General Relativity and Gravitation, Vol. 10, No. 3 (1979), pp. 227-230

Minicompact Objects According to the Bimetric Theory of Gravitation

D. FALIK 1 and R. OPHER

Department o f Physics, Technion-lsrael Institute o f Technology, Haifa, Israel

Received February 11,1978

Abstract

It is shown that, similarly to general relativity, the bimetric theory of gravitation predicts the folmalion of primordial minicompact objects. Contrary to general relativity, however, it predicts that such objects are stable. Observational consequences are discussed.

Hawking was the first to note that according to general relativity (GR), random fluctuations in the early universe might have formed mini-black-holes (MBHs) of masses 10 -s g and higher [1]. Such MBHs may have observational consequences if they are charged: (a) they may be seen in particle detectors as straight trajectories (even at small velocities their momentum is so high that even strong magnetic fields cannot bend them appreciably) of constant ioniza- tion (their loss of energy through ionization in the detector is insignificant); and (b) they will accumulate in the centers of stars up to some 1017 g. This accumu- lated central black hole may modify the conditions in the center of a star like the sun and thus reduce the flux of neutrinos from it (helping to explain perhaps the famous puzzle). When such a star collapses to a neutron star, it will be swallowed by the central black hole in about 10 million years.

Some years later, Hawking himself destroyed this nice picture: he showed [2] that black holes emit energy, and thus have a finite lifetime, which depends on their initial mass. The lifetime of black holes with initial mass of less than 101 s g is less than the age of the universe so that a primordial MBH cannot exist today if its initial mass was less than 10 is g.

1 In partial fulfillment of the requirements for the D.Se. degree.

227 0001-7701/79/0200-0227503.00/0 �9 1979 Plenum Pubfishing Corporation

228 F A L I K A N D O P H E R

Both these theories were developed in accordance with GR. Now the most serious challenger to GR is Rosen's bimetric (BM) theory of gravitation [3], and it is interesting to see whether this theory predicts similar effects.

Random fluctuations in the early universe are allowed also by BM. In GR, when a massMhas a radius smaller than 2 G M / c 2 it m u s t collapse. In BM, on the other hand, what happens depends on the equation of state of the material at ultrahigh densities-some equations of state cannot uphold the gravitational pull and let the matter collapse, but some (for instance t9 = const) may resist gravitation [4] and allow a static minicompact object (MCO) to form with a small, but finite, radius. Rosen and Rosen suggest [4] that the density of such compact objects may be the Planck density c S f l i G 2 (~1094 g cm-3) , so that their radius will be very sma l l -o f the general order of 10 -30 cm. 2

Although the radius of such a MCO is very small, it is still finite, and the metric around it will be regular, without any horizon forming near it.

An essential point in the quantum emission of energy from black holes is the existence of an horizon: a pair of positive and negative energy particles forms near the horizon because of quantum fluctuations, and the negative- energy particle can move inside the horizon because its Killing vector, which is spacelike outside the horizon (because of its negative energy), becomes timelike inside the horizon [5].

Since the MCOs formed in the early universe do not have any horizon around them, no particle creation by the Hawking process [2] can take place. The MCOs will be stable, and the early predictions about the abundance of MBHs apply to the abundance of MCOs according to BM.

Since MCOs do not radiate according to BM and are stable for the lifetime of the universe, they should be searched for. Besides the previously mentioned characteristics for charged MBHs [1] of constant ionization rate and straight line trajectories in a magnetic field, other characteristics for charged MCOs are to be noted. The ionization rate [6] for a charged MCO is ~10 GeV g-1 cm 2 for an incident velocity of 1000 km sec -1 . This is much higher than the ioniza- tion rate of any other highly penetrating particle (besides perhaps magnetic monopoles). For example, in a 1-m path length in a liquid scintilator a MCO will release ~1000 GeV.

An interesting coincidence is to be noted. The kinetic energy of an MCO of mass 10 -s g falling freely in the gravitational field of the sun is 1020 eV when it arrives at the earth. This is the approximate cutoff energy in the energy spec-

2For an object of mass m to be compact, its radius r has to be at least of the order of its gravitational radius r G = Gm/c 2 ; thus its average density has to be at least m/rG 3 or ( c6 /G3)m -2 . Minicompact objects, with mass less than 101 s g, have, therefore, densities

54 3 of more than 10 gcm- . Nothing can be said with certainty about the equation of state of matter at such high densities, not even whether the velocity of sound is smaller than the velocity of light [4] ; thus the equation of state p = const seems to be as realistic as any other.

MINICOMPACT OBJECTS IN BIMETRIC THEORY 229

trum of cosmic rays. The flux of cosmic rays [7] having energies greater than 102o eVis about 3 X 10 -16 cm -2 sec -1 Sr -1 , which is only five times larger than the flux of MBHs predicted by Hawking [1 ] . Since it is well known that there is great difficulty in explaining the flux of 1020 eV and its cutoff, the high-energy cosmic flux might possibly be connected with falling MCOs.

MCOs may accumulate at the center of a star, similar to the prediction for MBHs [1 ] , and when such a star collapses to a neutron star accretion may (or may not) occur onto the central compact object. Let us examine the amount of accretion that can occur onto the compact object in BM taking the equation of state as p = Po = const and examine what happens for different values of Po.

The line element about the central object will be [3]

ds2 =exp (2~r ) dt2 - exp (--~-)da2 (1)

where M is the mass of the compact object and M ' its "secondary mass." For a freely falling body we get, from the geodetic equation,

~t=exp[-(M~M')][1-exp(7-~-r )]a/2 (2)

The volume element is given by

do = 4rrr 2 exp (3M'/r) dr (3)

so the mass accreted per unit time will be

dm dv (3M'){dr) ~-~=p -~ = 47rR2p exp \ R ]\dt]r=R (4)

and finally

dm__= 47rR2p exp M 1 - exp (5) dt

where R is the radius of the compact object. As can be seen from the table in [4], when M > > c 3 G-3Dpo -1/2 (for Po =

eS/~G 2 this corresponds t o M > > 10 -s g) then M ' is negative,M/R > > 1, -M'/R > > 1 so that dm/dt < < < (dm/dt)cR, where (dm/dt)GR is the accretion rate in GR, equal to 16rrM2p.

We see that if Po ~ c s/~G 2 ~ 1094 g cm -a , a central compact object in a neutron star will not accrete any appreciable amount of matter even during the entire lifespan of the universe. If, however, P0 < 10 s~ g cm -3 , a central compact object of mass 101 ~ g will have M/R ~ 1, M'/R ~ 1, so that dm/dt ~ (dm/dt)cR. This shows that such a compact object will swallow a neutron star in the same period of 10 million years as predicted by Hawking [1 ].

In a recent article Jacobs an Seltzer [8] suggest that MBHs form today in

230 FALIK AND OPHER

the centers of neutron stars. This suggestion rests on the possibility that a mass M will have a radius smaller than 2 GM/c z , a situation which, according to GR, leads to a formation of a MBH. All the arguments for the occurance of such a situation apply also according to BM, and such a situation will cause the forma- tion of an MCO. Such an MCO may, or may not, accrete matter from the star (depending on its density), but will not radiate appreciably.

Note Added in Proof

An MCO may have already been observed and mistakenly identified as a magnetic monopole [9].

References

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