Caustics in gravitational theories

Abstract

The gravitational collapse of a spherically symmetric, pressure-free dust is an interesting problem in General Relativity for it can lead, under certain initial conditions, to the situation where infinitesimally neighbouring shells approach and cross each other. The curve generated by these points of crossing generate a particular case of a caustic. In the situation where we have matter associated with each shell the density becomes unbounded on the caustic and, in a General Relativistic framework, we have a singularity.The interest in these types of singularity is two-fold: they present a possible mechanism for galaxy formation and they represent a counter example to both the strong and weak versions of the cosmic censorship hypothesis. In fact, Yodzis and collaborators prove that an event horizon is generated to the future of the initial singularity, so that these types of singularity are naked. If, however, a solution to the field equations  describing a spacetime with a caustic represent points that are internal (rather than being excluded as is generally the case for singularities), then this would stop these solutions as being counter examples to the simplest form of the cosmic censorship hypothesis. In addition, this would reinforce the idea that only strong singularities are censored. The characteristic feature of shell crossing singularities is that at some point the worldlines of shells coincide, meaning that the fluid flow vector becomes non-unique. If, however, we lift the geodesies that our shells follow onto the tangent bundle, then the vector tangent to these curves is unique. This indicates that we might be able to use the methods of Rendall and collaborators as a way to obtain existence to a solution of the field equations but, unfortunately, the unbounded nature of the density functions which arises in our formulation precludes this. We are forced, instead, to take the direct approach and consider solving the equations that model several superimposed dusts. The critical factor in any existence proof is to determine the shape of the caustic close to the point of cusp formation. In Newtonian theory or General Relativity this becomes the question of whether or not gravity alters the shape that is predictedby the simple cubic which is well known from catastrophe theory. We shall refer to this as the zero gravity solution. In this thesis we present a rigorous investigation of the limiting behaviour of both the Newtonian and General Relativistic pictures, showing in both cases that it can be represented by a similarity solution. We also relate the Newtonian to the Relativistic case. To further our understanding we also investigate the dynamics of the situation by constructing a computer model based on the Relativistic formulation. This numerical solution corroborates the results previously obtained. In the Newtonian analysis we show that the similarity solution (based on simple scaling transformations) obtained in the limit as we approach the cusp describes unbounded densities on the axis of symmetry. To correct this we suppose that the Newtonian constant G must also be scaled. We find that the solution now obtained in the limit is one where G = 0 which describes the zero gravity case. Moreover, if the initial conditions are described by a cubic, then we find that the asymptotic shape of the caustic does not differ from that of the generic caustic. We check for any other, more general transformation group that leaves the Newtonian differential equations invariant whilst reducing to the gravity free equations in this asymptotic limit. The conclusion is that, subject to an arbitrary Galilean transformation, the scaling transformations are the only transformations that fit this description. A similar analysis is performed with the General Relativistic equations. In this case, to enable asymptotic solutions to exist, we find that c must also be scaled. The resultis that the geodesic and conservation of matter equations reduce again to the gravity free case. Thus even in the General Relativistic formulation of caustic formation wehave gravity playing no part. In the latter parts of this thesis, work is presented that goes some way towards an existence proof for the Newtonian problem. We formulate the differential equations using a Lagrangian coordinate system and then discuss the set-up of a contraction mapping proof of existence of the solution to these equations. In the set-up of the existence proof, we prove that the solution must be C². We assume that any solution corresponding to G ≠ 0 cannot deviate from the zero gravity solution by more than a certain parameter which we are able to chose. By considering a small neighbourhood containing the cusp, we write the solution as a double iterated integral in time away from t = 0. We find that the integrand is not integrable through the cocaustic thus excluding any proof of existence of an initial value problem using a contraction mapping type of argument. It did, however, prove possible to show existence for a family of solutions parameterised by two arbitrary functions based on using the Arzela-Ascoli theorem. This approach which has been published in collaboration with C.J.S. Clarke.

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