state and obtain the space-time geometry of quark and strange quarkmatter. We also discuss the features of the obtained solutions in the context of higher-dimensional general theory of relativity.

Keywords Strange quark matter · Bag constant · Higher dimensional space time

1 Introduction

In the last few decades the study of higher dimensional theories has been revived and con-siderably generalized after realizing that many interesting theories of particle interactionsneed more than four dimensions for their consistent formulation. On the other hand, gen-eral relativity was formulated in a space-time with just four space-time dimensions. Thusit is important to generalize the results obtained in four dimensional general relativity inthe framework of higher dimensions and look for the effects due to incorporation of extradimensions in the theory. This idea is particularly important in the field of cosmology, sincewe known that our Universe was much smaller in its early stage than it is today.

The latest development of super-string theory and super-gravitational theory have cre-ated interest among scientists to consider higher-dimensional space-times study of the earlyUniverse. Sahdev [1], Emelyanov et. al. [2], Chatterjee et. al. [3] have studied physics ofthe Universe in higher-dimensional space-time. Overduin and Wesson [4] have presentedan excellent review of higher-dimensional unified theories, in which the cosmological andastrophysical implications of extra-dimensions have been discussed. Randall and Sundrum[5] gave an intersting picture of gravity in which although the extra-dimension is not com-pact, four-dimensional Newtonian gravity is recovered in five-dimensional and Anti-disittar

G.S. Khadekar (�) · R. WanjariDepartment of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University,Mahatma Jyotiba Phule Educational Campus, Amravati Road, Nagpur 440033, Indiae-mail: gkhadekar@yahoo.com

Int J Theor Phys (2012) 51:1408–1415 1409

space-time in the low energy limit. Shen and Tan [6] obtained a global regular solution ofthe higher-dimensional Schwarzschild space-time. Paul [7] studied the mass to radius ratioof a uniform density star in the framework of higher-dimension.

Kaluza [8] and Klein [9] independently were the first who initiated the study of unifygravity with electromagnetic interaction by introducing an extra dimension. Kaluza-Kleintheory is essentially an extension of Einstein general theory of relativity in five dimensionswhich is of much interest in partical physics and cosmology.

Typically, quark matter is modeled with an equation of state (EOS) based on the phe-nomenological bag model of quark matter, in which quark confinement is described by anenergy term proportional to the volume. In the framework of this model the quark matteris composed of massless u, d quark, massive s quark and electrons [10]. In the simplifiedversion of the bag model, assuming the quarks are massless and non-interacting, we havethe quark pressure pq = ρq

3 (ρq is the quark energy density); the total energy density is

ρ = ρq + Bc, (1)

while the total pressure is

p = pq − Bc. (2)

One then obtains the EOS for strange quark matter [11–13],

p = 1

3(ρ − 4Bc) , (3)

where Bc is the difference between the energy density of the perturbative and non-perturbative Quantum Chropmpodynamics (QCD) vacuum (the bag constant). Equation (3)is essentially the EOS of a gas of massless particles with corrections due to the QCD traceanomaly and perturbative interactions.

Dey et al. [14] have obtained new sets of EOSs for strange matter based on a model ofinterquark potential which has the following features: (a) asymptotic freedom, (b) confine-ment at zero baryon density and deconfinement at high baryon density, (c) chiral symmetryrestoration and (d) gives stable uncharged β-stable matter. These EOSs have later been ap-proximated to the following linear form by Gondek et al. [15],

p = ε (ρ − ρ0) , (4)

where ρ0 denotes the energy density at a zero pressure and ε is a constant [16].In this study we will examine quark matter and strange quark matter in the framework of

higher-dimensional space-time.This paper is organized as follows: in Sect. 2 we have solved the Einstein field equations

for quark matter in higher dimension spherical symmetric space-time by using the features ofthe perfect fluid of quark matter. Section 3 deals with solution of the Einstein field equationsfor higher dimensional spherical symmetric space-time by using EOS (3) and (4) for strangequark matter. The concluding remark is given in Sect. 4.

2 Space-Time Geometry of Quark-Gluon Matter

We choose higher dimensional spherically symmetric metric of the form

ds2 = −e2νdt2 + e2ψdr2 + R2d�2, (5)

1410 Int J Theor Phys (2012) 51:1408–1415

where ν,ψ and R are function of r and t , d�2 = dθ21 + sin2 θ1dθ2

2 + sin2 θ1 sin2 θ2dθ23 . The

perfect fluid of quark-matter is taken as (Yilmaz [17])

Tik = (ρ + p)uiuk + pgik, (6)

where ρ = ρq + Bc and p = pq − Bc.Now the explicit form of the Einstein’s field equations for the metric (5) with matter field

given by (6) are (choosing 8πG = c = 1)

3

R2− e−2ψ

(3R′2

R2+ 3ν ′ R

′

R

)+ e−2ν

(3R

R+ 3

R2

R2− 3ν

R

R

)= −p, (7)

e−2ψ

(ν ′′ − ν ′2 + ψ ′ν ′ − 2

R′′

R− 2ν ′ R

′

R+ 2ψ ′ R

′

R− R′2

R2

)

+ e−2ν

(ψ + ψ2 − ψ ν + 2

R

R+ 2ψ

R

R− 2ν

R

R+ R2

R2

)= −p, (8)

3

R2− e−2ψ

(3ψ ′ R

′

R− 3

R′′

R− 3

R′2

R2

)+ e−2ν

(3R2

R2+ 3ψ

R

R

)= ρ, (9)

and

R′ − ψR′ − ν ′R = 0, (10)

where over dot (.) and prime (′) denote the partial derivatives with respect to t and r , respec-tively.

The scalar expansion and shear σ for the model (5) in higher dimension are given by

= ui;j ,

(t, r) = 1

eν

[ψ + 3

R

R

], (11)

and

σij = 1

2(ui;j + uj ;i ) −

4(gij − uiuj ),

σθ1θ1

= σθ2θ2

= σθ3θ3

= −1

3σ r

r = 1

4e−ν

(R

R− ψ

). (12)

We consider the situation of the experimental results of Brookhaven Lab (http://www.bnl.gov/bnlweb/pubaf/pr) the quark-gluon plasma has a very small viscosity, i.e., the mattershear vanishes identically. This situation like a perfect fluid [18–20]. This means

R

R= ψ, (13)

or

R = q(r)eψ, (14)

where q(r) is an arbitrary function of integration.

Int J Theor Phys (2012) 51:1408–1415 1411

We can use available freedom of scaling to rescale the radial coordinate r so that we takeq(r) = r . We then have

R = reψ . (15)

By using (15), (10) can be expressed after integration as

eν = a(t)ψ, (16)

where a(t) is an arbitrary function of integration.With the help of above equation the Einstein Field Equations (EFEs) gives

6

a2− 3

a

a3ψ−

(3

ψ ′

rψ+ 6

ψ ′

r+ 3

ψ ′ψ ′

ψ+ 3ψ ′2

)e−2ψ = −p, (17)

6

a2− 3

a

a3ψ−

(2

ψ ′

rψ+ 4

ψ ′

r+ ψ ′′

ψ+ 2ψ ′′ + ψ ′ψ ′

ψ+ ψ ′2

)e−2ψ = −p, (18)

6

a2−

(9ψ ′

r+ 3ψ ′′ + 3ψ ′2

)= ρ. (19)

From (17) and (18), we get

−2ψ ′

r− 2ψ ′2 − ψ ′

rψ− 2

˙ψ ′ψ ′

ψ+ 2ψ ′′ + ψ ′′

ψ= 0. (20)

For solving the above equation, we can choose ψ(r, t) to be separable:

eψ(r,t) = f (t)g(r), (21)

with f (t) = c1tn, and g(r) = 1

c2r−m, with m and n constants. By using (21) into (20), we

get

ψ(r, t) = log(c1tn) − log(c2r

m). (22)

For the value of m = 0, 2 (22) has to satisfy (20) and there is no any restriction on theconstant value n.

So, we have easily found the metric potentials [ν(r, t),ψ(r, t) and R(r, t)] exactly. Thismeans that we can determine the space-time geometry of the matter in the framework ofhigher dimensional general relativity.

In this case, by using (15), (16) and (22) the higher dimensional space-time geometry ofquark-gluon matter is given by

ds2 = −n2a2

t2dt2 + c2

1t2n

c22r

2m

(dr2 + d�2

r2

), (23)

where m = 0 and 2.From (17), (18), (19), (22) and (11), we obtain the following physical parameters

p = − 6

a2+ 3t a

na3, (24)

ρ = 6

a2, (25)

1412 Int J Theor Phys (2012) 51:1408–1415

and the expansion is

= 4

a(t). (26)

3 Space-Time Geometry of Strange Quark Matter

In this section we obtain the strange quark models by using the equations of state (3) and(4) for strange quark matter.

Case (i) Strange Quark Matter in the BAG Model: By using the values of p and ρ from(24) and (25), in the EOS (3) we get

a(t) = ± 6√6Bc + 6C0Bct

−16n3

. (27)

Using this value in (24), (25) and (26), we respectively get pressure, density andexpansion for the model.

p = −Bc + c0Bc

3t16n

3

, (28)

ρ = Bc + c0Bc

t16n

3

, (29)

and

= ±2√

6Bc + 6c0Bct−16n

3

3. (30)

Hence from (23), the higher dimensional model is given by

ds2 = − 36n2

(6Bc + 6c0Bct−16n

3 )t2dt2 + c2

1t2n

c22r

2m

(dr2 + d�2

r2

), (31)

where m = 0 and 2.Case (ii) Strange Quark Matter in the Case of Linear Equation of State: By using the

values of p and ρ from (24) and (25), in the EOS (4) we get

a(t) = ±√

6(ε + 1)√ερ0 + ερ0c0t−4n−4nε

. (32)

Using this value in (24), (25) and (26), we respectively get pressure, density andexpansion for the model.

p = εερ0c0t

−4n−4nε − ρ0

ε + 1, (33)

ρ = ερ0 + ερ0c0t−4n−4nε

ε + 1, (34)

Int J Theor Phys (2012) 51:1408–1415 1413

and

= ±4√

ερ0 + ερ0c0t−4n−4nε

√6(ε + 1)

. (35)

Hence again higher dimensional model from (23) is given by

ds2 = − 6(ε + 1)n2

[ερ0 + ερ0c0t−4n−4nε]t2dt2 + c2

1t2n

c22r

2m

(dr2 + d�2

r2

), (36)

where m = 0 and 2.

4 Conclusions

We have obtained the strange quark matter solution in the context of higher dimensionalgeneral theory of relativity. Our solutions are consistent with the result of Brookhaven Lab-oratory, i.e.vanishing shear of quark-gluon plasma, also our solutions do not contain the r

coordinates. When the radius of quark star is very small, the density and pressure of thequark star are depend on t and independent on r . So our solution are physically meaningfulin the context of higher-dimensional space-time. In the case of the quark and strange quarkmatter, the geometries of this matter have no shear i.e. responsible for the dissipation ofmatter.

We now check the mass in the case of strange quark matter. The total mass M of the starin five dimension is defined as M = 8π

∫ρr3dr = 2πρr4.

So, the mass function M(t, r) for the metric of (5) is given by

M(t, r) = R2(1 + e−2νR2 − e−2ψR′2), (37)

(for detail calculation see Appendix) which can be interpreted as the total mass inside thecomoving radius r at the time t .

(i) In the case of strange quark matter in the BAG Model, the mass is given as follows:For m = 0 and m = 2, from (15), (16), (22) and (29) we obtain respectively,

M(r, t) = c41r

4t4n

6c42

(B + c0Bt−16n

3 ) = c41r

4t4n

6c42

ρ, (38)

and

M(r, t) = c41t

4n

6c42r

4(B + c0Bt

−16n3 ) = c4

1t4n

6c42r

4ρ. (39)

From (38) and (39), it is easily seen that in addition to the fundamental constants,mass also depends on the bag constant.

(ii) In the case of strange quark matter in the linear equation of state, the mass is given asfollows:

For m = 0 and m = 2, from (15), (16), (22) and (34) we obtain respectively,

M(r, t) = c41r

4t4n

6c42

(ερ0 + ερ0c0t−4n−4nε)

(ε + 1)= c4

1r4t4n

6c42

ρ, (40)

1414 Int J Theor Phys (2012) 51:1408–1415

and

M(r, t) = c41t

4n

6c42r

4

(ερ0 + ερ0c0t−4n−4nε)

(ε + 1)= c4

1t4n

6c42r

4ρ. (41)

Appendix

For the line element (5) the non zero component of Rijij is

Rijij = (1 + e−2νR2 − e−2ϕR′2), (42)

where i = 3,4,5, . . . , n j = 2,3,4, . . . , n − 1 and i > j .Examination of this expression reveals that if R = r our line element (5) reduces to

ds2 = e−2νdt2 − dr2

1 − Rijij

− r2d�2. (43)

If curvature coordinates used in empty region then spherically symmetric five dimensions(5D) Schwarzschild’s like metric is given by

ds2 =(

1 − A

r2

)dt2 − dr2

(1 − A

r2 )− r2d�2, (44)

where A is a constant and from Newtonian approximation A = M . Comparison of (43) and(44) gives

(Rijij )b = M

r2b

, (45)

where the subscript b means evaluation at the boundary. We can, at this stage, follow thearguments of Cahill and McVittie [21] in defining intuitively a mass function in 5D as

m(r, t) = R2(1 + e−2νR2 − e−2ϕR′2). (46)

References

1. Sahdev, D.: Phys. Rev. D 30, 2495 (1984)2. Emelyanov, V.M., Nikitin, Y.P., Rozental, J.L., Berkov, A.V.: Phys. Rep. 143, 1 (1986)3. Chatterjee, S., Bhui, B.: Int. J. Theor. Phys. 32, 671 (1993)4. Overduin, J.M., Wesson, P.S.: Phys. Rep. 283, 303 (1987)5. Randall, L., Sundrum, R.: Phys. Rev. Lett. 83, 3370, 4690 (1999)6. Shen, Y., Tan, Z.: Phys. Lett. A 142, 341 (1989)7. Paul, B.C.: Class. Quantum Gravity 18, 2311 (2001)8. Kaluza, T.: Sitz.ber. Preuss. Akad. Wiss. Berl. Philos.-Hist. Kl. F1, 966 (1921)9. Klein, O.: Ann. Phys. 37, 895 (1926)

10. Gondek-Rosinska, D., Gourgoulhon, E., Haensel, P.: Astron. Astrophys. 412, 777 (2003)11. Kapusta, J.: Finite-Temperature Field Theory. Cambridge University Press, Cambridge (1994)12. Sotani, H., Kohri, K., Harada, T.: Phys. Rev. D 69, 084008 (2004)13. Xu, R.X.: Chin. J. Astron. Astrophys. 3, 33 (2003)14. Dey, M., Bombaci, I., Dey, J., Ray, S., Samanta, B.C.: Phys. Lett. B 438, 123 (1998)

Int J Theor Phys (2012) 51:1408–1415 1415

15. Gondek-Rosinska, D., Bulik, T., Zdunik, L., et al.: Astron. Astrophys. 363, 1005 (2000)16. Sharma, R., Karmakar, S., Mukherjee, S.: Int. J. Mod. Phys. D 15, 405 (2006)17. Yilmaz, I.: Gen. Relativ. Gravit. 38, 1397 (2006)18. Back, B.B., et al.: Nucl. Phys. A 757, 28 (2005)19. Admas, J., et al.: Nucl. Phys. A 757, 102 (2005)20. Adcox, K., et al.: Nucl. Phys. A 757, 184 (2005)21. Cahill, M.E., McVittie, G.C.: J. Math. Phys. 11, 1382 (1970)