Astrophys Space Sci (2011) 331: 645–650DOI 10.1007/s10509-010-0463-2

O R I G I NA L A RT I C L E

Horizon-free gravitational collapse of radiating fluid sphere

Neeraj Pant · B.C. Tewari

Received: 15 July 2010 / Accepted: 5 August 2010 / Published online: 19 August 2010© Springer Science+Business Media B.V. 2010

Abstract In this paper we present a detailed study of BCTIst solution Tewari (Astrophys. Space Sci. 149:233, 1988)representing time dependent balls of perfect fluid withmatter-radiation in general relativity. Assuming the life timeof quasar 107 years our model has initial mass ≈ 108M�

with an initial linear dimension ≈ 1015 cm. Our model is ra-diating the energy at a constant rate i.e. L∞ = 1047 ergs/secwith the gravitational red shift, z = 0.44637. In this modelwe have 2GM(u)/c2RS(u)) = 0.3191 i.e. the model is hori-zon free.

Keywords General relativity · Quasar · Black hole ·Gravitational collapse

1 Introduction

Ever since the formulation of Einstein’s field equations, therelativists have been proposing different models of immensegravity astrophysical objects like quasar, black-hole or othersuper-dense object with the result of gravitational collapse.The concept of gravitational collapse was first studied byOppenheimer and Snyder (1939) for spherically symmet-ric fluid distribution neglecting the components like dissi-pation of energy (radiation), pressure and rotation. How-ever, Vaidya (1951, 1953, 1966) suggested that the gravi-tational collapse is a highly dissipating energy process and

N. Pant (�)Department of Mathematics, National Defence Academy,Khadakwasla Pune 411023, Indiae-mail: neeraj.pant@yahoo.com

B.C. TewariDepartment of Mathematics, Kumaun University,SSJ Campus Almora 263601, Indiae-mail: drbctewari@yahoo.co.in

initiated his problem with the extension of Tolman (1939)taking account of out flowing radiation. As usual the dissi-pation of energy is described in two limiting cases. (i) Freestreaming approximation applies whenever, the mean freepath of the particle responsible for propagation of energyin stellar is larger than the typical length of the object anddissipation is expressed by mean of an outgoing null fluid.(ii) Diffusion approximation applies whenever the meanfree path of the particle responsible for propagation of en-ergy in stellar is much smaller than the typical length ofthe object. Some of the authors studied the radiating ballwith physically significant solutions in the free streamingcase (Tewari 1988, 1994, 2006, Pant and Tewari 1990;Pant et al. 2010) by solving the modified field equationsproposed by Misner (1965), Lindquist et al. (1965). How-ever, Herrera et al. (1980) presented a new approach to thestudy of a non static radiating fluid. They proposed a gen-eral method for obtaining collapsing radiating models fromexact static solutions of Einstein’s equations for a spheri-cally symmetric fluid distribution. The model proposed byGlass (1981) has been extensively studied by Santos (1985)for the junction conditions of a shear-free collapsing spher-ically symmetric non adiabatic fluid with radial heat flow.On similar grounds a number of studies have been proposedby de Oliveira et al. (1985, 1988), de Oliveira and Santos(1987), Bonnor et al. (1989), Kramer (1992), Maharaj andGovender (1997, 2005), Debnath et al. (2005), Herrera etal. (1998, 2004a, 2006, 2007, 2009), Herrera and Santos(2004), Mitra (2006), Naidu and Govinder (2007); see alsoreferences therein. One of the most vital component of theall the said models is mass-radius gradient, which distin-guishes one with other. If 0 < 2GM(u)/c2Rs(u) < 1, themodel is horizon free i.e. collapse process keeps on goingand left over core is a black hole of point dimension (nakedsingularity).

646 Astrophys Space Sci (2011) 331: 645–650

In this paper we present a model of Tewari (1988) first so-lution keeping in view of conditions proposed by Pant et al.(2010) along with most important phenomenon of immensegravity objects i.e. gravitational Red shift. Some cases theobserved red shift in their emission line spectrum is as largeas 3.6. The observed redshift is the resultant of gravitationalredshift, Doppler’s redshift (object receding with high ve-locity and due to rotational or spin motion of the object).

2 Field equations of a radiating fluid ball in isotropicco-ordinates

We consider the line element for a spherically symmetricdistributions of collapsing fluid under going dissipation inthe form of the free streaming limit

ds2 = c2eυ(r,t)dt − eω(r,t){dr2 + r2(dθ2 + sin2 θdφ2)

}(1)

Here υ and ω are functions of r and t such that we canassume the following variables to be separable forms for thegravitational field variables

eυ = f 2(r)g2(t) (2)

eω = h2(r)n2(t) (3)

The field equations of general relativity for a distribution ofa mixture of perfect fluid and radiation are

Rij − 1

2Rgij = −8πG

c4Tij (4)

here

Tij = (p + ρc2)vivj − pgij + q

cwiwj (5)

Where p,ρ respectively denote the isotropic pressure anddensity of the matter within the distributions and vi its fourvelocity

gij vivj = 1 (6)

q denotes the radiation flux density and wi its four velocitywhich is null:

gijwiwj = 0 (7)

We consider r as a co-moving coordinate so that vi =(e−υ/2,0,0,0) and we chose wi (Misner 1965) such that

wi ∂

∂xi= e−υ/2 1

c

∂

∂t+ e−ω/2 ∂

∂r(8)

Thus we find that for the metric (1) under these conditionswith the matter-radiation distributions the field equation (4)

reduces to the following:

f ′′

f+ h′′

h− 2

(h′

h

)2

− 2

(h′

h

f ′

f

)2

− 1

r

(f ′

f+ h′

h

)

− 2n

g

hf ′

f 2= 0 (9)

8πG

c4p = 1

h2n2

{h′′

h−

(h′

h

)2

+ h′

rh+ f ′′

f

f ′

rf

}

− 1

f 2g2

{2n

n+

(n

n

)2

− 2g

g

n

n

}(10)

8πG

c2ρ = − 1

h2n2

{h′′

h+

(h′

h

)2

+ 5h′

rh− f ′′

f+ f ′

rf

+ 2

(h′

h

f ′

f

)}+ 3

f 2g2

(n

n

)2

(11)

8πG

c5q = −2

n

g

f

f 2hn2(12)

here prime (′) denotes differentiation with respect to r and adot (·) denotes differentiation with respect to ct.

The luminosity L and the fluid collapse rate θ are givenby

L = 4πr2h2n2q (13)

θ = rhn

fg(14)

3 Boundary conditions for radiating fluid ball

The external space-time to a radiating fluid ball of massM(u) that is filled with pure radiation, is represented by theVaidya’s radiating metric i.e.

ds2 =(

1 − 2GM

c2R

)du2 + 2dudR

− R2(dθ2 + sin2 θdφ2) (15)

where R is the radial coordinate. The conditions describingthe boundary of (1) and (15) over the hyper surface r = rsor equivalently R = Rs(u) have been obtained by Misner i.e.the continuity of (1) and (15). We have.

p(rst) = 0 (16)

f (rs)g(t)cdt =[

1 + rhn

fg+ rh′

h

]

s

du (17)

Rs(u) = rsh(rs)n(t) (18)

M(u) = c2

2G

[r3h3n2

f 2g2− 2r2h′n − r3h′2n

h

]

s

(19)

Astrophys Space Sci (2011) 331: 645–650 647

The observed luminosity at rest at infinity L∞ is given by

L∞ = Ls

[1 + rhn

fg+ rh′

h

]2

s

(20)

Rate of contraction of the boundary of radiating fluid ball isgiven by

θs = rs

(hn

fg

)

s

(21)

If 2GM(u)/c2RS(u) < 1 the model is horizon free. For suit-able choice of arbitrary constants, one can get the collapsingbody without horizon.

The gravitational red shift z is given by

z = 1

c

du

dt− 1 = f (rs)g(t)

[1 + rhnfg

+ rh′h

]s− 1 (21a)

4 Properties of BCT solution I

In this section we shall study the properties BCT solution I(Tewari 1988), hence, our task is to obtain the fluid parame-ters p,ρ and q from (10), (11) and (12) for BCT solution.In order to solve (9) Tewari (1988) assumed

n = 1

2sg (22a)

f ′′

f− f ′

f r= 0 (22b)

And (9) transforms into a linear equation of first order in(h′/h2)

(h′

h2

)′−

(2f ′

r+ 1

r

)(h′

h2

)− s

f ′

r= 0 (23)

In view of (22b), the solution of (23) is (Tewari 1988)

f = A(B2 + r2) (24)

h = A

[D + C(B2 + r2)3

− s

144B6

{33B4r + 40B2r3 + 15r5

+ 1

B

(15(B2 + r2)3 − 48B6) tan−1

(r

B

)}]−1

(25)

Here A,B,C and D are the constants of integration.Here we assume that t is a proper time of the observer on

the hyper surface r = rs , we have

eν(rs ,t) = 1 (26)

In view of (2), (22a), (24) and (26), we get

g(t) = 1

A(B2 + r2s )

= constant (27)

n(t) = K + s

2A(B2 + r2s )

ct (28)

Where K is an arbitrary non-negative constant.In view of (22a), (24) and (25) we obtain from (10), (11)

and (12) the pressure, density and radiation flux density re-spectively.

8πG

c4pA2n2 = 4A2

h2(B2 + r2)+

{−6Cr(B2 + r2)2

+ s

24B6(B2 + r2)

(33B4r2 + 40B2r4

+ 15r6 + 15r(B2 + r2)3

Btan−1

(r

B

))}2

+ A

h

{−12C(B2 + r2)2 − 24Cr2(B2 + r2)

+ s

12B6(B2 + r2)2

(56B6r + 139B4r3

+ 15B2r5 + 45r7

+ 15(B2 + r2)3(B2 + 3r2)

Btan−1

(r

B

))}

− s2

4(B2 + r2)(29)

8πG

c2ρA2n2 = −3

{−6Cr(B2 + r2)2

+ s

24B6(B2 + r2)

(33B4r2 + 40B2r4

+ 15r6 + 15r(B2 + r2)3

Btan−1

(r

B

))}2

+ A

h

{−36C(B2 + r2)2 − 48Cr2(B2 + r2)

+ s

12B6(B2 + r2)2

(123B6r +351B4r3

+ 325B2r5 + 105r7

+ 15(B2 + r2)3(3B2 + 7r2)

Btan−1

(r

B

))}

− 3s2

4(B2 + r2)(30)

8πG

c5qA2n2 = −2sr

(B2 + r2)2(31)

648 Astrophys Space Sci (2011) 331: 645–650

Subjecting Pant et al. (2010) condition i.e. q > 0, throughoutthe distribution we have

s < 0 (32)

In view of (13) and (14), the luminosity L and rate of con-traction θ of the radiating ball are given by

L = −(

c5

G

)sr3h

A(B2 + r2)2(33)

θ = srh

2A(B2 + r2)(34)

In view of (32), (33) and (34), we observe that luminosityL is positive however, the rate of contraction (θ) is negativeat any time throughout the distribution. Thus, the radiatingbody agrees with a significant model satisfied by (32) (Pantet al. 2010 condition).

In view of (25), (29), (30) and (31), the central values ofthe physical parameters are given by

8πG

c4p0A

2n2 = −12CB4(D + CB6)

+ 4

B2(D + CB6)2 − s2

4B4(35)

8πG

c2ρ0A

2n2 = 36CB4(D + CB6) + 3s2

4B4(36)

q0 = 0 (37)

Thus, we can say that with suitable choice of constants cen-tral values of pressure and density can be made positive.Also, the central values of pressure and density are inverselyproportional to square of n(t). From (28) and (32), it is ob-vious that n(t) is a monotonically decreasing function of t

and for t → 107 years (life time of quasar) n(t) → 0. Thus,at the end of radiating process the central density of ball istending to infinite.

From (18), (25) and (28) we obtain the expression for theradius of sphere at any time

RS(u) = Arsn(t)

[D + C(B2 + r2

s )3

− s

144B6

{33B4rs + 40B2r3

s + 15r5s

+ 1

B

(15(B2 + r2

s )3

− 48B6) tan−1(

rs

B

)}]−1

(38)

The total energy-mass inside the surface is given by us-ing (19), (24), (25) and (29),

2GM(u)

c2= An(t)

[s2

4(B2 + r2s )2

−{

6Crs(B2 + r2

s )3

− s

24B6(B2 + r2s )

(33B4r2

s + 40B2r4s + 15r6

s

+ 1

B

(15rs(B

2 + r2s )3) tan−1

(rs

B

))}

− 2

{6CB2 + r2

s )2 − s

24B6(B2 + r2s )

×(

33B4r2s + 40B2r4

s + 15r6s

+ 1

B

(15rs(B

2 + r2s )3) tan−1

(rs

B

))}

× {D + C(B2 + r2

s )3}

− s

144B6

{33B4rs + 40B2r3

s + 15r5s

+ 1

B

(15(B2 + r2

s )3 − 48B6) tan−1(

rs

B

)}]

×[D + C(B2 + r2

s )3

− s

144B6

{33B4rs + 40B2r3

s + 15r5s

+ 1

B

(15(B2 + r2

s )3

− 48B6) tan−1(

rs

B

)}]−3

(39)

The luminosity observed on the surface is given by

LS = −(

c5

G

)s2r3

s

((B2 + r2

s )−2)[D + C(B2 + r2

s )3

− s

144B6

{33B4rs + 40B2r3

s + 15r5s

+ 1

B

(15(B2 + r2

s )3 − 48B6) tan−1(

rs

B

)}]−1

(40)

In view of (17), (24), (25), (26) and (28) we have

cdt = du

[1 + 1

2

{srs − r2

s

(12C(B2 + r2

s )3

− s

12B6

{33B4rs + 40B2r3

s + 15r5s

+ 1

B

(15(B2 + r2

s )3 − 48B6) tan−1(

rs

B

)})}

× (B2 + r2s )−1

{D + C(B2 + r2

s )3

Astrophys Space Sci (2011) 331: 645–650 649

− s

144B6

{33B4rs + 40B2r3

s + 15r5s

+ 1

B

(15(B2 + r2

s )3 − 48B6) tan−1(

rs

B

)}}−1](41)

In view of (20), (24), (25), and (41), the observed luminosityat rest at infinity L∞ is given by

L∞ = Ls

[1 + 1

2

{srs − r2

s

(12C(B2 + r2

s )3

− s

12B6

{33B4rs + 40B2r3

s + 15r5s

+ 1

B(15(B2 + r2

s )3 − 48B6) tan−1(

rs

B

)})}

× (B2 + r2s )−1

{D + C(B2 + r2

s )3

− s

144B6

{33B4rs + 40B2r3

s + 15r5s

+ 1

B(15(B2 + r2

s )3 − 48B6) tan−1(

rs

B

)}}−1]2

(42)

Which is constant for all time.Horizon free function is given by

2GM(u)/c2RS(u) = ζ(A,B,C, s) (42a)

5 Particular case B = C = 1, D = 5 and s = − 2 × 10−10

In this section we shall consider a particular radiating fluidball model of quasar for B = C = 1, D = 5 and s =−2 × 10−10 and calculate the various parameters like, mass,radius, luminosity, etc. Putting these values in (29) and using(16) we get rs ≈ 0.48234. Equations (29) and (30) then givethe march of p,ρ and p/ρc2 at any time t within the radi-ating fluid ball and using (32) we get the march of luminos-ity (Table 1). We observe that pressure, density, luminosityare positive and p/ρc2 < 1/3, everywhere within the ball.Also, pressure and pressure-density ratio are monotonicallydecreasing, however, luminosity is monotonically increasingwith the increase of r .

In view of (28), we have

n(t) = K − 2.43375A−1t (43)

Consequently, we have

M(u) = 1.51081 × 1027(AK − 2.43375t) gram (44)

R(u) = 0.7018(AK − 2.43375t) cm (45)

du = 1.44637cdt (46)

Table 1 The march of pressure, density, pressure—density ratio andLuminosity at any instant within the ball (p in dyne-cm−2, ρ in g-cm−3

and L in erg-s−1)

r/rs8πGc4 pA2n2 8πG

c2 ρA2n2 p

ρc2 L

72.0 216.0 0.3333 0

0.1 71.329 217.678 0.3276 1.35 × 1044

0.2 69.313 222.747 0.3111 1.06 × 1046

0.3 65.937 231.313 0.2850 3.47 × 1046

0.4 61.173 243.55 0.2512 7.92 × 1046

0.5 54.984 259.705 0.2117 1.47 × 1047

0.6 47.314 280.090 0.1689 2.38 × 1047

0.7 38.087 305.090 0.1248 3.52 × 1047

0.8 27.261 335.15 0.0813 4.84 × 1047

0.9 14.562 370.784 0.0392 6.29 × 1047

1.0 0 412.554 0.0 7.79 × 1047

2GM(u)/c2RS(u)) = 0.3191 < 1 (47)

In view of (21a) and (46) the gravitational red shift,z = 0.44637.

Assuming the life time of quasar 107 years our modelhas initial mass ≈108M� with an initial linear dimen-sion ≈1015 cm. Our model is radiating energy at a con-stant rate i.e. L∞ ≈ 1047 ergs/sec. In this model we have0 < 2GM(u)/c2RS(u)) < 1, the model is horizon free i.e.collapse process keeps on going and left over core is a blackhole of point dimension.

Acknowledgements (1) Authors are thankful to IUCAA Pune (In-dia) for providing library facilities.

(2) We are grateful to referee for pointing out the errors in the orig-inal manuscript and making constructive suggestions.

(3) One of us (N.P.) acknowledges his gratitude to Professor M. RoyPrincipal N.D.A. and Professor A.N. Srivastava, HOD (Mathematics),N.D.A., for their motivation and encouragement.

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