Can a Black Hole Collapse to a Space-time Singularity?
Noname manuscript No.
(will be inserted by the editor)
Can a Black Hole Collapse to a Space-time Singularity?
R. K. Thakur
Received: date / Accepted: date
Abstract A critique of the singularity theorems of Penrose, Hawking, and Geroch is
given. It is pointed out that a gravitationally collapsing black hole acts as an ultrahigh
energy particle accelerator that can accelerate particles to energies inconceivable in any
terrestrial particle accelerator, and that when the energy E of the particles comprising
matter in a black hole is ∼ 102GeV or more, or equivalently, the temperature T is
∼ 1015K or more, the entire matter in the black hole is converted into quark-gluon
plasma permeated by leptons. As quarks and leptons are fermions, it is emphasized that
the collapse of a black-hole to a space-time singularity is inhibited by Pauli’s exclusion
principle. It is also suggested that ultimately a black hole may end up either as a
stable quark star, or as a pulsating quark star which may be a source of gravitational
radiation, or it may simply explode with a mini bang of a sort.
Keywords black hole · gravitational collapse · space-time singularity · quark star
1 Introduction
When all the thermonuclear sources of energy of a star are exhausted, the core of
the star begins to contract gravitationally because, practically, there is no radiation
pressure to arrest the contraction, the pressure of matter being inadequate for this
purpose. If the mass of the core is less than the Chandrasekhar limit (∼ 1.44M⊙), the
contraction stops when the density of matter in the core, ρ > 2 × 106 g cm−3; at this
stage the pressure of the relativistically degenerate electron gas in the core is enough
to withstand the force of gravitation. When this happens, the core becomes a stable
white dwarf. However, when the mass of the core is greater than the Chandrasekhar
limit, the pressure of the relativistically degenerate electron gas is no longer sufficient
to arrest the gravitational contraction, the core continues to contract and becomes
R. K. Thakur
Retired Professor of Physics,
School of Studies in Physics,
Pt. Ravishankar Shukla University, Raipur, India
Tel.: +91-771-2255168
E-mail: rkthakur0516@yahoo.com
http://arxiv.org/abs/0704.1661v2
denser and denser; and when the density reaches the value ρ ∼ 107 g cm−3, the process
of neutronization sets in; electrons and protons in the core begin to combine into
neutrons through the reaction
p+ e− → n+ νe
The electron neutrinos νe so produced escape from the core of the star. The gravi-
tational contraction continues and eventually, when the density of the core reaches the
value ρ ∼ 1014 g cm−3, the core consists almost entirely of neutrons. If the mass of
the core is less than the Oppenheimer-Volkoff limit (∼ 3M⊙), then at this stage the
contraction stops; the pressure of the degenerate neutron gas is enough to withstand
the gravitational force. When this happens, the core becomes a stable neutron star. Of
course, enough electrons and protons must remain in the neutron star so that Pauli’s
exclusion principle prevents neutron beta decay
n→ p+ e− + νe
Where νe is the electron antineutrino (Weinberg 1972a). This requirement sets a
lower limit ∼ 0.2M⊙ on the mass of a stable neutron star.
If, however, after the end of the thermonuclear evolution, the mass of the core of a
star is greater than the Chandrasekhar and Oppenheimer-Volkoff limit, the star may
eject enough matter so that the mass of the core drops below the Chandrasekhar and
Oppenheimer-Volkoff limit as a result of which it may settle as a stable white dwarf
or a stable neutron star. If not, the core will gravitationally collapse and end up as a
black hole.
As is well known, the event horizon of a black hole of mass M is a spherical
surface located at a distance r = rg = 2GM/c
2 from the centre, where G is Newton’s
gravitational constant and c the speed of light in vacuum; rg is called gravitational
radius or Schwarzschild radius. An external observer cannot observe anything that is
happening inside the event horizon, nothing, not even light or any other electromagnetic
signal can escape outside the event horizon from inside. However, anything that enters
the event horizon from outside is swallowed by the black hole; it can never escape
outside the event horizon again.
Attempts have been made, using the general theory of relativity (GTR), to under-
stand what happens inside a black hole. In so doing, various simplifying assumptions
have been made. In the simplest treatment (Oppenheimer and Snyder 1939; Weinberg
1972b) a black hole is considered to be a ball of dust with negligible pressure, uniform
density ρ = ρ(t), and at rest at t = 0. These assumptions lead to the unique solution
of the Einstein field equations, and in the comoving co-ordinates the metric inside the
black hole is given by
2 −R2(t)
1− k r2
in units in which speed of light in vacuum, c=1, and where k is a constant. The require-
ment of energy conservation implies that ρ(t)R3(t) remains constant. On normalizing
the radial co-ordinate r so that
R(0) = 1 (2)
one gets
ρ(t) = ρ(0)R
(t) (3)
The fluid is assumed to be at rest at t = 0, so
Ṙ(0) = 0 (4)
Consequently, the field equations give
ρ(0) (5)
Finally, the solution of the field equations is given by the parametric equations of
a cycloid :
ψ + sin ψ
(1 + cos ψ) (6)
From equation (6) it is obvious that when ψ = π. i.e., when
t = ts =
8πGρ(0)
a space-time singularity occurs; the scale factor R(t) vanishes. In other words, a black
hole of uniform density having the initial values ρ(0), and zero pressure collapses from
rest to a point in 3 - subspace, i.e., to a 3 - subspace of infinite curvature and zero
proper volume, in a finite time ts; the collapsed state being a state of infinite proper
energy density. The same result is obtained in the Newtonian collapse of a ball of dust
under the same set of assumptions (Narlikar 1978).
Although the black hole collapses completely to a point at a finite co-ordinate
time t = ts, any electromagnetic signal coming to an observer on the earth from the
surface of the collapsing star before it crosses its event horizon will be delayed by its
gravitational field, so an observer on the earth will not see the star suddenly vanish.
Actually, the collapse to the Schwarzschild radius rg appears to an outside observer to
take an infinite time, and the collapse to R = 0 is not at all observable from outside
the event horizon.
The internal dynamics of a non-idealized, real black hole is very complex. Even
in the case of a spherically symmetric collapsing black hole with non-zero pressure
the details of the interior dynamics are not well understood, though major advances
in the understanding of the interior dynamics are now being made by means of nu-
merical computations and analytic analyses. But in these computations and analyses
no new features have emerged beyond those that occur in the simple uniform-density,
free-fall collapse considered above (Misner,Thorne, and Wheeler 1973). However, us-
ing topological methods, Penrose (1965,1969), Hawking (1996a, 1966b, 1967a, 1967b),
Hawking and Penrose (1970), and Geroch (1966, 1967, 1968) have proved a number
of singularity theorems purporting that if an object contracts to dimensions smaller
than rg, and if other reasonable conditions - namely, validity of the GTR, positivity of
energy, ubiquity of matter and causality - are satisfied, its collapse to a singularity is
inevitable.
2 A critique of the singularity theorems
As mentioned above, the singularity theorems are based, inter alia, on the assump-
tion that the GTR is universally valid. But the question is : Has the validity of
the GTR been established experimentally in the case of strong fields ? Actually,
the GTR has been experimentally verified only in the limiting case of week fields,
it has not been experimentally validated in the case of strong fields. Moreover, it
has been demonstrated that when curvatures exceed the critical value Cg = 1/L
where Lg =
h̄ G/c3
= 1.6 × 10−33 cm corresponding to the critical density
ρg = 5 × 1093 g cm−3, the GTR is no longer valid; quantum effects must enter the
picture (Zeldovich and Novikov 1971). Therefore, it is clear that the GTR breaks down
before a gravitationally collapsing object collapses to a singularity. Consequently, the
conclusion based on the GTR that in comoving co-ordinates any gravitationally col-
lapsing object in general, and a black hole in particular, collapses to a point in 3-space
need not be held sacrosanct, as a matter of fact it may not be correct at all.
Furthermore, while arriving at the singularity theorems attention has mostly been
focused on the space-time geometry and geometrodynamics; matter has been tacitly
treated as a classical entity. However, as will be shown later, this is not justified;
quantum mechanical behavior of matter at high energies and high densities must be
taken into account. Even if we regard matter as a classical entity of a sort, it can be
easily seen that the collapse of a black hole to a space-time singularity is inhibited
by Pauli’s exclusion principle. As mentioned earlier, a collapsing black hole consists,
almost entirely, of neutrons apart from traces of protons and electrons; and neutrons
as well as protons and electrons are fermions; they obey Pauli’s exclusion principle. If
a black hole collapses to a point in 3-space, all the neutrons in the black hole would be
squeezed into just two quantum states available at that point, one for spin up and the
other for spin down neutron. This would violate Pauli’s exclusion principle, according to
which not more than one fermion of a given species can occupy any quantum state. So
would be the case with the protons and the electrons in the black hole. Consequently,
a black hole cannot collapse to a space-time singularity in contravention to Pauli’s
exclusion principle.
Besides, another valid question is : What happens to a black hole after t > ts, i.e.,
after it has collapsed to a point in 3-space to a state of infinite proper energy density,
if at all such a collapse occurs? Will it remain frozen forever at that point? If yes, then
uncertainties in the position co-ordinates of each of the particles - namely, neutrons,
protons, and electrons - comprising the black hole would be zero. Consequently, accord-
ing to Heisenberg’s uncertainty principle, uncertainties in the momentum co-ordinates
of each of the particles would be infinite. However, it is physically inconceivable how
particles of infinite momentum and energy would remain frozen forever at a point.
From this consideration also collapse of a black hole to a singularity appears to be
quite unlikely.
Earlier, it was suggested by the author that the very strong ’hard-core’ repulsive
interaction between nucleons, which has a range lc ∼ 0.4× 10−13 cm, might set a limit
on the gravitational collapse of a black hole and avert its collapse to a singularity
(Thakur 1983). The existence of this hard-core interaction was pointed out by Jastro
(1951) after the analysis of the data from high energy nucleon-nucleon scattering ex-
periments. It has been shown that this very strong short range repulsive interaction
arises due to the exchange of isoscalar vector mesons ω and φ between two nucleons (
Scotti and Wong 1965). Phenomenologically, that part of the nucleon-nucleon potential
which corresponds to the repulsive hard core interaction may be taken as
Vc(r) = ∞ for r < lc (8)
where r is the distance between the two interacting nucleons. Taking this into account,
the author concluded that no spherical object of mass M could collapse to a sphere
of radius smaller than Rmin = 1.68 × 10−6M1/3 cm, or of the density greater than
ρmax = 5.0× 1016 g cm−3. It was also pointed out that an object of mass smaller than
Mc ∼ 1.21 × 1033 gm could not cross the event horizon and become a black hole; the
only course left to an object of mass smaller than Mc was to reach equilibrium as
either a white dwarf or a neutron star. However, one may not regard these conclusions
as reliable because they are based on the hard core repulsive interaction (8) between
nucleons which has been arrived at phenomenologically by high energy nuclear physi-
cists while accounting for the high energy nucleon-nucleon scattering data; but it must
be noted that, as mentioned above, the existence of the hard core interaction has been
demonstrated theoretically also by Scotti and Wong in 1965. Moreover, it is interesting
to note that the upper limitMc ∼ 1.21×1033 g = 0.69M⊙ on the masses of objects that
cannot gravitationally collapse to form black holes is of the same order of magnitude
as the Chandrasekhar and the Oppenheimer- Volkoff limits.
Even if we disregard the role of the hard core, short range repulsive interaction
in arresting the collapse of a black hole to a space-time singularity in comoving co-
ordinates, it must be noted that unlike leptons which appear to be point-like particles
- the experimental upper bound on their radii being 10−16 cm (Barber et al. 1979)
-nucleons have finite dimensions. It has been experimentally demonstrated that the
radius r0 of the proton is about 10
−13 cm(Hofstadter & McAllister 1955). Therefore,
it is natural to assume that the radius r0 of the neutron is also about 10
−13 cm.
This means the minimum volume vmin occupied by a neutron is
3. Ignoring the
“mass defect” arising from the release of energy during the gravitational contraction
(before crossing the event horizon), the number of neutrons N in a collapsing black
hole of mass M is, obviously, Mmn
where mn is the mass of the neutron. Assuming
that neutrons are impregnable particles, the minimum volume that the black hole can
occupy is Vmin = Nvmin = vmin
, for neutrons cannot be more closely packed
than this in a black hole. However, Vmin =
where Rmin is the radius of
the minimum volume to which the black hole can collapse. Consequently, Rmin =
. On substituting 10−13 cm for r0 and 1.67× 10−24 g for mn one finds that
Rmin = 8.40 × 10−6M1/3. This means a collapsing black hole cannot collapse to a
density greater than ρmax =
= Nmn
4/3πr3
= 3.99× 1014 g cm−3. The critical mass
Mc of the object for which the gravitational radius Rg = Rmin is obtained from the
equation
This gives
Mc = 1.35× 1034 g = 8.68M⊙ (10)
Obviously, for M > Mc, Rg > Rmin, and for M < Mc, Rg < Rmin.
Consequently, objects of mass M < Mc cannot cross the event horizon and become
a black hole whereas those of mass M > Mc can. Objects of mass M < Mc will,
depending on their mass, reach equilibrium as either white dwarfs or neutron stars. Of
course, these conclusions are based on the assumption that neutrons are impregnable
particles and have radius r0 = 10
−13cm each. Also implicit is the assumption that
neutrons are fundamental particles; they are not composite particles made up of other
smaller constituents. But this assumption is not correct; neutrons as well as protons and
other hadrons are not fundamental particles; they are made up of smaller constituents
called quarks as will be explained in section 4. In section 5 it will be shown how, at
ultrahigh energy and ultrahigh density, the entire matter in a collapsing black hole is
eventually converted into quark-gluon plasma permeated by leptons.
3 Gravitationally collapsing black hole as a particle accelerator
We consider a gravitationally collapsing black hole. On neglecting mutual interactions
the energy E of any one of the particles comprising the black hole is given by E2 =
p2 + m2 > p2, in units in which the speed of light in vacuum c = 1, where p is the
magnitude of the 3-momentum of the particle and m its rest mass. But p = h
, where λ
is the de Broglie wavelength of the particle and h Planck’s constant of action. Since all
lengths in the collapsing black hole scale down in proportion to the scale factor R(t)
in equation (1), it is obvious that λ ∝ R(t). Therefore it follows that p ∝ R−1(t), and
hence p = aR−1(t), where a is the constant of proportionality. From this it follows
that E > a/R. Consequently, E as well as p increases continually as R decreases. It is
also obvious that E and p, the magnitude of the 3-momentum, → ∞ as R → 0. Thus,
in effect, we have an ultra-high energy particle accelerator, so far inconceivable in any
terrestrial laboratory, in the form of a collapsing black hole, which can, in the absence
of any physical process inhibiting the collapse, accelerate particles to an arbitrarily
high energy and momentum without any limit.
What has been concluded above can also be demonstrated alternatively, without
resorting to GTR, as follows. As an object collapses under its selfgravitation, the in-
terparticle distance s between any pair of particles in the object decreases. Obviously,
the de Broglie’s wavelength λ of any particle in the object is less than or equal to s, a
simple consequence of Heisenberg’s uncertainty principle. Therefore, s ≥ h/p, where h
is Planck’s constant and p the magnitude of 3-momentum of the particle. Consequently,
p ≥ h/s and hence E ≥ h/s. Since during the collapse of the object s decreases, the
energy E as well as the momentum p of each of the particles in the object increases.
Moreover, from E ≥ h/s and p ≥ h/s it follows that E and p → ∞ as s → 0. Thus,
any gravitationally collapsing object in general, and a black hole in particular, acts as
an ultrahigh energy particle accelerator.
It is also obvious that ρ, the density of matter in the black hole, increases as it
collapses. In fact, ρ ∝ R−3, and hence ρ→ ∞ as R → 0.
4 Quarks: The building blocks of matter
In order to understand eventually what happens to matter in a collapsing black hole one
has to take into account the microscopic behavior of matter at high energies and high
densities; one has to consider the role played by the electromagnetic, weak, and strong
interactions - apart from the gravitational interaction - between the particles compris-
ing the matter. For a brief account of this the reader is referred to Thakur(1995), for
greater detail to Huang(1992), or at a more elementary level to Hughes(1991).
As has been mentioned in Section 2, unlike leptons, hadrons are not point-like parti-
cles, but are of finite size; they have structures which have been revealed in experiments
that probe hadronic structures by means of electromagnetic and weak interactions. The
discovery of a very large number of apparently elementary (fundamental) hadrons led
to the search for a pattern amongst them with a view to understanding their nature.
This resulted in attempts to group together hadrons having the same baryon number,
spin, and parity but different strangeness S ( or equivalently hypercharge Y = B + S,
where B is the baryon number) into I-spin (isospin) multiplets. In a plot of Y against
I3 (z- component of isospin I), members of I-spin multiplets are represented by points.
The existence of several such hadron (baryon and meson) multiplets is a manifestation
of underlying internal symmetries.
In 1961 Gell-Mann, and independently Neémann, pointed out that each of these
multiplets can be looked upon as the realization of an irreducible representation of an
internal symmetry group SU(3) ( Gell-Mann and Neémann 1964). This fact together
with the fact that hadrons have finite size and inner structure led Gell-Mann, and
independently Zweig, in 1964 to hypothesize that hadrons are not elementary particles,
rather they are composed of more elementary constituents called quarks (q) by Gell-
Mann (Zweig called them aces). Baryons are composed of three quarks (q q q) and
antibaryons of three antiquarks (q q q) while mesons are composed of a quark and
an antiquark each. In the beginning, to account for the multiplets of baryons and
mesons, quarks of only three flavours, namely, u(up), d (down), and s(strange) were
postulated, and they together formed the basic triplet
 of the internal symmetry
group SU(3). All these three quarks u, d, and s have spin 1/2 and baryon number
1/3. The u quark has charge 2/3 e whereas the d and s quarks have charge −1/3 e
where e is the charge of the proton. The strangeness quantum number of the u and d
quarks is zero whereas that of the s quark is -1. The antiquarks (u , d , s) have charges
−2/3 e, 1/3 e, 1/3 e and strangeness quantum numbers 0, 0, 1 respectively. They all
have spin 1/2 and baryon number -1/3. Both u and d quarks have the same mass,
namely, one third that of the nucleon, i.e., ≃ 310MeV/c2 whereas the mass of the s
quark is ≃ 500MeV/c2. The proton is composed of two up and one down quarks (p:
uud) and the neutron of one up and two down quarks (n: udd).
Motivated by certain theoretical considerations Glashow, Iliopoulos and Maiani
(1970) proposed that, in addition to u, d, s quarks, there should be another quark
flavour which they named charm (c). Gaillard and Lee (1974) estimated its mass to be
≃ 1.5GeV/c2. In 1974 two teams, one led by S.C.C. Ting at SLAC (Aubert et al. 1974)
and another led by B. Richter at Brookhaven (Augustin et al. 1974) independently
discovered the J/Ψ , a particle remarkable in that its mass (3.1GeV/c2) is more than
three times that of the proton. Since then, four more particles of the same family,
namely, ψ(3684), ψ(3950), ψ(4150), ψ(4400) have been found. It is now established
that these particles are bound states of charmonium (cc), J/ψ being the ground state.
On adopting non-relativistic independent quark model with a linear potential between
c and c, and taking the mass of c to be approximately half the mass of J/ψ, i. e. ,
1.5GeV/c2, one can account for the J/ψ family of particles. The c has spin 1/2, charge
2/3 e, baryon number 1/3, strangeness −1, and a new quantum number charm (c)
equal to 1. The u, d, s quarks have c = 0. It may be pointed out here that charmed
mesons and baryons, i. e. , the bound states like (cd), and (cdu) have also been found.
Thus the existence of the c quark has been established experimentally beyond any
shade of doubt.
The discovery of the c quark stimulated the search for more new quarks. An ad-
ditional motivation for such a search was provided by the fact that there are three
generations of lepton weak doublets:
, and
where νe, νµ, and ντ are elec-
tron (e), muon (µ), and tau lepton (τ ) neutrinos respectively. Hence, by analogy, one
expects that there should be three generations of quark weak doublets also:
. It may be mentioned here that weak interaction does not distinguish between
the upper and the lower members of each of these doublets. In analogy with the isopin
1/2 of the strong doublet
, the weak doublets are regarded as possessing weak isopin
IW = 1/2, the third component (IW )3 of this weak isopin being + 1/2 for the upper
components of these doublets and - 1/2 for the lower components. These statements ap-
ply to the left-handed quarks and leptons, i. e. , those with negative helicity (i. e. , with
the spin antiparallel to the momentum) only. The right-handed leptons and quarks,
i. e. , those with positive helicity (i. e. , with the spin parallel to the momentum), are
weak singlets having weak isopin zero.
The discovery, at Fermi Laboratory, of a new family of vector mesons, the upsilon
family, starting at a mass of 9.4GeV/c2 gave an evidence for a new quark flavour called
bottom or beauty (b) (Herb et al. 1997; Innes et al. 1977). These vector mesons are
in fact, bound states of bottomonium (bb). These states have since been studied in
detail at the Cornell electron accelerator in an electron-positron storage ring of energy
ideally matched to this mass range. Four such states with masses 9.46, 10.02, 10.35, and
10.58 GeV/c2 have been found, the state with mass 9.46GeV/c2 being the ground state
(Andrews et al. 1980). This implies that the mass of the b quark is ≃ 4.73GeV/c2. The
b quark has spin 1/2 and charge −1/3 e. Furthermore, the b flavoured mesons have
been found with exactly the expected properties (Beherend et al. 1983).
After the discovery of the b quark, the confidence in the existence of the sixth quark
flavour called top or truth (t) increased and it became almost certain that, like leptons,
the quarks also occur in three generations of weak isopin doublets, namely,
. In view of this, intensive search was made for the t quark. But the discovery
of the t quark eluded for eighteen years. However, eventually in 1995, two groups, the
CDF (Collider Detector at Fermi lab) Collaboration (Abe et al. 1995) and the Dφ
Collaboration (Abachi et al. 1995) succeeded in detecting toponium tt in very high
energy pp collisions at Fermi Laboratory’s 1.8TeV Tevetron collider. The toponium tt
is the bound state of t and t. The mass of t has been estimated to be 176.0±2.0GeV/c2 ,
and thus it is the most massive elementary particle known so far. The t quark has spin
1/2 and charge 2/3 e.
Moreover, in order to account for the apparent breaking of the spin-statistics the-
orem in certain members of the Jp = 3
decuplet (spin 3/2,parity even), e. g. , △++
(uuu), and Ω− (sss), Greenberg (1964) postulated that quark of each flavour comes
in three colours, namely, red, green, and blue, and that real particles are always colour
singlets. This implies that real particles must contain quarks of all the three colours
or colour-anticolour combinations such that they are overall white or colourless. White
or colourless means all the three primary colours are equally mixed or there should
be a combination of a quark of a given colour and an antiquark of the corresponding
anticolour. This means each baryon contains quarks of all the three colours(but not
necessarily of the same flavour) whereas a meson contains a quark of a given colour and
an antiquark having the corresponding anticolour so that each combination is overall
white. Leptons have no colour. Of course, in this context the word ‘colour’ has noth-
ing to do with the actual visual colour, it is just a quantum number specifying a new
internal degree of freedom of a quark.
The concept of colour plays a fundamental role in accounting for the interaction
between quarks. The remarkable success of quantum electrodynamics (QED) in ex-
plaining the interaction between electric charges to an extremely high degree of preci-
sion motivated physicists to explore a similar theory for strong interaction. The result
is quantum chromodynamics (QCD), a non-Abelian gauge theory (Yang-Mills theory),
which closely parallels QED. Drawing analogy from electrodynamics, Nambu (1966)
postulated that the three quark colours are the charges (the Yang-Mills charges) re-
sponsible for the force between quarks just as electric charges are responsible for the
electromagnetic force between charged particles. The analogue of the rule that like
charges repel and unlike charges attract each other is the rule that like colours repel,
and colour and anticolour attract each other. Apart from this, there is another rule in
QCD which states that different colours attract if the quantum state is antisymmetric,
and repel if it is symmetric under exchange of quarks. An important consequence of
this is that if we take three possible pairs, red-green. green-blue, and blue-red, then a
third quark is attracted only if its colour is different and if the quantum state of the
resulting combination is antisymmetric under the exchange of a pair of quarks thus
resulting in red-green-blue baryons. Another consequence of this rule is that a fourth
quark is repelled by one quark of the same colour and attracted by two of different
colours in a baryon but only in antisymmetric combinations. This introduces a factor
of 1/2 in the attractive component and as such the overall force is zero, i.e., the fourth
quark is neither attracted nor repelled by a combination of red-green-blue quarks. In
spite of the fact that hadrons are overall colourless, they feel a residual strong force
due to their coloured constituents.
It was soon realized that if the three colours are to serve as the Yang-Mills charges,
each quark flavour must transform as a triplet of SUc(3) that causes transitions between
quarks of the same flavour but of different colours ( the SU(3) mentioned earlier causes
transitions between quarks of different flavours and hence may more appropriately be
denoted by SUf (3)). However, the SUc(3) Yang-Mills theory requires the introduction
of eight new spin 1 gauge bosons called gluons. Moreover, it is reasonable to stipulate
that the gluons couple to left-handed and right-handed quarks in the same manner
since the strong interactions do not violate the law of conservation of parity. Just as
the force between electric charges arise due to the exchange of a photon, a massless
vector (spin 1) boson, the force between coloured quarks arises due to the exchange
of a gluon. Gluons are also massless vector (spin 1) bosons. A quark may change its
colour by emitting a gluon. For example, a red quark qR may change to a blue quark
qB by emitting a gluon which may be thought to have taken away the red (R) colour
from the quark and given it the blue (B) colour, or, equivalently, the gluon may be
thought to have taken away the red (R) and the antiblue (B) colours from the quark.
Consequently, the gluon GRB emitted in the process qR → qB may be regarded as
the composite having the colour R B so that the emitted gluon GRB = qRqB . In
general, when a quark qi of colour i changes to a quark qj of colour j by emitting a
gluon Gij , then Gij is the composite state of qi and qj , i.e., Gij = qiqj . Since there are
three colours and threeanticolours, there are 3×3 = 9 possible combinations (gluons)of
the form Gij = qiqj . However, one of the nine combinations is a special combination
corresponding to the white colour, namely, GW = qRqR = qGqG = qBqB . But there is
no interaction between a coloured object and a white (colourless) object. Consequently,
gluon GW may be thought not to exist. This leads to the conclusion that only 9−1 = 8
kinds of gluons exist. This is a heuristic explanation of the fact that SUc(3) Yang-Mills
gauge theory requires the existence of eight gauge bosons, i.e., the gluons. Moreover, as
the gluons themselves carry colour, gluons may also emit gluons. Another important
consequence of gluons possessing colour is that several gluons may come together and
form gluonium or glue balls. Glueballs have integral spin and no colour and as such
they belong to the meson family.
Though the actual existence of quarks has been indirectly confirmed by experiments
that probe hardronic structure by means of electromagnetic and weak interactions, and
by the production of various quarkonia (qq) in high energy collisions made possible by
various particle accelerators, no free quark has been detected in experiments at these
accelerators so far. This fact has been attributed to the infrared slavery of quarks, i.e.,
to the nature of the interaction between quarks responsible for their confinement inside
hadrons. Perhaps enormous amount of energy , much more than what is available in the
existing terrestrial accelerators, is required to liberate the quarks from confinement.
This means the force of attraction between quarks increases with increase in their
separation. This is reminiscent of the force between two bodies connected by an elastic
string.
On the contrary, the results of deep inelastic scattering experiments reveal an al-
together different feature of the interaction between quarks. If one examines quarks at
very short distances (< 10−13 cm ) by observing the scattering of a nonhadronic probe,
e.g., an electron or a neutrino, one finds that quarks move almost freely inside baryons
and mesons as though they are not bound at all. This phenomenon is called the asymp-
totic freedom of quarks. In fact Gross and Wilczek (1973 a,b) and Politzer (1973) have
shown that the running coupling constant of interaction between two quarks vanishes
in the limit of infinite momentum (or equivalently in the limit of zero separation).
5 Eventually what happens to matter in a collapsing black hole?
As mentioned in Section 3 the energy E of the particles comprising the matter in a
collapsing black hole continually increases and so does the density ρ of the matter
whereas the separation s between any pair of particles decreases. During the continual
collapse of the black hole a stage will be reached when E and ρ will be so large and
s so small that the quarks confined in the hadrons will be liberated from the infrared
slavery and will enjoy asymptotic freedom, i.e., the quark deconfinement will occur. In
fact, it has been shown that when the energy E of the particle ∼ 102 GeV (s ∼ 10−16
cm) corresponding to a temperature T ∼ 1015K all interactions are of the Yang-Mills
type with SUc(3)×SUIW (2)×UYW (1) gauge symmetry, where c stands for colour, IW
for weak isospin, and YW for weak hypercharge, and at this stage quark deconfinement
occurs as a result of which matter now consists of its fundamental constituents : spin 1/2
leptons, namely, the electrons, the muons, the tau leptons, and their neutrinos, which
interact only through the electroweak interaction(i.e., the unified electromagnetic and
weak interactions); and the spin 1/2 quarks, u, d, s, c, b, t, which interact eletroweakly
as well as through the colour force generated by gluons(Ramond, 1983). In other words,
when E ≥ 102 GeV (s ≤ 10−16 cm) corresponding to T ≥ 1015K, the entire matter
in the collapsing black hole will be in the form of qurak-gluon plasma permeated by
leptons as suggested by the author earlier (Thakur 1993).
Incidentally, it may be mentioned that efforts are being made to create quark-gluon
plasma in terrestrial laboratories. A report released by CERN, the European Organi-
zation for Nuclear Research, at Geneva, on February 10, 2000, said that by smashing
together lead ions at CERN’s accelerator at temperatures 100,000 times as hot as the
Sun’s centre, i.e., at T ∼ 1.5 × 1012K, and energy densities never before reached in
laboratory experiments, a team of 350 scientists from institutes in 20 countries suc-
ceeded in isolating tiny components called quarks from more complex particles such
as protons and neutrons. “A series of experiments using CERN’s lead beam have pre-
sented compelling evidence for the existence of a new state of matter 20 times denser
than nuclear matter, in which quarks instead of being bound up into more complex
particles such as protons and neutrons, are liberated to roam freely ” the report said.
However, the evidence of the creation of quark gluon plasma at CERN is indirect,
involving detection of particles produced when the quark-gluon plasma changes back
to hadrons. The production of these particles can be explained alternatively without
having to have quark-gluon plasma. Therefore, Ulrich Heinz at CERN is of the opinion
that the evidence of the creation of quark-gluon plasma at CERN is not enough and
conclusive. In view of this, CERN will start a new experiment, ALICE, soon (around
2007-2008) at its Large Hadron Collider (LHC) in order to definitively and conclusively
creat QGP.
In the meantime the focus of research on quark-gluon plasma has shifted to the
Relativistic Heavy Ion Collider (RHIC), the worlds newest and largest particle accel-
erator for nuclear research, at Brookhaven National Laboratory in Upton, New York.
RHIC’s goal is to create and study quark-gluon plasma. RHIC’s aim is to create quark-
gluon plasma by head-on collisions of two beams of gold ions at energies 10 times those
of CERN’s programme, which ought to produce a quark-gluon plasma with higher
temperature and longer lifetime thereby allowing much clearer and direct observation.
RHIC’s quark-gluon plasma is expected to be well above the transition temperature
for transition between the ordinary hadronic matter phase and the quark-gluon plasma
phase. This will enable scientists to perform numerous advanced experiments in order
to study the properties of the plasma. The programme at RHIC began in the summer
of 2000 and after two years Thomas Kirk, Brookhaven’s Associate Laboratory Director
for High Energy Nuclear Physics, remarked, “It is too early to say that we have dis-
covered the quark-gulon plasma, but not too early to mark the tantalizing hints of its
existence.” Other definitive evidence of quark-gluon plasma will come from experimen-
tal comparisons of the behavior in hot, dense nuclear matter with that in cold nuclear
matter. In order to accomplish this, the next round of experimental measurements at
RHIC will involve collisions between heavy ions and light ions, namely, between gold
nuclei and deuterons.
Later, on June 18, 2003 a special scientific colloquium was held at Brcokhaven
Natioal Laboratory (BNL) to discuss the latest findings at RHIC. At the colloquium,
it was announced that in the detector system known as STAR ( Solenoidal Tracker AT
RHIC ) head-on collision between two beams of gold nuclei of energies of 130 GeV per
nuclei resulted in the phenomenon called “jet quenching“. STAR as well as three other
experiments at RHIC viz., PHENIX, BRAHMS, and PHOBOS, detected suppression
of “leading particles“, highly energetic individual particles that emerge from nuclear
fireballs, in gold-gold collisions. Jet quenching and leading particle suppression are
signs of QGP formation. The findings of the STAR experiment were presented at the
BNL colloquium by Berkeley Laboratory’s NSD ( Nuclear Science Division ) physicist
Peter Jacobs.
6 Collapse of a black hole to a space-time singularity is inhibited by
Pauli’s exclusion principle
As quarks and leptons in the quark-gluon plasma permeated by leptons into which
the entire matter in a collapsing black hole is eventually converted are fermions, the
collapse of a black hole to a space-time singularity in a finite time in a comoving co-
ordinate system, as stipulated by the singularity theorems of Penrose, Hawking and
Geroch, is inhibited by Pauli’s exclusion principle. For, if a black hole collapses to
a point in 3-space, all the quarks of a given flavour and colour would be squeezed
into just two quantum states available at that point, one for spin up and the other
for spin down quark of that flavour and colour. This would violate Pauli’s exclusion
principle according to which not more than one fermion of a given species can occupy
any quantum state. So would be the case with quarks of each distinct combination of
colour and flavour as well as with leptons of each species, namely, e, µ, τ, νe, νµ and ντ .
Consequently, a black hole cannot collapse to a space-time singularity in contravention
to Pauli’s exclusion principle. Then the question arises : If a black hole does not collapse
to a space-time singularity, what is its ultimate fate? In section 7 three possibilities
have been suggested.
7 Ultimately how does a black hole end up?
The pressure P inside a black hole is given by
P = Pr +
Pij +
P ij +
P k (11)
where Pr is the radiation pressure, Pij the pressure of the relativistically degenerate
quarks of the ith flavour and jth colour, Pk the pressure of the relativistically degenerate
leptons of the kth species, P ij the pressure of relativistically degenerate antiquarks of
the ith flavour and jth colour, Pk that of the relativistically degenerate antileptons of
the kth species. In equation (11) the summations over i and j extend over all the six
flavours and the three colours of quarks, and that over k extend over all the six species
of leptons. However, calculation of these pressures are prohibitively difficult for several
reasons. For example, the standard methods of statistical mechanics for calculation of
pressure and equation of state are applicable when the system is in thermodynamics
equilibrium and when its volume is very large, so large that for practical purpose
we may treat it as infinite. Obviously, in a gravitationally collapsing black hole, the
photon, quark and lepton gases cannot be in thermodynamic equilibrium nor can their
volume be treated as infinite. Moreover, at ultrahigh energies and densities, because
of the SUIW (2) gauge symmetry, transitions between the upper and lower components
of quark and lepton doublets occur very frequently. In addition to this, because of the
SUf (3) and SUc(3) gauge symmetries transitions between quarks of different flavours
and colours also occur. Furthermore, pair production and pair annihilation of quarks
and leptons create additional complications. Apart from these, various other nuclear
reactions may as well occur. Consequently, it is practically impossible to determine the
number density and hence the contribution to the overall pressure P inside the black
hole by any species of elementary particle in a collapsing black hole when E ≥ 102 Gev
(s ≤ 10−16 cm), or equivalently, T ≥ 1015K. However, it may not be unreasonable
to assume that, during the gravitational collapse, the pressure P inside a black hole
increases monotonically with the increase in the density of matter ρ. Actually, it might
be given by the polytrope, P = kρ
(n+1)
n , where K is a constant and n is polytropic
index. Consequently, P → ∞ as ρ → ∞, i.e., P → ∞ as the scale factor R(t) → 0 (or
equivalently s→ 0). In view of this, there are three possible ways in which a black hole
may end up.
1. During the gravitational collapse of a black hole, at a certain stage, the pressure
P may be enough to withstand the gravitational force and the object may become
gravitationally stable. Since at this stage the object consists entirely of quark-gluon
plasma permeated by leptons, it means it would end up as a stable quark star. Indeed,
such a possibility seems to exist. Recently, two teams - one led by David Helfand of
Columbia University, NewYork (Slane, Helfand, and Murray 2002) and another led
by Jeremy Drake of Harvard-Smithsonian Centre for Astrophysics, Cambridge, Mass.
USA (Drake et al. 2002) studied independently two objects, 3C58 in Cassiopeia, and
RXJ1856.5-3754 in Corona Australis respectively by combining data from the NASA’s
Chandra X-ray Observatory and the Hubble Space Telescope, that seemed, at first, to
be neutron stars, but, on closer look, each of these objects showed evidence of being
an even smaller and denser object, possibly a quark star.
2. Since the collapse of a black hole is inhibited by Pauli’s exclusion principle, it can
collapse only upto a certain minimum radius, say, rmin. After this, because of the
tremendous amount of kinetic energy, it would bounce back and expand, but only
upto the event horizon, i.e., upto the gravitational (Schwarzschild ) radius rg since,
according to the GTR, it cannot cross the event horizon. Thereafter it would collapse
again upto the radius rmin and then bounce back upto the radius rg . This process of
collapse upto the radius rmin and bounce upto the radius rg would occur repeatedly. In
other words, the black hole would continually pulsate radially between the radii rmin
and rg and thus become a pulsating quark star. However, this pulsation would cause
periodic variations in the gravitational field outside the event horizon and thus produce
gravitational waves which would propagate radially outwards in all directions from just
outside the event horizon. In this way the pulsating quark star would act as a source
of gravitational waves. The pulsation may take a very long time to damp out since the
energy of the quark star (black hole) cannot escape outside the event horizon except
via the gravitational radiation produced outside the event horizon. However, gluons in
the quark-gluon plasma may also act as a damping agent. In the absence of damping,
which is quite unlikely, the black hole would end up as a perpetually pulsating quark
star.
3. The third possibility is that eventually a black hole may explode; amini bang of a sort
may occur, and it may, after the explosion, expand beyond the event horizon though it
has been emphasized by Zeldovich and Novikov (1971) that after a collapsing sphere’s
radius decreases to r < rg in a finite proper time, its expansion into the external space
from which the contraction originated is impossible, even if the passage of matter
through infinite density is assumed.
Notwithstanding Zeldovich and Novikov’s contention based on the very concept
of event horizon, a gravitationally collapsing black hole may also explode by the very
same mechanism by which the big bang occurred, if indeed it did occur. This can be
seen as follows. At the present epoch the volume of the universe is ∼ 1.5 × 1085 cm3
and the density of the galactic material throughout the universe is ∼ 2×10−31 g cm−3
(Allen 1973). Hence, a conservative estimate of the mass of the universe is ∼ 1.5 ×
1085 × 2× 10−31 g = 3× 1054 g. However, according to the big bang model, before the
big bang, the entire matter in the universe was contained in an ylem which occupied
very very small volume. The gravitational radius of the ylem of mass 3 × 1054g was
4.45× 1021 km (it must have been larger if the actual mass of the universe were taken
into account which is greater than 3× 1054 g). Obviously, the radius of the ylem was
many orders of magnitude smaller than its gravitational radius, and yet the ylem
exploded with a big bang, and in due course of time crossed the event horizon and
expanded beyond it upto the present Hubble distance c/H0 ∼ 1.5 × 1023 km where
c is the speed of light in vacuum and H0 the Hubble constant at the present epoch.
Consequently, if the ylem could explode in spite of Zeldovich and Novikov’s contention,
a gravitationally collapsing black hole can also explode, and in due course of time
expand beyond the event horizon. The origin of the big bang, i.e., the mechanism by
which the ylem exploded, is not definitively known. However, the author has, earlier
proposed a viable mechanism (Thakur 1992) based on supersymmetry/supergravity.
But supersymmetry/supergravity have not yet been validated experimentally.
8 Conclusion
From the foregoing three inferences may be drawn. One, eventually the entire matter
in a collapsing black hole is converted into quark-gluon plasma permeated by leptons.
Two, the collapse of a black hole to a space - time singularity is inhibited by Pauli’s
exclusion principle. Three, ultimately a black hole may end up in one of the three
possible ways suggested in section 7.
Acknowledgements The author thanks Professor S. K. Pandey, Co-ordinator, IUCAA Ref-
erence Centre, School of Studies in Physics, Pt. Ravishankar Shukla University, Raipur, for
making available the facilities of the Centre. He also thanks Sudhanshu Barway, Mousumi Das
for typing the manuscript.
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	Introduction
	A critique of the singularity theorems
	Gravitationally collapsing black hole as a particle accelerator
	Quarks: The building blocks of matter
	Eventually what happens to matter in a collapsing black hole?
	Collapse of a black hole to a space-time singularity is inhibited by Pauli's exclusion principle
	Ultimately how does a black hole end up?
	Conclusion
