Comment to: Corrections to the fine structure constant in the spacetime of a cosmic string from the generalized uncertainty principle

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by  Geusa de A. Marques

Comment to: Corrections to the fine structure constant in the spacetime of a cosmic
string from the generalized uncertainty principle
∗
Geusa de A. Marques
Departamento de F´ısica,
Universidade Federal de Campina Grande
58109-790 Campina Grande, Pb, Brazil.
(Dated: February 7, 2008)
In the paper [F. Nasseri, Phys. Lett. B 632 (2006) 151–154], F. Nasseri supposed that the value
of the angular momentum for the Bohr’s atom in the presence of the cosmic string is quantized in
units of ¯h. Using this assumption it was obtained an incorrect expression for Bohr radius in this
scenario. In this comment I want to point out that this assumption is not correct and present a
corrected expression for the Bohr radius in this background.
7
InarecentpaperF.Nasseri[1]analyzedthe fine struc- Intheabsenceofacosmicstring,thelowestorbit(n=1)
0
0 ture constant in the spacetime of a cosmic string from of the Bohr orbit has the following expression
2 the generalizeduncertainty principle. The author claims
Jan ttthhoreatoaftuhtoehrdoBreorchπorn’/ss4ird×aedr1iiu0ns−g6itn.haSthtuicsthhseyasstrteeamstuiolitnnacwrrayesaosoerbbstibatsyiniaendftabhcye- aB = 4mπǫe¯h22 =5.29×10−11m. (5)
Thus combining eqs.(4) and (5), we obtain
0 spacetime of a cosmic string have an integer number of
3 wavelengths in the interval from 0 to 2π. This is not
1 correct once for stationary orbits we have aB = 1− πGµ 1−4Gµ 2. (6)
aˆ (cid:18) 4 c2 (cid:19)(cid:18) c2 (cid:19)
v B
9 dS
6 =n; n=1,2,3,... (1)
I λ In the weak field approximation, eq.(6) turns into
1
1
whereS isthe lengthof the orbitandλthe wavelength.
0
7 As the metric of a cosmic string is given by aB ≈1− Gµ 8+ π . (7)
aˆ c2 4
0 B (cid:16) (cid:17)
/
c ds2 =c2dt2−dz2−dρ2−ρ2dϕ′2, (2)
q In the limit µ → 0, i.e., in the absence of a cosmic
gr- where ϕ′ = 1− 4cG2µ ϕ, which implies that ϕ′ varies string, aaˆBB → 1. It is worth noticing that there is a dif-
: from 0 to 2πb(cid:16). Theref(cid:17)ore dS = ρdϕ′, leads to a correc- ferencebetweeneq.(22)of1andeq.(6)ofthisComment.
v Taking into account the correction given by eq.(6) and
tion in the condition of quantization of the angular mo-
Xi mentum. Thus, instead of L =n¯h as considered by the inserting Gc2µ ≃10−6 we get
n
ar author in [1], we have Ln(b) = nb¯h, where b is the deficit
angle and is given by b = 1− 4Gµ[2]. With this consid-
c2 a
eration,the radiusofthenthBohrorbitofthe hydrogen aˆ = B . (8)
atom in the presence of a cosmic string becomes B 1− 8+ π4 ×10−6
(cid:0) (cid:0) (cid:1) (cid:1)
4πǫn2¯h2
ρn = 2. (3) Fromtheaboveequation,weconcludethatthenumer-
me2 1− π4Gc2µ 1−4Gc2µ icalfactorwhichcorrecttheBohrradiusisdifferentfrom
(cid:16) (cid:17)(cid:16) (cid:17) the one obtained in eq.(23) of reference [1].
Theeq.(17)of[1]iscorrectonlyforflatspacetime. Inthe
presence of a cosmic string, the radius of the nth Bohr
Acknowledgments: I would like to thank V. B.
orbit of the hydrogen atom is given by
Bezerra for the valuable discussions and to thank to
2 CNPq-FAPESQ-PB(PRONEX)forpartialfinancialsup-
4πǫ¯h
aˆ = . (4) port.
B 2
me2 1− πGµ 1−4Gµ
4 c2 c2
(cid:16) (cid:17)(cid:16) (cid:17)
[1] Forough Nasseri. Phys. Lett. B632 (2006) 151. [2] A. Vilenkin, E. P. S. Shellard, Cosmic string and Other
2
∗
Topological Defects, Cambridge Univ. Press, Cambridge, Electronic address: [email protected]
1994.