Ideal Black Hole Gas 

Larry L. Smalley 

Department of Physics 

University of Alabama in Huntsville 

Huntsville, AL 35899 

and 

Space Science Laboratory 

Marshall Space Flight Center, AL 35812 

7 August 1987 

(NASA-TB-107788) IDEAL BLACK HOLE GAS 

(NASA. Marshall Space Pliqht Center) 12 F 

CSCL 038 

-4 

1 

w 

Unclas 

G3/9Ct 0213096 
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The concept of primordial black hole creation in the early universe 

has been a common theme in early cosmological scenarios.' 

letter we put forward the concept of a primordial black hole fluid with 

intrinsic spin density and its consequence for supercluster-sized, i.e. 

large-scale voids, and the missing mass question. 

In this 

2 

3 It has been hypothesized that the mass of primordial black holes 

-5 

range as far down as the Planckian mass limit of about 10 gm. On 

the other extreme, primordial black holes with a mass of about lo1' gm 

should now be in the final stages of Hawking e~aporation.~ 

time, it was thought that these latter black holes were candidates for 

g-a-ray and x-ray bursts. However it appears more likely that 

these events are associated with the dynamics of solar remnants. 

Thus it would be likely, within the original scenario, that either these 

relic black holes were not created with sufficient mass to survive until 

the present epoch or that they were just simply not created at all. 

There is, however, the supposition that the lifetime of the inflationary 

era is associated with the evaporation time for the primordial mini black 

holes. 

been created with masses consistent with the evaporation time scale. 

What follows does not in principle conflict with this hypothesis although 

it probably modifies the time scale somewhat and makes our scenario 

more difficult. On the other hand, quantum mechanical arguments have 

been put forth that suggest that the ultimate remnants of an evaporatAt one 

5 

This would mean that the primordial black holes would have 

6 

ing black hole a degenerate gas of Planck mass black holes called 

planckons.which are stable against further decay due to the onset of 

quantum stability of the "lowest" state of a black hole.' If this is the 
3 

case, our scenario would be easier to construct. However, the consistency between the temperature of the evaporating mini black holes and 

that of the planckons seems to contradict the principle of Hawking 

evaporation and the existence of the planckons themselves. That is, 

the planckons must be relatively cool, or they would interfere with the 

big bang relic 3 K black body radiation. We do not know how this can 

be overcome, but the thermodynamic description must somehow be 

replaced with a quantum mechanical decay process especially in the 

latter stages of evaporation. 

from the planckian limit and thus well within the classical realm. 

In either case, what we present is far 

Recent observations supports the existence of significant large 

scale structures8 with extent greater than 80 Mpc. It is in fact likely 

that some of these objects may not even be visible, such as the recently discovered object in the constellation Leo which, it is conjectured, 

supposedly lenses a quasar over 1.6 Gpc distant.’ The existence of 

large scale voids with diameters of the order of 100 Mpc seem to be a 

consistent with this structure.” The general features of the universe 

shows a vast network of clusters, filaments, and voids as is evident in 

the analysis of the Shane-Wirtanen survey. l1 

general features is the question of the missing mass which is usually 

attributed to massive neutrinos, axions, strings, l4 Higgs boson 

decay,” or other generally unspecified cool or even hot dark mass. 

The existence of large quantities of dark matter seems to be indicted 

for instance by infall in the Virgo cluster;” however this dark matter 

is probably not composed of baryons. l8 

this missing mass be attributed to an ideal black hole gas with 

Combined with these 

12 13 

16 

In this note we propose that 
4 

sufficient intrinsic spin density to avoid collapse and collocated with the 

large scale voids. 

In order to avoid confrontation with the 3 K black body background radiation field, which appears from experimental measurements 

to be highly isotropic with quadrapole or higher multipole moments 

consistent with zero," we assume that the temperature of objects within 

the "voids" have an average temperature close to the background 

blackbody temperature. 

fluid which must be met: 

librium with the back ground radiation field, and the surface temperature of the mini black holes must also be near 3 K. 

This imposes two constraints on a black hole 

the fluid temperature itself must be in equi20 

We assume that during the early stages of the big bang, mini 

black holes, with possibly some intrinsic spin will be formed. 

ter the black holes can grow by accretion of other black holes. 

the accretion process, the spin of the black holes will tend to increase 

both through the union of individual spins and the absorption of the 

relative orbital angular momentum of colliding black holes (and perhaps 

ordinary matter). "his tendency for the intrinsic spin of the daughter 

black holes in black hole coalescence to increase is due to the randomness of both the initial spin distributions and the collisions between 

pairs of blackholes. 

intrinsic spin of the newly formed black holes after each union will 

favor an increase in spin angular momentum of succeding daughters. 

ThereafDuring 

This means that the random walk absorption of the 

Consider a large scale, supercluster-sized void with diameter 100 

Mpc. 

same as elsewhere in the universe with a density close to the critical 

density p 

If we assume that the initial matter density in the void is the 

3 = loe2' gm/cm , then the mass in the void would be 

C 
5 

50 approximately 8.2 X 10 gm. This assumption follows closely the 

results of a recent measurement by Loh and Spillar of the mass density 

of the universe based upon the redshift and fluxes of 1000 field galaxies. They find that the density of matter is 0.9 (+0.7,-0.5) the critical 

mass density at the 95% confidence level. 

sensitive to any matter, dark or luminous.21 

Their method is suposedly 

It is interesting to note 

for later comparison that if all this matter had coalesced into a single 

gigantic Schwarzschild black hole, it would have a radius 1.2 X cm 

(or about 40 kpc) with an "internal density" of about 1.1 X 10 

gm/cm . 

-19 

3 

The surface temperature of a black hole is given by 4,22 

T = (8nMl-l = 1.225 X lo1' gm-K/M (1) 

where M is given in grams. 

only approximate since the mass should be corrected for rotation. 

(In what follows, this should be taken as 

18 

The temperature and mass could be iterated in what follows, but this 

would not change our qualitative conclusion.) For a surface temperature of 3 K, the mass would be M3K = 4.1 X 10l8 gm with radius R 

6.1 X 10-l' cm with internal density 4.2 X 10 

= 3K 

gm/cm . Note that 51 3 

this radius is larger than the radius of a nucleus. Thus if all the mass 

of the cluster were concentrated in such black holes, there would be 

2.0 X of them in the black hole fluid! 

The spin of matter should give rise to a repulsion (or bounce) 

23 during the "final" collapse of the universe towards a singularity. 

The same would be true for any fluid with sufficient spin density. 

Thus if the 3 K black holes have sufficient spin, they would be stable 

against further coalescence. 

growth problem even for stiff matter.24 

This repulsion then avoids the further 

Suppose that the relaxation 
6 

process has continued until the black hole fluid has attained a density 

comparable with nuclear density, p = 7 X 10 gm/cm . The 3 K black 

n 

holes would have a relative separation of the order of 5.2 X 10 

a root mean square fluid velocity of 1.7 X 10 

9 3 

2 

cm and 

-17 cm/s. Such numbers 

give a new meaning to the concept of "stiff" matter used below. 

close-packed "radius" of the fluid would be about 30 X 1013 cm, and 

The 

thus the fluid could be contained in a region within the orbit of Jupiter. This should be compared with the gigantic supercluster-sized 

black hole mentioned above. Although such a large black hole is possible, the surface temperature is far below the background 3 K black 

body radiation and thus should give rise to some dectectable multipole 

components to the uniform blackbody radiation. Experimental data, 

however, seems to exclude this possibility19 although there is now 

reasonable evidence that large black holes with mass of the order of 

106'8 solar masses are contained in the nuclei of gala~ies.~' Such 

masses are consistent with previously observed mass distributions in 

spiral galaxies providing the mass of the black hole is less than 10% of 

the mass of the galaxy. 26 Within the context of our arguments here, 

the spin was not able to prevent coalescence. 

We now investigate how spin can prevent the fluid from collapsing. 

It can be shown that for a fluid with randomly oriented spin density, 

the renormalized pressure and energy density is given by 

22 p' = p - 2rG s /C 

and 

24 

p' = p - 2nG s /C 
7 

where G is the gravitationa constant, c is the speeu of light, and s is 

the spin density. Let the renormalized density be of the order of 

nuclear density and let the fluid be ideal. Then 

p - (2rG/cL)sL = pn kT/mgK 

2 For a stiff fluid, p = pnc , then 

22 pnc2 - (2nG/c )s = pn kT/m3K 

and 

4 2 s2 = (pnc /2nG)[1 - kT/rn3Kc ] 

For T = 3K and M3K = 4.1 X 10l8 gm, we note that 

kT/m3Kc2 = 1.1 X 10 -55 

(4) 

(5) 

(7) 

and the spin density 

s = 1.2 x gm/cm-s. (8) 

This is the spin density that gives an energy density comparible with 

the nuclear energy density and thus prevents collapse of the fluid into 

a larger black hole. 

We note the "fine tuning" of the energy associated with the fluid 

temperature to that of the rest mass energy of the fluid "particles" to 

one part in los5. Such fine tuning between present epoch astronomical 

data compared with intial conditions in the big bang has been noticed 

consistently before. 

Experimental evidence indicates that large compact objects are 

contained in the cores of some galaxies. Thus galaxies (and necessarily 

clusters) may have passed beyond fluid state so that they contain a 

large black hole core or even a small black hole fluid core. This is 

precisely what one would expect on the average for a system so finely 

tuned. Thus for galaxies the initial state was also a black hole fluid; 

however the "particle" mass of the fluid did not grow sufficiently to 
a 

avoid the Hawking evaporation up to the present epoch. 

masses of the order of 10 

and source for the emission of energy by a quasar. 

hole fluid is much more concentratdthen a single black hole of comparaFor particle 

gm, we now have a possible mechanism 

Since the black 

14-15 

ble mass, the size of the emission regions would no longer be a source 

problem for quasar emission. If this hypothesis is correct, then some 

quasars could be relatively close providing the particle masses were 

large enough. For example, quasars with z < 0.2 are known but are 

rare compared with those with z > 1. It is interesting to speculate that 

the most spectacular event that we might observe would be the "turning 

on" of a quasar, or even a galactic nucleus, e.g. the brightening by a 

factor of two of 3C147 over a period of six years although this is 

normally attributed to relativistic motion within the core. 27 We also 

speculate that the spin axes of the black hole fluid would tend to align 

over the accretion period period of the fluid since anti-alignment would 

favor particle coalescence. 

orbital angular momentum) of the black hole fJuid would then be S = 

The total spin (not including any overall 

70 2 V s = 1.4 X 10 gm cm /s. Muradian has shown that astronomical 3K 

objects seem to group into two angular momentum, J, classes:28 J = fi 

(M/m )4'3 for planets and stars, and J = d (M/mp)3/2 for galaxies and 

clusters where m is the mass of a proton. For the 4/3 exponent, J = 

1.1 x 10 gm cm /s, whereas for the 3/2 exponent, J is twelve orders 

P 

P 

72 2 

of magnitude larger. Thus the total spin of the black hole fluid compares well with the observed angular momentum relationship for planets 

and stars. We speculate that this implies that local black hole evaporation provides the seed perturbation for stars instead of galaxies whereas the overall orbital motion is imparted to the galaxy (or cluster) as a 
9 

whole. 

polarized medium (object) which could explain the directionality and 

perhaps the strength of jets (radio lobes) from quasars. 

Also the fluid would represent in its mature stages a compact 

Finally the overall consequence is that the dark matter in the 

universe which is concentrated in the ''voids" should be of the same 

order of magnitude as visible matter in the universe. 

this scenario, the universe would probably be closed. 

On the basis of 
10 

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