Abstract:
In this talk I will review the evidence that generic asymptotically Minkowskian
radiating space-times will have a conformal completion at null infinity which is polyhomogeneous rather than
smooth. I will start by recalling the definition of polyhomogeneous functions. I will review the
Bondi - van der Burgh - Metzner - Sachs analysis, pointing out that generic solutions of the
characteristic constraint equations are polyhomogeneous in 1/r, where r is an area
coordinate. I will review the results of the analysis by Lars Andersson, myself, and Helmut Friedrich,
which show that solutions of the constraint equations on hyperboloidal hypersurfaces obtained from
generic smooth seed fields are polyhomogeneous. Finally I will point out the results, obtained in collaboration
with Olivier Lengard, on existence of Scri for solutions of the vacuum Einstein equations
with polyhomogeneous hyperboloidal initial data. The references to my work on those problems are:
L. Andersson, P.T. Chrusciel, H. Friedrich:
``On the Regularity of Solutions to the Yamabe Equation and
the Existence of Smooth Hyperboloidal Initial Data for Einstein's
Field Equations", Commun. Math. Phys. 149,
587-612 (1992) [tex,dvi,ps files].
``On ``hyperboloidal" Cauchy data for vacuum Einstein
equations and obstructions to smoothness of Scri'',
preprint TRITA-MAT-1992-0038; CMA ANU Research Report
CMA-MR35-92; Commun. Math. Phys.161, 533-568 (1994)
[tex,dvi,ps files].
P.T. Chrusciel, M.A.H. MacCallum, D.
Singleton:
``Gravitational Waves in General Relativity. XIV: Bondi
Expansions and the ``Polyhomogeneity'' of Scri'', ANU
Research Report CMA-MR14-92 (SMS-53-92), Phil. Trans. Royal
Soc. of LondonA350, 113-141 (1995) [http://xxx.lanl.gov/abs/gr-qc/9305021].
L. Andersson, P.T. Chrusciel:
``On asymptotic behaviour of solutions of the constraint
equations in general relativity with ``hyperboloidal boundary
conditions'' '', Tours preprint 103/95, Dissertationes
Mathematicae355, 1-100 (1996).
Copyright 29.VIII.02 by P.T.Chrusciel and A.Chopin