ABSTRACT

This
article makes use of the Zeldovich definition of the cosmological constant in a De-Sitter
universe in order to quantize the torsion effects with or without a black
hole.
For the De-Sitter metric without a black hole, we found that
the spin has the values s^{2} = ±1
/ 9 _{
}
, where in the case of the black hole being present in the metric,
s^{2}
= ±
2 / 9 _{
}
, ±4
/ 9 _{
}
. Furthermore, and for
Planck mass black holes, we found a connection of the spin effects to
the maximum curvature of the universe, when the quantum effects were
important, and again with the help of the Zeldovich’s definition
of the cosmological constant, which is another result of the quantum
field theory.

KEYWORDS: torsion effect, De Sitter
universe, Zeldovich, cosmological constant, theory.

INTRODUCTION
__
____
__

A
natural way to talk about spin effects in gravitation is through
torsion. Its
introduction becomes significant for the understanding of the last
stage in black hole evaporation.
It could be the case that an evaporating black hole of mass M_{H} disappears via an explosion burst which can last for 10^{-43} sec, when it reaches a mass
of the order of Planck mass m_{P} = _{
}
.

If
it happens, then there might be three distinct possibilities for the fate
of the evaporating black hole:^{1}

(i)
The black hole may evaporate completely leaving no residue,
in which case it would give rise to a serious problem of quantum
consistency.

(ii)
If the final state of evaporation leaves a naked singularity
behind, then it might violate cosmic censorship at the quantum
level.

(iii)
If a stable remnant of residue of appropriately the Planck
mass remains, then the emission process might stop.

**
****
**

THEORY
AND CALCULATIONS

To
understand spin effects in gravitation we can use torsion.
Therefore, let us first write the two modified De Sitter
metrics including torsion effects^{2} and a De Sitter metric with
torsion also containing a black hole:^{3}

_{
}
(1)

and
also:

(2)

where
is the cosmological constant, and s is the
torsion, which could also be written as

s = r^{3}, where is the spin density, and M is the mass of the
black hole. The surface
gravity of a black hole in general is given by:^{4}

_{
}
,
(3)

where: F(r) = g_{ij}
are the metric element functions.
We now are going to work with the first De Sitter metric
elements:

_{
}
(4)

and
also with:

_{
}
.
(5)

The
above relationships would imply that the spin s _{
}
, and is parallel or
anti-parallel to gravitation. For
such a black hole, the surface gravity and hence the temperature,
vanishes. Therefore, in
this case, the torsion effects entering with the same or opposite
sign to gravitation cancel those of gravity.

Now let us consider the metric element of the De-Sitter
universe with one black hole in it.
Going through the same analysis, we obtain the two expressions
below: the first for torsion anti-parallel to gravity, and the
second for torsion parallel to gravity.

For
anti-parallel spin:

_{
}
(6)

and
for parallel spin:

_{
}
(7)

which
now give that s is given by:

_{
}
(8)

_{
}
(9)

Upon
substitution for r = r_{ plaanck}=_{}
and M = m_{Planck} = _{
}as before, we obtain for anti-parallel spin:

_{
}
(10)

and
for parallel respectively:

_{
} .
(11)

For
a mass m = m_{Planck} and radius r = r_{Planck} ,
zero
surface gravity would correspond to an expression given by (3).
Therefore, for the De Sitter metric without the black hole,
we
obtain from the anti-parallel torsion to gravitation:

_{
}
(12)

and
for the parallel one:

_{
} .
(13)

Therefore,
the corresponding spin expression for torsion effects, in the same
direction and also opposite to gravity, are:

_{
}
(14)

where
here the negative sign is for anti-parallel spin and the positive
for parallel.

Next,
let us write the cosmological constant in-terms of the Zeldovich
definition,^{5 }which arises in the theory of quantum vacuum.
Then we have that:

_{
} .
(15)

If
we substitute it into (14), along with m = m_{Planck}, and r
= r_{Planck} ,we then obtain

_{
}
(16)

which,
simplifies to the following equation:

_{
} .
(17)

Similarly
working out the details in (10) and (11),
we have the following
values of s for parallel and anti-parallel spin:

_{
}
(18)

and:

_{
} .
(19)

From
the equations above we can see that s is quantized in units of _{
}
,
and corresponding to
torsion effects parallel and anti-parallel to gravitation.

Next,
if we choose to work with the same metric element as before along
with Zeldovich’s definition of the cosmological constant, we can
easily obtain an expression for the spin density effects of the
black hole in terms of the maximum curvature of the universe. Starting with the metric element g_{44} we have:

_{
}
.
(20)

Now,
writing s = sr^{3},
the metric element above becomes:

_{
}
.
(21)

Next,
calculating the expression for the surface gravity for (21), we
have:

_{
} .
(22)

Solving
for s, we further obtain:

_{
} . (23)

Next,
substituting for = _{Z}
and m = m_{Planck} as before, we obtain the following
expression:

_{
}
(24)

since
quantum effects on the geometry of space-time would imply a maximum
curvature, and it is shown that _{
}
= 10^{66} cm^{-2}
.^{6}

Now,
taking
into account the combinations of the signs for opposite and same
with gravity effects, we obtain all the separate spin density cases
in terms of _{max}:

_{
}
(25)

and
also:

_{
} .
(26)

Looking
at the above expressions, we can also see how the spin density of a
quantum black hole can be related to the maximum curvature of the
space-time, or, in another way, that the maximum curvature of the
space-time, where quantum effects are
important, can be expressed in
terms of the spin density of the quantum black holes which are
present. This also
means that:

_{
}
(27)

or:

_{
}
(28)

CONCLUSIONS

Making
use of the Zeldovich definition of the cosmological constant in a De-Sitter
universe, where torsion effects were present in the space-time
metric, and also in the same metric including a black-hole,
we could
obtain expressions for the torsion and show that in both cases this
torsion is quantized in units of _{
}
. We can also see that
in a
De-Sitter
universe without a black hole, the torsion effects are quantized in
less units of _{
}
. In the case of the
addition of a black hole in the metric, the torsion effects increase
in units of _{
}
, and for the same De-Sitter metric. For the De-Sitter metric without a black hole, we found that
the spin has the values s^{2} = ±1
/ 9 _{
}
, where in the case of the black hole being present in the metric,
s^{2}
= ±
2 / 9 _{
}
, ±4
/ 9 _{
}
. Furthermore, and for
Planck mass black holes, we found a connection of the spin effects to
the maximum curvature of the universe, when the quantum effects were
important, and again with the help of the Zeldovich’s definition
of the cosmological constant, which is another result of the quantum
field theory.

REFERENCES

1.
V. De Sabbata, D. Wang Annalen der Physik 7, Band 47, Heft 6, 1990,
S. 508-510.

2.
Venzo de Sabbata and Z. Zang: Black Hole Physics p: 225-282, Kluwer
Academic Publishers, 1992.