# Discreteness of Curved Spacetime from GUP

Authors: Ahmad Adel Abutaleb
Journal: Advances in High Energy Physics
 (2013)

---

## 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.


---

## Body

## 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 [xi,xj]=0=[pi,pj]:
(1)[xi·pj]=ih[δij-a(pδij+pipjp)+a2(p2δij+3pipj)],ΔxΔp≥h2[1-2a〈p〉+4a2〈p2〉]≥h2[1+(ap2+4a2)Δp200000+4a2〈p〉2-2a〈p2〉(ap2+4a2)],
where a=a0/Mplc=a0Lpl/h, Mpl = Planck mass, Lpl≃10-35 m = Planck length, and Mplc2 = Planck energy ≈1019 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]:
(2)Δx≥(Δx)min≈a0Lpl,Δp≤(Δp)max≈Mplca0.
It is natural to take a0=1; for more details see [17].The following definitions are proposed in [1] and used in [1, 17]:(3)xi=x0i,pi=p0i(1-ap0+2a2p02)
with x0i and p0j satisfying the canonical commutation relations [x0i,p0j]=ihδij, such that p0i=-ih(∂/∂x0i), p02=∑j=13p0jp0j.In [1], it was shown that any nonrelativistic Hamiltonian of the form H=p2/2m+V(r⃑) can be written as H=p02/2m-(a/m)p03+V(r⃑)+O(a2) 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]:
(4)(c(α⃑·p⃑)+βmc2)ψ=Eζψ,
where m is the rest mass of the particle, ψ is the Dirac spinor, α⃑ and β are Dirac matrices, p⃑≡(pr,pθ,pφ) are momentum operators, ζ=1-rs/r, rs is the Schwarzschild radius of massive body, related to its mass M by rs=2GM/C2, G is the gravitational constant, and c is the speed of light in free space. Using ψ=(χ1χ2), (4) can be written as
(5)c(α⃑·p⃑)χ2+mc2χ1=Eζχ1,c(α⃑·p⃑)χ1-mc2χ2=Eζχ2.
Now, using GUP correction (3) and (5) take the form
(6)c(α⃑·p0⃑)χ2+mc2χ1-cap02χ1=Eζχ1,c(α⃑·p0⃑)χ1-mc2χ2-cap02χ2=Eζχ2,
where
(7)p02=-h2[ζr2∂∂r(r2ζ∂∂r)+1r2sinθ000000×∂∂θ(sinθ∂∂θ)+1r2sin2θ∂2∂φ2ζr2].
We study Dirac equations in (6) in Schwarzschild metric with spherical cavity with radius R defined by the potential
(8)U(r)=0,r≤R,U(r)=U0⟶∞,r>R,
so, we can write the corrected GUP Dirac equations with spherical cavity defined by (8) in Schwarzschild metric as
(9)c(α⃑·p0⃑)χ2+(mc2+U)χ1-cap02χ1=Eζχ1,c(α⃑·p0⃑)χ1-(mc2+U)χ2-cap02χ2=Eζχ2.
Notice that, when a=0, ζ=1, equations in (9) are usual Dirac equations in flat spacetime. When a≠0, ζ=1, (9) are Dirac equations with GUP in flat spacetime proposed in [17]. When a=0, ζ≠1, 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(10)ψ=(χ1χ2)=(gk(r)γjlj3(r^)ifk(r)γjl-j3(r^)),(11)γjlj3(r^)=(l12j3-1212∣jj3)×Ylj3-(1/2)(r^)(10)+(l12j3+12-12∣jj3)×Ylj3+(1/2)(r^)(01),
where Ylj3-(1/2)(r^) and Ylj3+(1/2)(r^) are spherical harmonics and (j1j2m1m2∣jj3) are Clebsh-Gordon coefficients, χ1, χ2 are eigenstates of L2 (L⃑is the angular momentum operator) with eigenvalues h2l(l+1) and h2l-(l-+1), respectively, such that the following hold:if(12)k=j+12>0,then(13)l=k=j+12,l-=k-1=j-12,and if(14)k=-(j+12)<0,then(15)l=-(k+1)=j-12,l-=-k=j+12.We use (α⃑·A⃑)(α⃑·B⃑)=A⃑·B⃑+iα⃑·(A⃑×B⃑) and the related identity (α⃑·r⃑)(α⃑·r⃑)=r2; so we have
(16)(α⃑·p0⃑)=(α⃑·r⃑)(α⃑·r⃑)(α⃑·p0⃑)r2=α⃑·r⃑r2(r⃑·p0⃑+iα⃑·(r⃑×p0⃑))=α⃑·r⃑r2(r⃑·p0⃑+iα⃑·L⃑).
But from the definition of the momentum operators in Schwarzschild metric [18], we can write (16) as
(17)(α⃑·p0⃑)=(α⃑·r^)(-ihζ∂∂r+irα⃑·L⃑).
Also, we have
(18)(α⃑·L⃑+1)χ1,2=∓χ1,2,(α⃑·r^)γjlj3(r^)=-γjl-j3(r^),(α⃑·r^)γjl-j3(r^)=-γjlj3(r^).
Next, from the definition of p02 in Schwarzschild metric [18] we can write
(19)p02F(r)Ylm=h2[-ζr2ddr(r2ζ∂∂r)+l(l+1)r2]F(r)Ylm.
So, using (17), (18), and (19), we can obtain from (9) the following equations:
(20)-chζdfkdr+c(k-1)rfk+(mc2+U)gk+cah2[ζr2ddr(ζr2dgkdr)-l(l+1)r2gk]=Eζgk,chζdgkdr+c(k+1)rgk-(mc2+U)fk+cah2[ζr2ddr(ζr2dfkdr)-l-(l-+1)r2fk]=Eζfk.
It can be shown that MIT bag boundary condition (at r=R) is equivalent to [19, 20]
(21)ψ~ψ=0.
As in [17], we can expect new nonperturbative solutions of the forms fk=FK(r)eiϵr/ah and gk=GK(r)eiϵr/ah (where ϵ=O(1)) for which (20) simplifies to
(22)ahd2gkdr2=ζdfkdr,ahd2fkdr2=-ζdgkdr,
where we have dropped terms which are ignorable for small a.Whenζ=1, equations in (22) are identical to the (60)-(61) in reference [17], and in this case we have the following solutions: fkN=iNeir/ah, gkN=Neir/ah, N is constant; so we can assume the solutions of (22) as
(23)fkN=iNeir/ah,gkN=NB(r)eir/ah.
By applying (23) on (22), we find that
(24)a2h2d2B(r)dr2+B(r)+ζ-2ζ=0.
Consider that rs is very small, so we can approximate (24) to
(25)a2h2d2B(r)dr2+B(r)-1-3rs2r=0.
The solution of (25) takes the form
(26)B(r)=c1sin(rah)+c2cos(rah)+1.5rsah[Ci(rah)sin(rah)00000000-Si(rah)cos(rah)]+1,
where c1 and c2 are constants,  Si(r/ah) and Ci(r/ah) are the sine integral function and cosine integral function defined as Si(y)=∫0y(sint/t)dt and Ci(y)=-∫y∞(cost/t)dt. For more details about this functions see [22, 23].Therefore, the solutions of (22) take the form
(27)fkN=iNeir/ah,gkN=N[1+σ(r)]eir/ah,
where
(28)σ(r)=c1sin(rah)+c2cos(rah)+1.5rsah[Ci(rah)sin(rah)0000000000-Si(rah)cos(rah)].
Here, one must have limrs→oc1=limrs→oc2=0, and, in this case (σ(r)=0, rs=0), the results are the same of [17].Now, the boundary condition (21) gives
(29)|gk(R)+gkN(R)|2=|fk(R)+fkN(R)|2,
which to O(a) translates to
(30)(gk2-fk2)+2N[gk(1+σ(R))cos(Rah)-fksin(Rah)]+N2[(1+σ(R))2-1]=0.
From (30), we have
(31)fk=gk,(32)tan(Rah)=1+σ(R),σ(R)=0,orσ(R)=-2.
Observing (28), we can write
(33)σ(R)cos(R/ah)=c1tan(Rah)+c2+1.5rsah[Ci(Rah)tan(Rah)-Si(Rah)].From (32) we have
(34)ifσ(R)=0thentan(Rah)=1,ifσ(R)=-2thentan(Rah)=-1.For the case ofσ(R)=0 and rs=0 we have the same result of discreteness of space in flat spacetime [17].For the second caseσ(R)=-2, we have, from (33),
(35)4ah3rscos(R/ah)+2Cah3rs=Ci(Rah)+Si(Rah),00000000000000000000000000Cisconstant.
If we take the constant  C=0, then we have
(36)4ah3rscos(R/ah)=Ci(Rah)+Si(Rah).Suppose that we choosers=(4/3)ah (very small as assumption); then we have
(37)1cos(R/ah)=Ci(Rah)+Si(Rah).Equation (37) has infinite number of solutions; we write some numerical values of it, (R/ah)=5,565, (R/ah)=7.159, (R/ah)=11, and 755, (R/ah)=13.448. So, the radius of cavity R has been quantized in terms of ah 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.



---

*Source: 124543-2013-12-25.xml*