There exist several reviews of the subject from different
points of view, e.g. by Geroch [66], by Penrose [110], by Schmidt [133], by Newman and Tod [103], by Ashtekar [6
,
7], and by Friedrich [53,
56].
A large part of the literature on null-infinity is concerned
with ``conserved quantities''. There exist several ways to derive
the Bondi-Sachs energy-momentum expression. It can be defined in
terms of limits of integrals, called linkages [69], over spheres which approach a cut of null-infinity
, where the integrals are taken over certain vectorfields in the
physical space-time which suitably approximate the infinitesimal
generators of asymptotic symmetries. Penrose, who had
earlier [111] reexpressed the original Bondi-Sachs expressions in terms of
genuine geometric quantities at
, has also derived them from his quasi-local mass
proposal [114]. They can also be obtained by ``helicity lowering'' of the
rescaled Weyl tensor at
using a two-index asymptotic twistor [138]. Other approaches (see [72] for a review) start from a Hamiltonian or Lagrangian
formulation of the theory and derive the energy-momentum
expressions via Noether theorems or the moment-map of symplectic
geometry (see e.g. [9,
12]). These formulations also provide a framework for ``asymptotic
quantization'', a scheme which is geared towards a
scattering-matrix description for gravity. The universal
structure of
provides the necessary background structure for the definition
of a phase-space of the radiative modes of the gravitational
field and its subsequent quantization [8].
While the energy-momentum expressions all coincide, there is
still disagreement about the various angular-momentum expressions
(see e.g. the review article by Winicour [147]). This difficulty is caused by the group structure of the BMS
group which does not allow to single out a unique Lorentz
subgroup (it is obtained only as a factor group). Hopefully these
discrepancies will be resolved once the structure of the
gravitational fields at
is completely understood.
All the ``conserved quantities'' are associated with a
(space-like) cut of null-infinity which is used for evaluation of
the surface integrals, and an infinitesimal generator of the
asymptotic symmetry group used in defining the integrand. They
are not conserved in a strict sense because they depend on the
cut. The prime example is again the Bondi-Sachs energy-momentum,
which obeys the famous Bondi-Sachs mass-loss formula which
relates the values of the energy-momentum at two given cuts with
a negative definite ``flux integral'' over the part of
between the two cuts.
Furthermore, there exist the somewhat mysterious
Newman-Penrose constants [102], five complex quantities which are also defined by surface
integrals over a cut of
. In contrast to the previous conserved quantities, the NP
constants are absolutely conserved in the sense that they do not
depend on the particular cut which is used for the evaluation of
the integrals. In space-times which have a regular point
, the NP constants turn out to be the value of the gravitational
field at
. If
is singular, then the NP constants are still well-defined,
although now they should probably be considered as the value of
the gravitational field at an ideal point
. Other interpretations relate them to certain combinations of
multipole moments of the gravitational field [102,
118
]. People have tried to give an interpretation of the NP
constants in terms of a Lagrangian or symplectic framework [71,
70,
122], but these results are still somewhat unsatisfactory. Very
recently, Friedrich and Kánnár [58
] were able to connect the NP constants defined at null-infinity
to initial data on a space-like asymptotically Euclidean
(time-symmetric) hypersurface.
Finally, we want to mention the recent formulation of general
relativity as a theory of null hypersurfaces, see [89]. This theory has its roots in the observation that one can
reconstruct the points of Minkowski space-time from structures
defined on null-infinity. The future light-cone emanating from an
arbitrary point in Minkowski space-time is a shear-free null
hypersurface intersecting
in a cut. The shear-free property of the light-cone translates
into the fact that the cut itself is given as a solution of a
certain differential equation, the ``good cut equation'' on
. Conversely, it was realized that
in flat space
the solution space of the good cut equation is isometric to
Minkowski space-time (in particular, it carries a flat metric).
Attempts to generalize this property led to Newman's
-space construction [99] which associates with each (complexified) asymptotically flat
and (anti-)self-dual space-time a certain complex
four-dimensional manifold which carries a Ricci-flat metric. It
is obtained as the solution space of the complex good cut
equation. Trying to avoid the unphysical complexification has
finally led to the above mentioned null surface formulation of
general relativity.
At this point the connection to Penrose's theory of twistors
is closest. Newman's
-spaces were the motivation for the ``non-linear graviton''
construction [112] which associates with each anti self-dual vacuum space-time a
certain three-dimensional complex manifold. The interpretation of
these manifolds at the time was that they should provide the
one-particle states of the gravitational field in a future
quantum theory of gravity. For a recent review of twistor theory,
we refer to [115]. The non-linear gravitons themselves have led to remarkable
developments in pure mathematics (see e.g. the contributions
in [83]).
![]() |
Conformal Infinity
Jörg Frauendiener http://www.livingreviews.org/lrr-2000-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |