2.5 Going further

There exist several reviews of the subject from different points of view, e.g. by Geroch [79], by Penrose [126], by Schmidt [149], by Newman and Tod [119], by Ashtekar [7, 8], and by Friedrich [63, 66].

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 [127] re-expressed the original Bondi–Sachs expressions in terms of genuine geometric quantities at , has also derived them from his quasi-local mass proposal [130]. They can also be obtained by “helicity lowering” of the rescaled Weyl tensor at using a two-index asymptotic twistor [154]. 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 [164]). 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 [118], 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 [118, 134]. People have tried to give an interpretation of the NP constants in terms of a Lagrangian or symplectic framework [84, 83, 138], 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 [115], 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 [128], 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 [131]. The non-linear gravitons themselves have led to remarkable developments in pure mathematics (see e.g. the contributions in [96]).