J. Modern Physics, 2010, 1, 143-146
doi:10.4236/jmp.2010.12020 Published Online June 2010 (http://www.SciRP.org/journal/jmp)
On Collapse of Uniform Density Sphere with
Pressure
Mahesh Chandra Durgapal1, Pratibha Fuloria2
1Retired Professor of Physics, Kumaun University, Naintal, India; 2Department of Physics, SSJ Campus, Kumaun University, Al-
mora, India.
Received March 1st, 2010; revised April 18th, 2010; accepted May 10th, 2010.
ABSTRACT
Adiabatic collapse solutions of uniform density sphere have been discussed by so many authors. An analysis of these
solutions has been done by considering the baryonic conservation law and the no heat transfer condition. We have ex-
amined whether the pressure can remain finite or not during the collapse.
Keywords: Genral Relativity, Astrophysics, Collapse
1. Introduction
Radial adiabatic motion of perfect fluid spheres of uni-
form density, E = E(t), but non-uniform pressure were
discussed by Bonnor and Faulkes [1], Thompson and
Whitrow [2,3] and Bondi [4] under various assumed re-
lationships between central pressure and density. These
authors discussed the problem of collapse and bounce
under two assumptions: first, that the motion is isotropic
or shear-free; and second, that the density is uniform. But
Mishra and Shrivastava [5] showed that the condition of
uniform density and regularity at the centre necessarily
lead to the isotropic motion.
The theme of this paper is rather different from that of
the other authors. We have examined whether the pres-
sure can remain finite or not. We have considered the
no-heat transfers (NHT) conditions (explained in the text)
and baryon conservation law during the collapse. It is
shown that if the fluid is isentropic or (and) the surface
temperature remains constant during the collapse the
pressure can not remain finite (it vanishes). On the other
hand if the fluid is neither isentropic nor the surface
temperature remains constant during the collapse, then
the results obtained by earlier authors (Bondi, 1969) are
found to be inconsistent with the baryonic conservation
and NHT condition.
2. The Metric and Uniform Density Sphere
Vanishing shear implies that one can simultaneously
introduce isotropic and co-moving coordinates
222222
(+ )dsy dtRdrr d

 (1)
2222
(,), (,),sin
y
yrt RRrt ddd

 
It is assumed that the fluid’s viscosity vanishes, and
the adiabatic flow condition makes T10 component of
energy momentum tensor vanish in the co-moving coor-
dinates. The energy momentum tensor can thus be writ-
ten as
()TPEUUPg




(2)
where E and P are energy density and pressure, respec-
tively and the four-velocity,
(, 0, 0, 0)Uy


(3)
The hydrodynamic equations,
;0T



and ;0UT



, and the equation of baryon
conservation, ;
()0nU



(where n = number density)
give us (Misner and Sharp) [6], (Demianski) [7]
(/)/( )
y
yPPE



 (4)
and
,0Us



or 0s

 and 0s (5)
()


partial differentiation w.r.t. r;;( )r partial dif-
ferentiation w. r. t. t.

On Collapse of Uniform Density Sphere with Pressure 144
3. The Boundary Condition and
Thermodynamic Relation
For the exterior solution some authors have chosen
Schwarzschild vacuum solution while others have cho-
sen Vaidya’s radiative solutions in the exterior. In the
later case the heat flow is given by Kramer [8]
2
(/ )()qKyRTy


Here, K is thermal conductivity. But in the cases
where the exterior solution is chosen as Schwarzschild
solution we get NHT conditions (q = 0) given by either
() 0,Ty  that is, bb TyT y where ()
bb
TTrr(6)
Or 0T (for cold stars) (7)
Or 0K (8)
The basic law of thermodynamic change is
(1 /)Tds dU Pdn (9)
nTdsdE hdn and (/)
n
Es nT (10)
where, Uspecific internal energy,
s
specific en-
tropy and ()/hPEn  specific enthalpy. The units
of n are chosen so that,0,,PEn
 and 1h.
Writing Bondi’s results (1969) in the present notations,
one gets
3()nRBr (11)
2
/(1 )Rr   and 3()nRBr (12)
/
y
FR R (13)
22
/(/)/(1 )RRr r   
 (14)
/( )/3
b
ER
yy PEEER  

 (15)
And
22
22
()
/(1 )[()]
b
b
rr
PE rr

 




(16)
(),(),(),( ),
bbb
ttFFtyyrrrr

  at the
boundary.
Since =()EEt or 0E, we write [using Equation
(10)]
2
(/ )/()nnTsPE

 (17)
(/)[using Equation (15)]
b
Ty Eys  (18)
4. Collapse of Uniform Density Sphere
The collapse of uniform density sphere is discussed un-
der various physical conditions. [We have assumed that
()any arbitrary function of
A
tt

and B(r)  any arbi-
trary function of r]
4(a) using NHT condition (6a):
Using Equation (6) in (18) one gets,
2
/(/)
b
nnT Es


 or
(/)1 ()
bb
EnTs s

 (19)
()
bb
s
sr r

.
It is obvious from Equation (19) that the entropy of
an adiabatic uniform density sphere is minimum at the
boundary.
4(a) (i): Isentropic case: Let the entropy be constant
throughout the sphere, that is, s = constant = sb. Equation
(19) gives
En

(20)
[Using Equation (11)]
3()ERBr ()( )RAtBr

 (21)
/()RRAt

[From Equation (13)]
y =A(t) or 0y (22)
[From Equation (4)]
0,P


or P=P(t) (23)
Since, ()0()
b
Pr rPt

 , the pressure vanishes
within the sphere. Hence, an isentropic uniform sphere
undergoes a collapse with vanishing pressure only.
4(a) (ii) Non-isentropic case with constant surface
temperatures: We assume that the surface temperature
remains constant during the collapse. This is very likely
because there is no energy loss to the surrounding from
the surface of the sphere. With Tb = constant during the
collapse one gets
/[1()]( )()
bb
nETssAt Br

  (24)
[From Equation (11)]
() ()RArBr

 (25)
Arguments similar to those in 4(i) show that the pres-
sure vanishes inside the sphere.
Hence, an adiabatic uniform density sphere with con-
stant surface temperature collapses with vanishing pres-
sure.
4(a) (iii) General case: Neither the fluid is isentropic
nor the temperature of the surface remains constant. In
this case
[1 ()]
bb
En Tss


On differentiating with respect to time we obtain

On Collapse of Uniform Density Sphere with Pressure 145
()
1( )
bb
bb
Tss En
En Tss



 
(26)
For an adiabatic motion the total mass energy is a con-
stant of motion, that is,
3
(4/ 3)constant
b
MER   or /3/
b
EE RR

(27)
Using Equations (11), (14), (26) and (27) we get
02 2
22
22
22
22
22 222
() 3 3
1( ) 11
3( )
(1 )(1 )
3( )
1()
bb b
bb b
b
b
b
bb
Tss r r
Ts srr
rr
rr
rr
rr rr
 






 




 


(28)
No choice of functions(),
s
sr ()t

  and Tb =
()
b
Tt can satisfy this equation. The solutions obtained
by various authors for collapsing/expanding uniform
density [with Schwarzschild exterior solutions] are in-
consistent with the conservation law and NHT.
4(a) (iv) Explanation of inconsistency: Equation (10)
shows that (/),
n
Es nT but from Equation (19) we
see that (/)
nb
Es nT . Therefore, b nT nT or T =
b
T. Since, bb TyT y [from Equation (6)] we get y =
()
b
yAt.
Hence, 0y or [from Equation(4)] 0P P


()0
b
Pr r
The pressure vanishes throughout the sphere.
4(b) using NHT condition (7):
When T = 0 Equation (14) gives0n or ()nnt or,
(t)( )RA Br [from Equation (9)]. As shown in 4(a) (i)
the pressure vanishes inside the sphere.
4(c) using NHT condition (8):
When thermal conductivity K = 0, it seems that all the
relations of Bondi’s paper are consistent. However, let us
analyse this condition in some details. From Equation
(10) we can see that
//()nnE PE
 and //()(/)nn EPEThs
 

(29)
And for ()EEt, (/)/Thsn n


 (30)
When K = 0, no heat enters or leaves any layer within
the structure during the collapse that is we can consider
temperature of each layer to be independent of time or T
= T (r).
Eliminating n from the twin Equations (29) we obtain
(Nariai ) [9]
0
TEPPE
s
hPE





 


for ()
EP EET
PE




(31)
or
(/)( /)nTTh PEPs



(32)
It can be seen from Equation (16), that the right hand
side of Equation (32) can not be made zero in any case.
Now, we consider a hypothetical case that during the
collapse, though K = 0, somehow the temperature of
each layer changes with time making T = T (r, t), but at
the surface the temperature will not change with time,
that is, 0
b
T

. It can be seen that
[(/)]b rr
b
ETEPsP T


 (33)
The right hand side of equation can not be made zero.
5. Conclusions
After studying adiabatic collapse of a uniform density
sphere using baryon conservation law and NHT condi-
tion it is concluded that, a uniform density sphere [with
Schwarzschild geometry in the exterior] always collapses
adiabatically with vanishing pressure. Collapse with pre-
ssure will involve violation of either the baryonic con-
servation law or the no-heat flow condition. Or we can
say that when the exterior geometry is defined by Sch-
warzschild vacuum solution then the solution given by
Oppenheimer and Snyder [10] is the only valid solution
for the collapse of a uniform density sphere.
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On Collapse of Uniform Density Sphere with Pressure 146
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