**Grant EP/E063896/1**

Gravitational Waves and Gravitational Collapse (2007)

Visiting Researcher: Prof. N.O. Santos

Principal Investigator: Prof. M.A.H. MacCallum

## Summary of Final Report

The existence of gravitational waves was disputed for a long time, but
in recent years their existence has been generally accepted. Waves in
the broad sense are characterized by time dependent fields
transporting energy. However this point leads to a difficulty in
General Relativity since the theory lacks a localized definition of
energy and hence gives no simple account of how energy is extracted
from the sources. Furthermore, up to now there are no exact solutions
to Einstein's field equations that explicitly make manifest the
production of these waves.
We aim to look in detail at the simplest collapse situation capable of
radiating, namely cylindrical collapse. We shall try to find new
shear-free solutions and match them to a radiating exterior, and
similarly study dust solutions (it has been conjectured that such
solutions cannot radiate). We also aim to study limitations on
equations of state for such bodies, and to examine dissipative and
charged spherical collapse.
Under the grant, we completed and published a paper on the dissipative
and charged spherical collapse (see
`http://arxiv.org/abs/0707.2794`). The main results concerned the
roles of different terms in the dynamical equation. The dynamical
equation was coupled to a causal transport equation in the context of
Israel-Stewart theory. The decrease of the inertial mass density of
the fluid, by a factor which depends on its internal thermodynamic
state, was reobtained, with viscosity terms included. In accordance
with the equivalence principle, the same decrease factor was obtained
for the gravitational force term. The effect of the electric charge on
the relation between the Weyl tensor and the inhomogeneity of energy
density was discussed.
On the main aim, we obtained the general governing equations and
matching conditions for the case of a fluid without hear transfer,
imposed the shearfree condition, and found we could integrate one of
the field equations in terms of a sum of a function of time and a
function of radius. The boundary conditions then enabled the time
dependence to be given as (in general) an elliptic function. We also
proved that a static exterior implies a static interior. This shows
that a collapsing interior must have a time-dependent exterior, but
leaves open the issue of whether this time dependence is properly
described as gravitational radiation. We will complete the study of
the remaining equations and boundary conditions after we have done the
special case of dust.
For dust, we are now working on the general solution, knowing from
work of Cocke that there is at least a special solution in which the
interior is a spatially homogeneous Robertson-Walker cosmology. We
showed Cocke's is the most general such solution, and we noted that
the time dependence of C-energy he found, a result agreeing with our
general result above, suggests further work is required on definitions
of energy and radiation, since one does not expect an RW solution to
radiate. We are also investigating the behaviour of the conformal
curvature and the possible gravitomagnetic effects.
This short grant enabled us to make substantial progress towards our
main aim and we should soon complete this work.

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On 10 Nov 2007, 20:05.