Caustics in gravitational theories
Caustics in gravitational theories
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 world
lines 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 predicted
by 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 result
is 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 we
have 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.
Swatton, Damon John Ridgley
b645bc9c-73ca-4c95-b4c9-2c0e3d4de2e1
February 1999
Swatton, Damon John Ridgley
b645bc9c-73ca-4c95-b4c9-2c0e3d4de2e1
Swatton, Damon John Ridgley
(1999)
Caustics in gravitational theories.
University of Southampton, Department of Mathematics, Doctoral Thesis, 218pp.
Record type:
Thesis
(Doctoral)
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 world
lines 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 predicted
by 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 result
is 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 we
have 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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Published date: February 1999
Organisations:
University of Southampton
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Local EPrints ID: 50636
URI: http://eprints.soton.ac.uk/id/eprint/50636
PURE UUID: eaec91f2-ace1-4fb6-9482-10b426ce785b
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Date deposited: 06 Apr 2008
Last modified: 13 Mar 2019 20:50
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Author:
Damon John Ridgley Swatton
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