Physics Letters B 308 (1993) 237-239 North-Holland PHYSICS LETTERS B
No glory in cosmic string theory
G. W. Gibbons
DAMTP, Silver Street, Cambridge CB3 9EW. UK
Received 24 March 1993; revised manuscript received 13 April 1993 Editor: P.V. Landshoff
Light deflection by non-axisymmetric cosmic strings is studied. Various properties of the geodesics are obtained including the non-existence of closed or almost closed geodesics. It is shown that the behaviour of the geodesics resembles the case of the axisymmetric cosmic strings usually studied. In particular no exotic behaviour such as glory scattering is possible. The usual formula for the deflection of distant geodesics is obtained. An application to the slow motion of SlJ(3) monopoles, considered as geodesic motion on a moduli space, is also given.
It is usual to model the metric of a static, straight cosmic string [ I] by using a flat cone, or more so- phisticatedly using an axisymmetric snub nosed cone. The axisymmetry means that the geodesics are com- pletely integrable by virtue of a constant of the mo- tion whose utility was first realised in the context of surfaces of revolution by Clairault [ 21.
However, it is not always true that cosmic string spacetimes need be axisymmetric. Consider one or more parallel strings for example or more realistically a complicated model with many gauge fields and Higgs fields, possibly non-abelian. Non-axisymmetric vor- tices are certainly encountered in liquid 3He [ 31.
If the spacetime is not axisymmetric one cannot expect to find the geodesics exactly and one might wonder to what extent the simple behaviour of the geodesic in the axisymmetric case continues to hold or whether exotic behaviour such as glories might result from closed or almost closed geodesics, i.e. geodesics which circle the cosmic string many times before es- caping to infinity. Remarkably it turns out that despite our complete inability to integrate the geodesic equa- tions in the general non-axisymmetric case one can, using some known results in global differential geom- etry, say a great deal about the geodesics. In particu- lar one can definitely rule out the existence of closed or almost closed geodesics, trapped geodesics or other exotic behaviour. In fact the overall behaviour may be shown to resemble closely the simple behaviour seen in the axisymmetric case. In other words the standard
predictions of cosmic string are robust under relax- ation of the axisymmetry assumption.
A cosmic string spacetime has topology R2 x Z and we shall take the metric to be of the form
ds2 = -dt2 + dz2 + gijdxdx, (1)
i = 1,2. For metrics of the form ( 1) the behaviour of massive and massless particles is governed entirely by the geodesics of the 2-metric gij on the 2-manifold X.
The Einstein field equations require that
Tg = T$ (2)
Tj =O, (3)
K = &CT%, (4)
where I( is the Gauss curvature of the metric (4). From (4) we have, assuming positive energy,
(A) K > 0. In what follows we shall insist that (A) holds every-
where, even outside the string. We also assume that the cosmic string has not closed up the universe i.e.
(B) C is non-compact. Assumptions (A) and (B) are just those made by
Cohn-Vossen in his classic studies of non-compact surfaces with positive Gauss-curvature [4-61. It fol- lows that
0370-2693/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved. 237