Advances in High Energy Physics

Volume 2013 (2013), Article ID 124543, 4 pages

http://dx.doi.org/10.1155/2013/124543

Research Article

Discreteness of Curved Spacetime from GUP

Ahmad Adel Abutaleb

Department of Mathematics, Faculty of Science, University of Mansoura, Elmansoura 35516, Egypt

Received 9 October 2013; Revised 13 December 2013; Accepted 16 December 2013

Academic Editor: Elias C. Vagenas

Copyright © 2013 Ahmad Adel Abutaleb. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Diverse theories of quantum gravity expect modifications of the Heisenberg's uncertainty principle near the Planck scale to a so-called Generalized uncertainty principle (GUP). It was shown by some authors that the GUP gives rise to corrections to the Schrodinger , Klein-Gordon, and Dirac equations. By solving the GUP corrected equations, the authors arrived at quantization not only of energy but also of box length, area, and volume. In this paper, we extend the above results to the case of curved spacetime (Schwarzschild metric). We showed that we arrived at the quantization of space by solving Dirac equation with GUP in this metric.

1. Introduction

Diverse approaches to quantum gravity expect a minimum measurable length and a modification of the Heisenberg uncertainty principle to a so-called generalized uncertainty principle or GUP. This implies a modification of the commutation relations between position coordinates and momentum. In [1], the following proposed GUP is consistent with Doubly special relativity or DSR theories and black hole physics which ensure that :

where , = Planck mass,  m = Planck length, and = Planck energy  GeV. GUP-induced terms become important near the Planck scale (for earlier version of GUP motivated by string theory, black hole physics, and DSR, see [2–14], and for some phenomenological implications, see [1, 15, 16]).

Equation (1) implies the following minimum measurable length and maximum measurable momentum [1, 17]:

It is natural to take ; for more details see [17].

The following definitions are proposed in [1] and used in [1, 17]:

with and satisfying the canonical commutation relations , such that , .

In [1], it was shown that any nonrelativistic Hamiltonian of the form can be written as using (3). This corrected Hamiltonian implies not only the usual quantization of energy, but also that the box length is quantized. In [17], the above results were extended to a relativistic particle in two and three dimensions. In this paper we study Dirac equation in Schwarzschild metric using GUP and show that we arrive at the quantization of space.

2. GUP Dirac Equations in Schwarzschild Metric

Dirac equation in Schwarzschild metric without (GUP) can be written as follows [18]:

where is the rest mass of the particle, is the Dirac spinor, and are Dirac matrices, are momentum operators, , is the Schwarzschild radius of massive body, related to its mass by , is the gravitational constant, and is the speed of light in free space. Using , (4) can be written as

Now, using GUP correction (3) and (5) take the form

where

We study Dirac equations in (6) in Schwarzschild metric with spherical cavity with radius defined by the potential

so, we can write the corrected GUP Dirac equations with spherical cavity defined by (8) in Schwarzschild metric as

Notice that, when , , equations in (9) are usual Dirac equations in flat spacetime. When , , (9) are Dirac equations with GUP in flat spacetime proposed in [17]. When , , equations in (9) are Dirac equations in Schwarzschild metric without GUP defined in [18]. We follow the analysis of [17, 19] and related references [20, 21].

We assume the form of Dirac spinor as

where and are spherical harmonics and are Clebsh-Gordon coefficients, , are eigenstates of (is the angular momentum operator) with eigenvalues and , respectively, such that the following hold:

 if

 then

 and if

 then

We use and the related identity ; so we have

But from the definition of the momentum operators in Schwarzschild metric [18], we can write (16) as

Also, we have

Next, from the definition of in Schwarzschild metric [18] we can write

So, using (17), (18), and (19), we can obtain from (9) the following equations:

It can be shown that MIT bag boundary condition (at ) is equivalent to [19, 20]

As in [17], we can expect new nonperturbative solutions of the forms and (where ) for which (20) simplifies to

where we have dropped terms which are ignorable for small .

When , equations in (22) are identical to the (60)-(61) in reference [17], and in this case we have the following solutions: , , is constant; so we can assume the solutions of (22) as

By applying (23) on (22), we find that

Consider that is very small, so we can approximate (24) to

The solution of (25) takes the form

where and are constants, and are the sine integral function and cosine integral function defined as and . For more details about this functions see [22, 23].

Therefore, the solutions of (22) take the form

where

Here, one must have , and, in this case (, ), the results are the same of [17].

Now, the boundary condition (21) gives

which to translates to

From (30), we have

Observing (28), we can write

From (32) we have

For the case of and we have the same result of discreteness of space in flat spacetime [17].

For the second case , we have, from (33),

If we take the constant , then we have

Suppose that we choose (very small as assumption); then we have

Equation (37) has infinite number of solutions; we write some numerical values of it, , , , and 755, . So, the radius of cavity has been quantized in terms of and we again arrived at the quantization of space in Schwarzschild-like metric.

3. Conclusion

Dirac equations with GUP in Schwarzschild metric have been studied. We showed that the assumption of existence of a minimum measurable length and a corresponding modification of uncertainty principle yields discreteness of space in this metric. But the question now arises, what is the guarantee that this result will continue to hold for more generic curved spacetimes? We expect that this discreteness will always appear provided that generalized uncertainty principle enters into the theory, but in fact we have no mathematical proof of existence of such discreteness of space if we work on the general metric of the general relativity theory.

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