There exist several reviews of the subject from
different points of view, e.g. by Geroch [79], by
Penrose [127], by
Schmidt [150], by Newman
and Tod [120], by
Ashtekar [7
, 8], and by
Friedrich [64, 67].
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 [82], over spheres that
approach a cut of null-infinity , where the
integrals are taken over certain vector fields in the physical
space-time that suitably approximate the infinitesimal generators
of asymptotic symmetries. Penrose, who had earlier [128] re-expressed
the original Bondi-Sachs expressions in terms of genuine geometric
quantities at
, has also derived them from his
quasi-local mass proposal [131].
They can also be obtained by “helicity lowering” of the rescaled
Weyl tensor at
using a two-index asymptotic
twistor [155]. Other approaches
(see [85] 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. [10, 13]). These
formulations also provide a framework for “asymptotic
quantization”, a scheme that 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 [9].
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 [165]). This difficulty is
caused by the group structure of the BMS group, which does not
allow one 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 [119], five complex
quantities that 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 used for the evaluation of the
integrals. In space-times that 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 [119, 135
]. People have tried
to give an interpretation of the NP constants in terms of a
Lagrangian or symplectic framework [84, 83, 139], but these
results are still somewhat unsatisfactory. Very recently, Friedrich
and Kánnár [71
] 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 [105]). 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 [116],
which associates with each (complexified) asymptotically flat and
(anti-)self-dual space-time a certain complex four-dimensional
manifold that 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 [129], 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 [132]. The non-linear
gravitons themselves have led to remarkable developments in pure
mathematics (see e.g. the contributions in [96]).