Journal of Theoretics

   Quantization of Torsion Effects in a De Sitter Universe

With or Without a Black-Hole:

Using the Zeldovich Definition of the Cosmological Constant

 

Author: Ioannis Iraklis Haranas

York University
Department of Physics and Astronomy
128 Petrie Science Building
4700 Keele Street North
Toronto, Ontario, Canada M3J-1P3

YHaranas@yahoo.com

                                     

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 s2 = ±1 / 9 , where in the case of the black hole being present in the metric, s2 = ± 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 MH  disappears via an explosion burst which can last for 10-43 sec, when it reaches a mass of the order of Planck mass mP = .

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 effects2 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  = r3, 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)  = gij 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 = mPlanck = as before, we obtain for anti-parallel spin:  

                                                 (10)

and for parallel respectively:

                .                                   (11)  

For a mass m = mPlanck and radius r = rPlanck , 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 = mPlanck, and r = rPlanck ,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 g44 we have:  

            .                                                                  (20)

Now, writing s  = sr3, 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 = mPlanck 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  = 1066 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 s2 = ±1 / 9 , where in the case of the black hole being present in the metric, s2 = ± 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.

3. G. W. Gibbons, S. W. Hawing, S.T. C. Siklos, The Very Early Universe, Cambridge University Press, 1985, p:459-463.

4. N. D. Birrel: Quantum Fields in Curved Space, p:225.

5. Ya. B. Zeldovich: Soviet Physics Uspeki vol. 11, 1968, p: 381-393.

6. Sivaran, V. De Sabbata, Astrophysics and Space Science, 176, 145-148, 1991.

 

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