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 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.