More on the analogy between disclinations and cosmic strings

Transcription

1 More on the analogy between disclinations and cosmic strings Fernando Moraes Grupo de Matéria Condensada Mole e Física Biológica Departamento de Física Universidade Federal da Paraíba João Pessoa, PB Brazil UFCG p.1/17

2 genesis in the early universe, a sequence of symmetry-breaking phase transitions G H... SU(3) SU(2) U(1) SU(3) U(1) left behind topological defects (global monopoles, cosmic strings, domain walls, textures and other hybrid creatures) as relics of the higher temperature phases UFCG p.2/17

3 nematics symmetry-breaking phase transitions in liquid crystals SO(3) T(3) SO(2) T(3) SO(2) T(2) T 1 T 2 <T c T 3 TEMPERATURA also generate topological defects UFCG p.3/17

4 hedgehogs, disclinations, textures z y x they have been used as a laboratory for cosmology with respect to defect formation, evolution and dynamics T. Kibble Phase-transition dynamics in the lab and the universe Physics Today, Sept. 2007, p. 47 UFCG p.4/17

5 geometrical optics in nematics Kline & Kay: light rays in inhomogeneous anisotropic media are extremals of Fermat s functional Born & Wolf: light rays obeying Fermat s principle are geodesics in some Riemannian space N = n 2 ocos 2 (β) + n 2 esin 2 (β) UFCG p.5/17

6 effective geometry for light propagating near defects in nematics F = B A Ndl = B A i,j g ijdx i dx j C. Sátiro and F. Moraes Lensing effects in a nematic liquid crystal with topological defects, Eur. Phys. J. E 20 (2006)173 On the deflection of light by topological defects in nematic liquid crystals, Eur. Phys. J. E 25 (2008) 425 UFCG p.6/17

7 assorted line defects ϕ(θ) = kθ + c where k is the winding number and c = ϕ(0) UFCG p.7/17

8 line defect geometry we found ds 2 2D = {n2 o cos 2 [(k 1)θ + c] + n 2 e sin 2 [(k 1)θ + c]}dr 2 +{n 2 o sin 2 [(k 1)θ + c] + n 2 e cos 2 [(k 1)θ + c]}r 2 dθ 2 {2(n 2 e n 2 o) 2 sin[(k 1)θ + c] cos[(k 1)θ + c]}rdrdθ which, for n o = n e, reduces to the Euclidean line element ds 2 2D = dr2 + r 2 dθ 2 UFCG p.8/17

9 conical defect for k = 1, c = π/2 the effective geometry is conical: ds 2 2D = dr2 + α 2 r 2 dθ 2 where α = n o n e so is the spacetime of a cosmic string: ds 2 4D = dt2 dz 2 dr 2 α 2 r 2 dθ 2 where α = 1 4Gµ C. Sátiro and F. Moraes A liquid crystal analogue of the cosmic string Mod. Phys. Lett. A 20 (2005) 2561 UFCG p.9/17

10 curvature from the metric ds 2 2D = {n2 o cos 2 [(k 1)θ + c] + n 2 e sin 2 [(k 1)θ + c]}dr 2 +{n 2 o sin 2 [(k 1)θ + c] + n 2 e cos 2 [(k 1)θ + c]}r 2 dθ 2 {2(n 2 e n 2 o) 2 sin[(k 1)θ + c] cos[(k 1)θ + c]}rdrdθ we found that the only nonzero Riemann curvature tensor components are R rθrθ = R θrθr = R θrrθ = R rθθr = k(k 1)(n 2 e n 2 o) cos{2[(k 1)θ + c]} UFCG p.10/17

11 curvature and, for ξ(θ) = (k 1)θ + c, the curvature (Ricci) scalar is R = 2 α (1 α)δ(ρ) ρ for k = 1 and R = 2k α [ ( )] α + 1 δ(ρ) + 2k(k 1)(α2 1) cos(2ξ) for k 1 α ρ α 2 ρ 2 the idealized defects have a δ-function curvature singularity at ρ = 0 and for ρ 0 there is a ρ- and θ- dependent curvature as one goes around the defects the curvature alternates between positive and negative values UFCG p.11/17

12 curvature profile disclinations with k 1: k = 3 2, 1, 1 2, 1 2 and 2 UFCG p.12/17

13 asymmetric strings inspired by disclinations in nematics, we propose a new family of cosmic strings by writing ds 2 4D = dt2 dz 2 ds 2 2D Einstein equation R µν 1 2 g µνr = 8πGT µν gives us the energy-momentum tensor { Tt t = Tz z = 1 8πG k α + 2k(k 1)(1 α2 ) α 2 ρ 2 [ ( )] α + 1 δ(ρ) α ρ } cos {2 [(k 1)θ]} which is the source of the geometry associated to the string UFCG p.13/17

14 asymmetric strings contrasting to ordinary cosmic strings, the mass distribution profile of the asymmetric strings are not limited to an infinitely thin core alternate between positive and negative values as one circles the string they also lead to the same light paths as the disclinations in nematics C. Sátiro, A. M. de M. Carvalho and F. Moraes An asymmetric family of cosmic strings Int. J. Mod. Phys. D, under review UFCG p.14/17

15 conclusions the analogy between cosmic defects and defects in nematics goes beyond their formation, evolution and dynamics they are also analogous from the point of view of geometrical optics yet unknown cosmological defects have been proposed inspired by their nematic counterparts UFCG p.15/17

16 conclusions the geometrical approach is very handy when it comes to determine global properties like topological phases A. M. de M. Carvalho, C. Sátiro and F. Moraes Aharonov-Bohm-like effect for light propagating in nematics with disclinations Europhys. Lett. 80 (2007) UFCG p.16/17

17 acknowledgements Caio Sátiro (UFRPE, Brazil) Alexandre Carvalho (UEFS, Brazil) CNPq, CAPES/PROCAD, FAPESQ-PB, PRONEX (Brazilian agencies) UFCG p.17/17