Self gravitating boson system equilibrium configuration

Ruffini, R.

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•System of Self-Gravitating Bosons
^e
Remo Ruffini
Institute for Advanced Study, Princeton, New Jersey
and
Palmer Physical Laboratory, Princeton, New Jersey
To appear in the Proceedings
of the Fifth International
Conference on Gravitation and
General Relativity, Tiblisi,
September 1 968
N70-12538
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M Talk given in Section II of GR5. Chairman, P rofessor A. Z.- Petrov.
This work has been si;pport(-u by an F,. S. It. O, Postdoctoral Fellowship
and a ?1. 2,.,
 
S. A. International Fellnw-!hip.
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We will examine in the following the problem of the possible equili-
brium configurations of a system of many self-gravitating bosons in their
ground state ( temperature zero) . Treating Bose particles allows us to put
all the particles in the same ground state.
One can treat a system of many bosons at constant temperature T
es a fluid with an equation of state p = p (p) derived from quantum statistics
in flat space If we extend this treatment to T = 0 we run into difficulties:
the pressure is proportional to T5/2 and therefore vanishes in the limit
T = 0. Proceeding to this limit we could never obtain an equilibrium con-
figuration and we would inevitably have gravitational collapse 2. We there-
fore approach the problem from a quite different point of view: instead of
using the concept of an equation of state at all we treat the system of many
particles in a Hartree-Fock type approximation, looking for a self-consis-
tent solution. We first approach this problem in a Newtonian approxima-
tion and we then go to a full relativistic treatment. In the latter case we
discover the remarkable fact that Tll # T22 = T 3
 . The results also show
that we have an equilibrium configuration corresponding to every value
of the central density. The use of an equation of state is therefore in-
Nk.
-3-
1. Newtonian Approximation
In the Newtonian treatment the gravitational potential V satisfies
the Poisson equation
A V = - 4nGp	 (1)
'where p is the matter density and G Newton' s gravitational constant.
The Schrtidinger equation for a particle of mass m, in the presence of
a gravitational potential V, is
	
A + 2m41 (E+mV)kP= 0	 (2)
we are interested in a system of self-gravitating bosons all of which are
in the same quantum state. We therefore assume that the gravitational
potential V satisfies equation (1) with p = N^*ttm where N is the num-
ber of bosons and 4 is the wave function of the quantum state under con-
sideration. The wave function is normalized to one.
f J,*^d3x = 1	 (3)
We shall consider only the ground state n = 1 t = 0 which we may assume
to be spherically symmetric. Consequently the resulting system of equation
in dimensionless quantities is
E	 r- ld2/dr-2 (r"(p) + (' +	 (p = 0	 (4)
r -1d 2 /dr 2 (rV)
t`
^i
. .
	
-4-
We have carried out a numerical integration of the system (4) by the Runge-
AKutta method. We have found the eigenvalue E by looking at the behaviour
of the wave function at infinity. As usual we have determined the ground
state by requiring that the eigenfunction have no nodes . We obtain ( central
density) - 0. 9G3N4m1011 10 -3 ( distance from the center to the point at which
.the potential falls to half value) , 6. 24h 2G -lN -1m 3 ( energy to remove all the
particles to infinite separation) - 0. 246G2m5N3f 2.	 Knowing the solution
of the universal system of equation (4) and thanks to relations ( 5) it is
possible to obtain a solution relative to an arbitrary number of bosons
simply making appropriate scale changes. This argument could suggest
that for any number of bosons in the ground state there always exists a
position of equilibrium. However considering the total energy of the
system as a function of the number of particles
Etot - Nmc 2 - 2EG2N3m5fi- 
2	 (6)
We see that in a first region an increase in the particle number corres-
ponds to an increase in the total energy of the system. In a second region
the total energy of the system decreases in correspondence to an increase
.Of the particle number and can even reach negative values. It is clear that
in this second region special and general relativity .play a very important
role and the use of the Newtonian approximation is absolutely meaningless.
`C =,h2 ( 2m) -1 ( gwv814* 8v ^ - µ 2^*0	 (7)
where µ = mch -1 . By usual variational methods we can obtain the field
equations
	
Va Va0+µ20= 0
	 (8)
where Pd and S7d are respectively the operators of covariant and con-
travariant differentiation. We can also obtain the energy momentum tensor
T ,, - 2 ( 1 9 1) -1 / 2	 a d	 a V I g f	 _ a v	 e--- 	 (9)µ	 8x	 a a µv	 agµ
8X Cr
and the current vector
Jµ = ffl -l c ( ( a^e/a( awl*)
	
a^e/a( aµ^)) )	 (10)
where g = det g dR . We consider only the ground state n = 1, k = 0 and we
suppose our particle distribution to be spherically symmetric and static.
We assume, therefore, a metric of the form
ds 2
 = B(r) c 2dt 2 - A(r)dr 2 - r 2 ( sin2 0 dcp 2 + d8 2)	 (11)
determined by our distribution of bosons 3
	
Rµ;. - 2gµv R _ K <T µl, >	 (12)
We indicate by <T
µv
> the expectation values of the operator (9) relative
to the ground state. The normalization condition adopted is
F	
f j7g<J0>d3x=N	 (13
the final system of equations is
`,	 1
-6-
B' /(ABr- ) - 1/r" 2(1 - 1/A) = E [ (B-1E 01 -1) R Ol + A 1R O1 ' I
A'/(A 2 f) + 1/r" 2(1 - 1/A) _ "e [(B-1E
-0', + 1) R 01. + A-1R' O 1 1
Rai + ( 2 /r + B' /2B -A' /2A) R'01 + A(E0 lB -1 -1) Rol = 0
f0 R01 B Z A Z r" 2dr = 1	 (14)
where
r = rµ
R =	 3/2
E01 = E01/mc2
A = L2µ 2N/E01
1
and L is the Planck length L = (,iG/c3) a.
The integration of the system has been carried out by a Runge-Kutta method
and some results of the numerical integration are reported in Fig. 1 and
Table I.
The introduction of special relativity ( Klein-Gordon equation) and
general relativity eliminates completely some difficulties present in the non-
relativistic Newtonian approximation. An increase ( decrease) in the number
of particles always corresponds to an increase ( decrease) in the mass at
infinity. On the other hand the mass at infinity and the number of particles}
expressed as a function of the central density reach a critical value of gra -
vitational coliap! c for 
`•*Zcrit (0 ' 311 x 10 -9 /xn) gr, N crit __ ( 3. 01 x 10-10/m2)
corresponding to a central density pcril (5. 26 x 1097 x ,m2) g /cm3 where
m is the bobon' s mass in grata.
iOf
-7-
3. Case of One Particle
Some authors 4 have recently looked at the equations (14) in con-
nection with the..  problem of one self-gravitating particle. We have solved
this problem and we give some typical order of magnitudes in Table II.
However use believe that the treatment valid for a large number of parti-
cles is a very poor approximation for a single particle as well as for a
system of only two or three particles. In this case, in fact, the metric
is computed in correspondence not to a real density of particles but only
to a density of probability. The deviation from the real solution can
therefore be enorinous.
Quite apart from these considerations, in the neighborhood of one
elementary particle of 10 5 g and 10 -33 cm there is an extremely strong
vacuum polarization due to the gravitational field of the particle s . This
effect has not been considered in equations (14) . Aside from these ob-
jections the solutions seem to suggest an upper limit to the mass of one
elementary particle corresponding to m - 10 -5gr ( Planck mass) . This
result could be intuitively justified from the fact that for m - Planck
.mass the Compton wavelength is equal to the Schwarzschild radius of
the particle
MG _	 4^	 (15)
c
2	 me
For a particle of larger mass the Compton wavelength would be shorter
than the Schwarzschild radius and the particle would unavoidably collapse.
i3
- 8 -
Acknowledgements
1 thank Professor John A. Wheeler for his help during the accom-
plishment of this work and for his financial support from Grant No. GP7669
of the N. S.F. that made possible the presentation of this work at the Tiblisi
Conference. Particular thanks to Professor Tullio Regge for enlightening
discussions on this work and for the hospitality at the Institute for Advanced
Study.
1.
2.)
3.
4.
5_
REFERENCES
L. Landau and E. Lifschitz, Statistical Mechanics, Addison Wesley 1962.
Oppenheimer and Snyder, Phys. Rev. 56, 655 (1939).
The idea of using this self-consistent approach was given by R. Potier-
Comptes Rendus 243, 939 (1956) and it was again proposed by C. MPer,
Royaumont Conference - 1959. The explicit field equations also in the
more general case of a complex scalar field have been written by A. Das,
Journal Math. Phys. 4, 45 (1963). See also R. Ruffini, unpublished thesis,
University of Rome 1965 - 66. S. Bonazzola and F. Pacini, Phys. Rev.
AL 1269 (1966). D. Feinblum and W. McKinley, Phys. Rev. 168, 1445 (1968).
D. 1: einblum and W. McKinley, Phys. Rev. 168, 1445 (1968) - M. J.
Stakvilevieius and Ya. P. Terletsky - Tiblisi - GR5 - M. Markov, 24,
584 (1967).
important results in the problem of vacuum polarization due to a gravitational
field have been obtained by L. Halpern, Arkiv for F'ysik 34, 539 (1967).
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System of self-gravitating bosons

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