Discreteness Of Curved Space-Time From GUP

Diverse theories of Quantum Gravity expect modification of the Heisenberg Uncertainty Principle near the Planck scale to a so-called Generalized Uncertainty Principle.It was shown by some authors that the Generalized uncertainty principle gives rise to corrections to Schrodinger,Klien-Gordon and Dirac equations.By solving the Generalized uncertainty principle corrected equations,the authors arrived at quantization not only of energy but also of box length,area and volume. We extend the above result to the case of curved Space-Time(Schwarzschild metric). We showed that we arrived at the Quantization of Space by solving Dirac equation with Generalized uncertainty principle in this metric.

In [1] it was shown that any non-relativistic 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.

GUP Dirac equations in Schwarzschild metric
Dirac equation in Schwarzschild metric without (GUP) can be written as follow [18] ⃑ ⃑ √ (4) 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, is the speed of light in free space. using ( ) , (4) can be written as Now, using GUP correction (3) and (5) where, We study Dirac equations in (6) in Schwarzschild metric with spherical cavity with radius R 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 space-time. When equations (9) are Dirac equations with GUP in flat space-time proposed in [19]. 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 (12) then (13) and if (14) 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 equations (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 form where we have dropped terms which are ignorable for small .

When
, equations in (22) are identical to the equations (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 equations (23) on (22), we find that Consider that is very small, so we can approximate equation (24) to .
The solution of (25) take the form where, are constants, ⁄ ⁄ are the sine integral function and cosine integral function defined as 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]. has been quantized in terms of and we again arrived at the quantization of space in Schwarzschild-like metric.

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.