Reply to"Comment on Fine Structure Constant in the Spacetime of a Cosmic String"

In this Reply, using E.R. Bezerra de Mello's comment, I correct calculations and results presented in Phys. Lett. B 614 (2005) 140-142 about fine structure constant in the spacetime of a cosmic string.

In a recent letter [1], I analysed the Bohr's atom in the spacetime of a cosmic string. I concluded that in the presence of a cosmic string the fine structure constant reduces by a factor π 2 ǫ 0 Gµ, or as numerical results by a factor 8.736 × 10 −17 , see Eqs.(18) and (19) of [1].
Based on E.R. Bezerra de Mello's comment [2], the conclusions presented in [1] about the fine structure constant in the spacetime of a cosmic string are not completely correct in the sence that the results presented in [1] are valid only in a special situation. As explained in [2], the conclusions presented in [1] are correct if we assume as ideal not only the proton is placed on the cosmic string but also the electron and the proton are in the plane orthogonal to the cosmic string lying along the z-axis.
The purpose of this Reply is to correct calcultions presented in [1] in accordance with the observation made in [2] and also to present the correct value of the fine structure constant in the spacetime of a cosmic string. As pointed out in [2], our approach is correct only if we consider the proton and the electron placed in a plane perpendicular to the cosmic string which lies along the z-axis, with the proton on the cosmic string.
It should be emphasized that the mistake concerning the value of the fine structure constant is due to the fact that we have used two different unit systems which we corrected in this Reply.
Linet in [3] has shown that the electrostatic field of a charged particle is distorted by the cosmic string. For a test charged particle in the presence of a cosmic string the electrostatic self-force is repulsive and is perpendicular to the cosmic string lying along the z-axis 2 where f ρ is the component of the electrostatic self-forcs along the ρ-axis in cylindrical coordinates and ρ 0 is the distance between the electron and the cosmic string.
For the Bohr's atom in the absence of a cosmic string, the electrostatic force between an electron and a proton is given by Coulomb law 3 F = −e 2 4πǫ 0 r 2r . ( As discussed in [2], to obtain the fine structure constant in the spacetime of a cosmic string we assume that the proton located on the cosmic string lying along the z-axis. We also assume that the proton located in the origin of the cylindrical coordinates and the electron located at ρ = ρ 0 , z = 0 and φ = 0. This means that the electron and the proton are in the plane orthogonal to the cosmic string.
The orbital speed of the electron in the first Bohr orbit is .
The ratio of this speed to the speed of light, v 1 /c, is known as the fine structute constant when µ → 0. Indeed we can put the fraction 2.5 π to be approximately equal to π 4 . With this substitution we obtain (1) of this Reply. 3 In fact in [1] I have used the same letter r to define the distance from the electron to the cosmic string, and from the electron to the proton. The force in Eq.(11) of [1] is perpendicular to the string, so we may use ρ = x 2 + y 2 . As to the force in Eq.(12) of [1], it is radial: r = x 2 + y 2 + x 2 . I thank E.R. Bezzera de Mello for pointing this out to me.
To calculate the fine structure constant in the spacetime of a cosmic string we consider a Bohr's atom in the presence of a cosmic string. For a Bohr's atom in the spacetime of a cosmic string, we take into account in Eq.(1) the sum of two forces, i.e. the electrostatice force for Bohr's atom in the absence of a cosmic string, given by Eq.(2), plus the electrostatic self-force of the electron in the presence of a cosmic string. Because we assume that the proton located at the origin of the cylindrical coordinates and on the cosmic string and also the plane of electron and proton is perpendicular to the cosmic string lying along the z-axis, the induced electrostatic self-force and the Coulomb force are at the same direction, i.e. the direction of the ρ-axis in cylindrical coordinates. Therefore, we can sum these two forces It can be easily shown that this force has negative value and is an attractive force ( πGµ 4c 2 < 1). The numerical value of the fine structure constant in the spacetime of a cosmic string can be computed by Eq.(5). The orbital speed of the electron in the first Bohr orbit in the spacetime of a cosmic string has positive value and is given byv .
The ratio of this speed to the speed of light,v 1 /c, is presented by the symbol α which is the fine structure constant in the spacetime of a cosmic strinĝ Gµ c 2 e 2 4πǫ 0h c .
From (4) and (7) we obtain This means that the presence of a cosmic string causes the value of the fine structure constant reduces by a factor πGµ 4c 2 . In the limit Gµ c 2 → 0, i.e. in the absence of a cosmic string, α/α → 1. From Eq.(8) we obtain (α−α)/α = πGµ 4c 2 . The dimensionless parameter Gµ c 2 plays an important role in the physics of cosmic strings. In the weak-field approximation Gµ c 2 ≪ 1. The string scenario for galaxy formation requires Gµ c 2 ∼ 10 −6 while observations constrain Gµ c 2 to be less than 10 −5 . For more details about the range of value of Gµ c 2 see [4]. Inserting Gµ c 2 ∼ 10 −6 in the right-hand side of Eq.(8) yieldŝ α ∼ 1 − π 4 × 10 −6 α.