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Masters Degrees (Applied Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7108

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Browsing Masters Degrees (Applied Mathematics) by Subject "Calculus of tensors."

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    Conformal motions in Bianchi I spacetime.
    (1992) Lortan, Darren Brendan.; Maharaj, Sunil Dutt.
    In this thesis we study the physical properties of the manifold in general relativity that admits a conformal motion. The results obtained are general as the metric tensor field is not specified. We obtain the Lie derivative along a conformal Killing vector of the kinematical and dynamical quantities for the general energy-momentum tensor of neutral matter. Equations obtained previously are regained as special cases from our results. We also find the Lie derivative of the energy-momentum tensor for the electromagnetic field. In particular we comprehensively study conformal symmetries in the Bianchi I spacetime. The conformal Killing vector equation is integrated to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. These conditions place restrictions on the metric functions. A particular solution is exhibited which demonstrates that these conditions have a nonempty solution set. The solution obtained is a generalisation of the results of Moodley (1991) who considered locally rotationally symmetric spacetimes. The Killing vectors are regained as special cases of the conformal solution. There do not exist any proper special conformal Killing vectors in the Bianchi I spacetime. The homothetic vector is found for a nonvanishing constant conformal factor. We establish that the vacuum Kasner solution is the only Bianchi I spacetime that admits a homothetic vector. Furthermore we isolate a class of vectors from the solution which causes the Bianchi I model to degenerate into a spacetime of higher symmetry.
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    On Stephani universes.
    (1992) Moopanar, Selvandren.; Maharaj, Sunil Dutt.
    In this dissertation we study conformal symmetries in the Stephani universe which is a generalisation of the Robertson-Walker models. The kinematics and dynamics of the Stephani universe are discussed. The conformal Killing vector equation for the Stephani metric is integrated to obtain the general solution subject to integrability conditions that restrict the metric functions. Explicit forms are obtained for the conformal Killing vector as well as the conformal factor . There are three categories of solution. The solution may be categorized in terms of the metric functions k and R. As the case kR - kR = 0 is the most complicated, we provide all the details of the integration procedure. We write the solution in compact vector notation. As the case k = 0 is simple, we only state the solution without any details. In this case we exhibit a conformal Killing vector normal to hypersurfaces t = constant which is an analogue of a vector in the k = 0 Robertson-Walker spacetimes. The above two cases contain the conformal Killing vectors of Robertson-Walker spacetimes. For the last case in - kR = 0, k =I 0 we provide an outline of the integration process. This case gives conformal Killing vectors which do not reduce to those of RobertsonWalker spacetimes. A number of the calculations performed in finding the solution of the conformal Killing vector equation are extremely difficult to analyse by hand. We therefore utilise the symbolic manipulation capabilities of Mathematica (Ver 2.0) (Wolfram 1991) to assist with calculations.