Corrections to the Fine Structure Constant in the Spacetime of a Cosmic String from the Generalized Uncertainty Principle

We calculate the corrections to the Fine Structure Constant in the spacetime of a cosmic string. These corrections stem from the generalized uncertainty principle. In the absence of a cosmic string our result here is in agreement with our previous result.

The gravitational properties of cosmic strings are strikingly different from those of non-relativistic linear distributions of matter.To explain the origin of the difference, we note that for a static matter distribution with energymomentum tensor, the Newtonian limit of the Einstein equations becomes where Φ is the gravitational potential.For non-relativistic matter, p i ≪ ρc 2 and ∇ 2 Φ = 4πGρ.Strings, on the other hand, have a large longitudinal tension.For a straight string parallel to the z-axis, p 3 = −ρc 2 , with p 1 and p 2 vanish when averaged over the string cross-section.Hence, the right-hand side of (2) vanishes, suggesting that straight strings produce no gravitational force on surrounding matter.This conclusion is confirmed by a full generalrelativistic analysis.Another feature distinguishing cosmic strings from more familiar sources is their relativistic motion.As a result, oscillating loops of string can be strong emitters of gravitational radiation.
The analysis in this letter is based on thin-string and weak-gravity approximations.The metric of a static straight string lying along the z-axis in cylindrical coordinates (t, z, ρ, φ) is given by2 where G is Newton's gravitational constant, µ the string mass per unit length and Introducing a new radial coordinate ρ ′ as we obtain to linear order in Gµ c 2 , Finally, with a new angular coordinate the metric takes a Minkowskian form So, the geometry around a straight cosmic string is locally identical to that of flat spacetime.This geometry, however is not globally Euclidean since the angle φ ′ varies in the range Hence, the effect of the string is to introduce an azimuthal 'deficit angle' implying that a surface of constant t and z has the geometry of a cone rather than that of a plane [1].
As shown above, the metric (6) can be transformed to a flat metric (8) so there is no gravitational potential in the space outside the string.But there is a delta-function curvature at the core of the cosmic string which has a global effect-the deficit angle (10).
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 [1].
Linet in [2] 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 3 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 As discussed in [3,4,5], 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 3 Linet in [2] has used the mks units and in Eqs.( 15) and ( 16) of [2] has obtained q 2 4πǫ 0 ρ 2 0 when µ → 0. Indeed we can put the fraction 2.5  π to be approximately equal to π 4 .With this substitution we obtain (11) of this article.and φ = 0.This means that the electron and the proton are in the plane orthogonal to the cosmic string.
To calculate the Bohr radius 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 the sum of two forces, i.e. the electrostatice force for Bohr's atom in the absence of a cosmic string, given by Eq.( 12), 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 Bohr radius in the spacetime of a cosmic string can be computed by (13).Using Newton's second law, we obtain where m is the mass of the electron.Cancelling one ρ 0 and rearranging gives There is a relationship between the radius and the momentum The product of the radius and the momentum in the left-hand side of ( 16) is the angular momentum.According to Bohr's hypothesis, the angular momentum L is quantized in units of h.This means that Substituting ( 16) into (15) gives or This equation obtains the radius of the n th Bohr orbit of the hydrogen atom in the presence of a cosmic string.In the absence of a cosmic string, the lowest orbit (n = 1) has a special name and symbol: the Bohr radius Using ( 19), the Bohr radius âB in the presence of a cosmic string is From ( 20) and ( 21) we obtain In the limit µ → 0, i.e. in the absence of a cosmic string, a B /â B → 1.
This means that the presence of a cosmic string causes the value of the Bohr radius increases (â B > a B ).
Our aim is now to obtain the effective Planck constant ĥeff in the spacetime of a cosmic string by using the generalized uncertainty principle.In doing so, we use the modified Bohr radius, âB , in the presence of a cosmic string.
The general form of the generalized uncertainty principle is where β is a dimensionless constant of order one and L P = (hG/c 3 ) 1/2 is the Planck length.In the case β = 0, (24) reads the standard Heisenberg uncertainty principle There are many derivations of the generalized uncertainty principle, some heuristic and some more rigorous.Eq.( 24) can be derived in the context of string theory and noncommutative quantum mechanics.The exact value of β depends on the specific model.The second term in the right-hand side of (24) becomes effective when momentum and length scales are of the order of the Planck mass and of the Planck length, respectively.This limit is usually called quantum regime.From (24) we solve for the momentum uncertainty in terms of the distance uncertainty, which we again take to be the radius of the first Bohr orbit.Therefore we are led to the following momentum uncertainty The maximum uncertainty in the position of an electron in the ground state in hydrogen atom is equal to the radius of the first Bohr radius, a B .In the spacetime of a cosmic string, the maximum uncertainty in the position of an electron in the ground state is equal to the modified radius of the first Bohr radius, âB , see (21).
Recalling the standard uncertainty principle ∆x i ∆p i ≥ h, we define an "effective" Planck constant ∆x i ∆p i ≥ heff .From (24), we can write So we can generally define the effective Planck constant from the generalized uncertainty principle in (26) and using (28) give us the effective Planck constant, ĥeff , in the spacetime of a cosmic string From ( 21) and M P = (hc/G) 1/2 which is the Planck mass, we have Using m = 9.11 × 10 −31 kg, e = 1.6 × 10 −19 C, c = 3.00 × 10 8 m/s, ǫ 0 = 8.85 × 10 −12 C 2 N −1 m −2 , G = 6.67 × 10 −11 m 3 s −2 kg −1 and M P = 2.1768 × 10 −8 kg we obtain the value of L P âB ≃ 10 −33 10 −9 ≃ 10 −24 is much less than one, we can expand (30).Therefore we have ĥeff So the effect of the generalized uncertainty principle in the presence of a cosmic string can be taken into account by using ĥeff instead of h.In the absence of a cosmic string, i.e. µ → 0, Eq.(32) leads us to our previous result in [6].In Ref. [3], we obtained the fine structure constant, α, in the spacetime of a cosmic string where α is the fine structure constant, α = e 2 4πǫ 0 hc .Substituting the effective Planck constant ĥeff from (32) into (33) we obtain the effective and corrected fine structure constant in the presence of a cosmic string by using the generalized uncertainty principle (37) This equation shows the corrections to the fine structure constant in the spacetime of a cosmic string from the generalized uncertainty principle.In the absence of a cosmic string the expression inside the parenthesis in the right-hand side of ( 37) is equal to one and we are led to our previous result in [6].In other words, in the absence of a cosmic string our result here, given by (37), is in agreement with our previous result in [6].