Summary Report of EPSRC Grant GR/R 53685
Global models describing isolated bodies in equilibrium in general relativity
Grant-holder: Prof. M.A.H. MacCallum

Isolated rotating objects in equilibrium are important as models of compact rotating stars, galactic nuclei and other rotating astrophysical bodies. However, their description in General Relativity is still surprisingly poorly understood: we still lack complete non-spherically symmetric models for a self-gravitating (finite) body together with its exterior.

The main focus of the work was the construction of global models by means of the matching of spacetimes: the whole configuration is composed of two regions, one spacetime describing the interior of the body and another describing the vacuum exterior, matched across a hypersurface. In order to model equilibria both are usually assumed stationary and also axisymmetric. It is also often assumed that an energy-flux-free fluid in the interior simply rotates around the axis (circularity), i.e. there is no convection.

One objective was the study of the relations between the boundary conditions at the matching surface and the stationarity, axial and circularity conditions. A first paper was devoted to motivating and presenting a rigorous definition of symmetry-preserving matchings, together with the consequences such a definition has on the preserved group of symmetries. Then, from the complete set of matching conditions for a general interior, it was shown that non-circularity in the interior has no effect on the exterior provided there is circularity at the boundary: this condition itself might be thought physically plausible as convective motions would not be present at the boundary.

The arguments, originally for vacuum exteriors and uncharged interiors, were generalized to the case with electromagnetic fields.

On the general problem of finding exteriors matching known non-vacuum interiors, compatibility conditions were derived but proved to be harder to apply analytically to examples than we had foreseen. The specific problem of matching the Wahlquist metric to a vacuum exterior led us to an extensive treatment of perturbed matchings which had not initially been planned. In particular we found, contrary to earlier claims, that in this example the matching could be done to second order.

Progress was thus made towards the general objective of obtaining restrictions on interior boundary data giving a unique exterior solution regardless of the identification at the boundary but a full resolution of this problem awaits further work.

The PDRA employed on this grant was Dr. Raul Vera, and Prof. José Senovilla and Dr. Marc Mars were Visiting Fellows. A total of 7 papers and conference reports have been or are being produced, and three related papers were partially supported.

(A fuller version of this report, and preprints of the associated papers, are available on request to

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