Sjödin, Robert
(2001)
Cosmic strings in general relativity.
*University of Southampton, Department of Mathematics, Doctoral Thesis*, 177pp.

## Abstract

In this thesis we examine the properties of Cosmic Strings in the theory of General Relativity. We begin by considering static Cosmic Strings in flat space-time. We derive the field equations for the Cosmic String and show that the solution depends upon a single scaling parameter a which is constructed from the physical constants. Using this result we construct 1-parameter families of solutions which depend on an auxiliary parameter e and which describe the thin-string limit of a Cosmic String. By interpreting these solutions as elements of the simplified Colombeau algebra we may interpret the relativistic energy density Too of the thin string as an element of the Colombeau algebra with delta-function mass-per-unit-length. Furthermore, for a critically coupled Cosmic String the energymomentum tensor in the thin-string limit may be given a distributional interpretation. We also solve the string equations numerically for various values of a. This is done by compactifying the space-time to include infinity as part of the numerical grid and then using a relaxation method to suppress exponentially growing un-physical solutions. In curved space-time we derive the equations for the scalar and vector fields which are now coupled to the geometric variables through Einstein's equations. We again examine the thin-string limit in the Colombeau algebra by considering a 1-parameter family of solutions. W'e derive an expression for the deficit angle in terms of the distributional energy-momentum tensor of the thin string. We use this result to investigate the gravitational lensing properties of the string and relate this to the deficit angle. In the special case of a cone we find the scattering angle is equal to the deficit angle. We also solve the coupled equations numerically using techniques similar to those used in flat space-time. The second part of the thesis involves the dynamics of Cosmic Strings. Einstein's equations then lead to wave equations for both the matter and metric variables. However, the space-time is not asymptotically flat and this leads to problems in applying the appropriate boundary conditions. By using a Geroch transformation it is possible to reformulate the equations in terms of geometrical variables defined on an asymptotically flat (2+l)-dimensional space-time. Three exact vacuum solutions describing gravitational radiation due to Weber-Wheeler, Xanthopoulos and Piran et al. are used to excite the string which is found to oscillate with frequencies which are proportional to the masses of the scalar and vector fields of the string. This is in agreement with the exact results obtained using the linearised equations of the thin dynamic string. The behaviour of the dynamic string is studied by solving the equations numerically using an implicit fully characteristic scheme. The use of the Geroch transformation allows us to compactify the space-time and include null infinity as part of the numerical grid. This enables us to use the correct boundary conditions at infinity and hence suppress un-physical divergent solutions. The code is tested by comparing the results with exact solutions, by checking that it agrees with the static code and by undertaking a time dependent convergence test. The code is found to be accurate, stable and exhibit clear second order convergence.

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