Warped Angle-deficit of a 5 Dimensional Cosmic String
WWarped Angle-deficit of a 5 Dimensional CosmicString.
Case I: The General Non-Abelian Case
R J Slagter and D Masselink
Institute of Physics, University of AmsterdamandASFYON, Astronomisch Fysisch Onderzoek Nederland, Bussum, The NetherlandsE-mail: [email protected]
Abstract.
We present a cosmic string on a warped five dimensional space timein Einstein-Yang-Mills theory. Four-dimensional cosmic strings show some seriousproblems concerning the mechanism of string smoothing related to the string massper unit length, Gµ ≈ − . A warped cosmic string could overcome this problemand also the superstring requirement that Gµ must be of order 1, which is far aboveobservational bounds. Also the absence of observational evidence of axially symmetriclensing effect caused by cosmic strings could be explained by the warped cosmic stringmodel we present: the angle deficit of the string is warped down to unobservable valuein the brane, compared to its value in the bulk. It turns out that only for negativecosmological constant, a consistent numerical solution of the model is possible.
1. Introduction
Recently, there is growing interest in the Randall-Sundrum(RS) warped 5D geometry[1,2]. One of the interesting outcomes of this idea is the solution of the large hierarchyproblem between the weak scale and the fundamental scale of gravity. The predictedKaluza-Klein particles in the model could be detected with the LHC at CERN. In theoriginal RS scenario, it was proposed that our universe is five dimensional, describedby the metric ds = e − | y | ky c g µν dx µ dx ν + y c dy . (1)The extra dimension y makes a finite contribution to the 5D volume because of theexponential warp factor, where y c is the size of the extra dimension. At low energies,gravity is localized at the brane and general relativity is recovered. At high energygravity ”leaks” into the bulk. The 4D Planck scale will be an effective scale which canbecome much larger than the fundamental Planck scale M P if the extra dimension ismuch larger than M − P . Further, the self-gravity of the brane must be incorporated.This will protect the 3 dimensional space from the large extra dimensions by curvaturerather than straightforward compactification. Also matter fields in the bulk can beincorporated. This will lead to a kind of ”holographic” principle, i.e., the 5D dynamicsmay be determined from knowledge of the fields on the 4D boundary. For an overview,see [3]. We will consider here the 5D model with a general Yang-Mills field, dependentof r, y and t . In a following article we investigate the interplay of the 4D and 5Dcoupled equations with the junction conditions. a r X i v : . [ g r- q c ] N ov arped Angle-deficit of a 5 Dimensional Cosmic String.
2. The model
We will consider here the RS2 model with two branes at y = 0, the weak visible braneand at y = y c , the gravity brane (in the RS1 model one let y c → ∞ ). The action ofthe model under consideration is [4] S = 116 π (cid:90) d x (cid:112) − (5) g (cid:104) G ( (5) R − Λ ) + κ (cid:16) (5) R (5) µναβ R µναβ − (5) R (5) αβ R αβ + (5) R (cid:17) − g T r F (cid:105) + (cid:90) d x (cid:112) − (4) g (cid:104) G Λ + S (cid:105) (2)with G the gravitational constant, Λ the cosmological constant, κ the Gauss-Bonnet coupling, g the gauge coupling, Λ the brane tension and S the effective4D Lagrangian, which is given by a generic functional of the brane metric and matterfields on the brane and will also contain the extrinsic curvature corrections due to theprojection of the 5D curvature. For the moment we will consider here only the 5Dequation in a general setting and with a Yang-Mills matter field. The 4D inducedequations together with the junction conditions will be presented in part 2 of a nextarticle.The coupled set of equations of the EYM-GB system will then become( from nowon all the indices run from 0..4)Λ g µν + G µν − κGB µν = 8 πG T µν , (3) D µ F µνa = 0 , (4)with the Einstein tensor G µν = R µν − g µν R, (5)and Gauss-Bonnet tensor GB µν = 12 g µν (cid:16) R γδλσ R γδλσ − R γδ R γδ + R (cid:17) − RR µν + 4 R µγ R γν +4 R γδ R γµδν − R µγδλ R νγδλ . (6)Further, with R µν the Ricci tensor and T µν the energy-momentum tensor T µν = Tr F µλ F λν − g µν Tr F αβ F αβ , (7)and with F aµν = ∂ µ A aν − ∂ ν A aµ + g(cid:15) abc A bµ A cν , and D α F aµν = ∇ α F aµν + g(cid:15) abc A bα F cµν where A aµ represents the YM potential.We will consider the warped axially symmetric space time ds = − F ( t, r, y )[ dt − dz − dr − A ( t, r, y ) dϕ ] + dy , (8)with y the bulk dimension and the YM parameterization A ( a ) t = (cid:16) , , Φ( t, r, y ) (cid:17) , A ( a ) r = A ( a ) z = A ( a ) y = 0 ,A ( a ) φ = (cid:16) , , W ( t, r, y ) (cid:17) . (9)So the metric and YM components depend t and the two space dimensions r and y.The set of PDE’s become, for κ = 0 for the time being, F tt = F rr + 12 F F yy + 34 F ( F t − F r )+ 12 Λ F − πGA (cid:104) W r − W t + F W y (cid:105) , (10) A tt = A rr + F A yy − A ( A r + A y − A t ) + 1 F ( F r A r − F t A t + 2 F F y A y ) − πGF (cid:104) W t − W r − F W y (cid:105) , (11) W tt = W rr + F W yy + 12 A W t A t + W y ( F y − F A A y ) − A W r A r . (12)
3. The Static case
In the static case the resulting PDE’s become A rr + F A yy +2 F y A y + F r A r F − F A y + A r A + 16 πGF (cid:16) W r + F W y (cid:17) = 0 , (13) F rr + F F yy + F y + F r A r + F F y A y A + 23 Λ F − πG A (cid:16) W r + F W y (cid:17) = 0 , (14) arped Angle-deficit of a 5 Dimensional Cosmic String. W rr + F W yy − W r A r A + W y ( F y − F A y A ) = 0 , (15)Φ rr + F Φ yy + Φ r A r A + Φ y ( F y + F A y A ) = 0 . (16)We also have the two constraints F Φ y + Φ r = 0 , W r Φ r + F W y Φ y = 0 . (17)When we substitute the equations for Φ and W into the conservation equation ∇ µ T νµ = 0, we obtain identically zero, as it should be. When we introduce thequantities θ i defined by θ ≡ F √ A A r , θ ≡ √ AF r , θ ≡ F √ A A y , θ ≡ F √ AF y , (18)then the equations can be written as ∂∂r θ + ∂∂y θ = − πG √ A ( W r + F W y ) , (19) ∂∂r θ + ∂∂y θ = 16 πG √ A ( W r + F W y ) −
23 Λ F √ A, (20) (cid:104) W r √ A (cid:105) r + (cid:104) F W y √ A (cid:105) y = 0 , (21) (cid:104) √ A Φ r (cid:105) r + (cid:104) F √ A Φ y (cid:105) y = 0 . (22)The Ricci scalar (5) R becomes: (5) R = 8 πG AF (cid:0) F W y + W r (cid:17) + 53 Λ . (23)We now investigate the properties of the static solution for large values of r and y . Wewill assume that (cid:90) ∞ √ Aσdr (24)converges, where σ is the energy density T = − πG W r + F W y AF . Further,lim r →∞ √ Aσ = 0 . (25)
4. Analysis of the Angle Deficit
The angle deficit can be calculated for a class of static translational symmetric spacetimes which are asymptotically Minkowski minus a wedge. If we denote with l thelength of an orbit of (cid:16) ∂∂ϕ (cid:17) a in the brane, then the angle deficit is given by[5, 6, 7](2 π − ∆ ϕ ) = lim r →∞ dldr , (26)with l = (cid:90) π (cid:115) g ab (cid:16) ∂∂ϕ (cid:17) a (cid:16) ∂∂ϕ (cid:17) b dϕ. (27)One better can use the Gauss-Bonnet theorem to obtain the angle deficit bycalculating the integral of the Gaussian curvature over the surface of S ( t, z ) = const.If one transports a vector around a closed curve, then the angle rotation α will begiven[7] by the area integral over of the subsurface of S α = (cid:90) d x (cid:112) (3) g (3) K, (28)with (3) K = 12 (3) g ik (3) g jl (5) R ijkl . (29)For our case, we obtain (cid:112) (3) g (3) K = − (cid:16) F r √ AF (cid:17) r − (cid:16) A r √ A (cid:17) r − (cid:16) F A y √ A (cid:17) y − ( √ AF y ) y + 14 √ AF y (cid:16) F y F + A y A (cid:17) = − (cid:104) F y F + F r F −
23 Λ − πG F A ( W r + F W y ) (cid:105) . (30) arped Angle-deficit of a 5 Dimensional Cosmic String. Then Eq.(28) becomes α = − π (cid:110)(cid:104)(cid:16) √ AF r F (cid:17) + (cid:16) A r √ A (cid:17)(cid:105) ∞ r =0 + (cid:104) F A y √ A + 2 √ AF y (cid:105) ∞ y =0 − (cid:90) ∞ (cid:90) ∞ √ AF y ( F y F + A y A ) drdy (cid:111) . (31)Or, using the second expression in Eq.(30) α = − π (cid:90) ∞ (cid:90) ∞ F √ A (cid:16) σ −
23 Λ (cid:17) drdy − π (cid:90) ∞ (cid:90) ∞ F √ A (cid:104) F y F + F r F (cid:105) drdy. (32)If one assumes that in the 4 dimensional case, for r → ∞ : F → √ A → br and for r → F → √ A → r , than the first term of Eq.(31) represents thewell-known result [6] that α = 2 π (1 − b ) in de brane, so S is asymptotically a conicalsurface. In the 5 dimensional case the results depend on the boundary values of ourwarp factor F, i.e., the last two terms in Eq. (31). From Eq.(32) one observes that thefirst term represents the proper mass per unit length of the string plus a contributionfrom the cosmological constant. The second term is the correction term.Now we try to obtain for the asymptotic warped metric ds = F c e k y + a (cid:104) − dt + dz + dr + ( k r + a ) dϕ (cid:105) + dy . (33)For y = 0 we recover de 4D result of a flat space time minus a wedge by thetransformation[6] r (cid:48) = r + a k , ϕ (cid:48) = k ϕ (0 ≤ ϕ ≤ π ) , (34)i.e., ds = − dt + dz + d ( r (cid:48) ) + ( r (cid:48) ) d ( ϕ (cid:48) ) + dy , (35)where now ϕ (cid:48) has a different range then ϕ . For y (cid:54) = 0 we have the warped metric ds = F c e k y + a (cid:2) − dt + dz + d ( r (cid:48) ) + ( r (cid:48) ) d ( ϕ (cid:48) ) (cid:105) + dy . (36)The angle deficit is determined by k F c e k y + a .Let us consider now ∂∂r ( θ + θ ) + ∂∂y ( θ + θ ) = − πG √ A (cid:16) W r + F W y (cid:17) −
23 Λ √ AF = − πG (cid:104) ∂∂r (cid:16) W W r √ A (cid:17) + ∂∂y (cid:16) F W W y √ A (cid:17)(cid:105) −
23 Λ √ AF , (37)where we used the Eq.’s (19)-(22). After rearranging we then obtain ∂∂r (cid:16) θ + θ + 323 πG W W r √ A (cid:17) = − ∂∂y (cid:16) θ + θ + 323 πG F W W y √ A (cid:17) −
23 Λ √ AF . (38)So we notice that the Φ-field disappears from the equation. It will have only acontribution on the brane. It is quite easy to obtain a particular solution of thisequation, Eq.(38). For F = F c e ±√− Λ y + a A = A c ( k r + a ) , (39)we obtain for W a solution of the form W ( r, y ) = W ( r ) W ( y ), where W and W are given by Bessel functions. This oscillatory behavior of W is not uncommon forgravitating YM vortices.So it seems to be possible to find the desired asymptotic warped form for the conicalspace time, i.e., Eq.(33).The next task is to obtain from the junction condition and the brane-bulk splitting,relations between the several constants in the model.
5. Numerical solutions
For a given set of initial conditions, these PDE’s determine the behavior of F, A, Wand Φ. We will impose particular asymptotic conditions, in order to obtain acceptablesolutions of the cosmic string. First, we have lim r →∞ Φ( r, y ) = 1 , lim r →∞ W ( r, y ) = arped Angle-deficit of a 5 Dimensional Cosmic String. Figure 1.
Typical solution of F, A, W and Φ for negative Λ with initial values: F = e ( − r + y ) , A = r , W = e ( − r − y ) , Φ = 1 − e ( − r − y ) and Dirichlet boundaryconditions on the outer boundaries. We also plotted g ϕϕ , F (0 , y ) = 1 , A (0 , y ) = 0 , ∂∂r A (0 , y ) = 1. Further, lim r →∞ g ϕϕ r = 1.We solve the system using the numerical code CADSOL-FIDISOL. The asymptoticform of the metric component g ϕϕ behaves as expected.
6. Conclusions
In earlier attempts[8, 9, 10], we tried to build a 5-dimensional cosmic string withouta warp factor and investigated the causal structure. Here we considered a differentapproach. It seems possible that the absence of cosmic strings in observational datacould be explained by our model, where the effective angle-deficit resides in the bulkand not in the brane. In this part we considered the 5D equations in general form,without the splitting of the energy-momentum tensor in a bulk and brane part. Wefind a consistent set of equations in the bulk. The asymptotic behavior of the metricoutside the core of the string seems to have the desired form. This solution must beconsistent with the system of equations obtained by the bulk-brane splitting. It is alsointeresting to investigate holographic ideas in our model. These subjects are understudy by the authors and will be published in a followup article. arped Angle-deficit of a 5 Dimensional Cosmic String. Figure 2.
The 5-D cosmic string
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