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J. Modern Physics, 2010, 1, 143-146 doi:10.4236/jmp.2010.12020 Published Online June 2010 (http://www.SciRP.org/journal/jmp) Copyright © 2010 SciRes. 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. Email: garciluiz@gmail.com 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 Copyright © 2010 SciRes. JMP 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, Uspecific 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 Copyright © 2010 SciRes. JMP 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. REFERENCES [1] W. B. Bonnor and M. C. Faulkes, “Exact Solutions for Oscillating Spheres in General Relativity,” Monthly No- tices of the Royal Astronomical Society, Vol. 137, 1967, pp. 239-251. [2] I. H. Thompson and G. J. Whitrow, “Time-Dependent Internal Solutions for Spherically Symmetrical Bodies in General Relativity-I. Adiabatic collapse,” Monthly No- tices of the Royal Astronomical Society, Vol. 136, 1967, pp. 207-217. [3] I. H. Thompson and G. J. Whitrow, “Time-dependent internal solutions for spherically symmetrical bodies in general relativity-II. Adiabatic radial motions of uni- formly dense spheres,” Monthly Notices of the Royal As- tronomical Society, Vol. 139, 1968, pp. 499-513. [4] H. Bondi, “Gravitational Bounce in General Relativity,” Monthly Notices of the Royal Astronomical Society, Vol. 142, 1969, pp. 333-353. [5] R. M. Misra and D. C. Srivastava, “Relativity-Bounce of Fluid Spheres,” Nature Physical Science, Vol. 238, 1972, p. 116. [6] C. W. Misner and D. H. Sharp,“Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Col- lapse,” Physical Review B, Vol. 136, No. 2B, 1964, pp. On Collapse of Uniform Density Sphere with Pressure Copyright © 2010 SciRes. JMP 146 B571-576. [7] M. Demianski, “Relativistic Astrophysics,” Pergamon Press, New York, 1985. [8] D. Kramer, “Spherically Symmetric Radiating Solution with Heat Flow in General Relativity,” Journal of Mathematical Physics, Vol. 33, No. 4, 1992, pp. 1458- 1462. [9] H. Nariai, “A Simple Model for Gravitational Collapse with Pressure Gradient,” Progress of Theoretical physics, Vol. 38, No. 1, 1967, pp. 92-106. [10] J. R. Oppenheimer and H. Snyder, “On Continued Gravi- tational Contraction,” Physical Review, Vol. 56, No. 5, 1939, pp. 455-459. |