Asymptotic structure session: some references


Some standard sources

R.Penrose 1965 Zero-rest mass fields including gravitation: asymptotic behaviour Proc.Roy.Soc.Lond A284, 159-203


R.Geroch 1977 Asymptotic structure of space-time in ‘Symposium on the asymptotic structure of space-time’ eds F.P.Esposito and L.Witten, Plenum: NY


A.Ashtekar and R.O.Hansen 1978 A unified treatment of null and spatial infinity in general relativity J.Math.Phys.19 1542-66


R.Penrose and W.Rindler 1984, 1986 Spinors and space-time 1 and 2 CUP:Cambridge


R.P.A.C.Newman 1989 The global structure of simple space-times Comm.Math.Phys. 123 17-52


For something on uniqueness see:


B.Schmidt 1991 On the uniqueness of boundaries at infinity of asymptotically flat spacetimes Class.Quant.Grav. 8 1491-1504


Existence theorems

D.Christodoulou and S.Klainerman 1993 The global nonlinear stability of Minkowski space Princeton Mathematical Series 41 PUP


(I have used the 1999 review article of Christodoulou in Surveys in Differential Geometry VI: Essays on Einstein Manifolds eds C.LeBrun and M.Wang , International Press.)


For the work of Helmut Friedrich, see his lectures here or the following recent review:


H.Friedrich 2002 Conformal Einstein evolution to appear in ‘The conformal structure of space-times: geometry, analysis, numerics’ eds J.Frauendiener and H.Friedrich


L.Anderssen and P.T.Chrusciel 1993 On hyperboloidal Cauchy data for vacuum Einstein equations and obstructions to smoothness of null infinity Phys.Rev.Lett. 70 2829-32 (references to related later work are in Friedrich 2002).


P.T.Chrusciel and E.Delay 2002 Existence of non-trivial, vacuum, asymptotically simple space-times gr-qc/0203053


The theorem of Mason on the non-existence of asymptotically-simple, algebraically-special vacuum solutions is in:


L.J.Mason 1998 Class.Quant.Grav. 15 1019-1030


Polyhomogeneous solutions

As arising from hyperboloidal data see Andersson and Chrusciel 1993 and later works, referenced in Friedrich 2002; for general theory see


P.T.Chrusciel, M.A.H.MacCallum and D.Singleton 1995 Gravitational waves in general relativity XIV: Bondi expansions and the polyhomogeneity of Scri Phil.Trans.Roy.Soc. A350 113-141


Papers of J.A.V.Kroon including:

J.A.V.Kroon  1999 Polyhomogeneity and zero-rest-mass fields with applications to the Newman-Penrose constants gr-qc/9907097

          2001 Can one detect a non-smooth null infinity? Class.Quant.Grav. 18 4311-16


Total mass

Defined in the standard sources; for proofs of positivity at null infinity see e.g. Ludvigsen and Vickers 1982  J..Phys.A15 L67-70, Horowitz and Perrry 1982 Phys.Rev.Lett.48 371-4, Reula and Tod 1984 J.Math.Phys. 25 1004-8. For uniqueness, and the case for a modified nomenclature, see


P.T.Chrusciel, J.Jezierski and M.A.H.MacCallum 1998 Uniqueness of the Trautman-Bondi mass gr-qc/9803010


Total angular momentum

There is a large literature on this, and no final consensus for the definition in the presence of radiation . See the discussion in Penrose and Rindler volume 2 (1986).


Infinity as a causal boundary

Discussed in Penrose and Rindler and in Hawking and Ellis. First done in

R.Geroch, E.Kronheimer and R.Penrose 1972 Ideal points of space-times Proc.Roy.Soc.Lond. A327 545-67


Newman-Penrose constants

Originally defined in

E.T.Newman and R.Penrose 1965 10 exact gravitationally conserved quantities Phys.Rev.Lett. 15 231-3


See also the discussion in Penrose and Rindler.


Defined for some polyhomogeneous solutions by Chrusciel., MacCallum and Singleton 1995 and Kroon 1999 as above.


H.Friedrich and J.Kannar 2000 Bondi systems near space-like infinity and the calculation of the NP constants J.Math.Phys.41 2195-232


See also the section  ‘Nonconservation of the NP conserved quantities’ in Misner, Thorne and Wheeler.


The Penrose Inequality

Originally derived as a prediction of his cosmic censorship hypothesis by Penrose (1973 Ann.NY.Acad.Sci.224 125-34); various special cases or related inequalities have been proved. See e.g. Ludvigsen and Vickers 1983 J.Phys.A16 3349-3353, Tod 1985 Class.Quant.Grav.2 L65-8, Malec and O Murchadha 1994 Phys.Rev.D14 6931-34, Gibbons 1997 Class.Quant.Grav.14 2905-15, Huisken and Ilmanen 1997 Int.Math.Res.Not. 20 1045-1058 and the lectures of Hubert Bray at this meeting.



                                                                                                                                                Paul Tod

                                                                                                                                                July 2002