3 The Regular Conformal Field 2 General Background2.4 Example: Minkowski space

2.5 Going further 

As we indicated already in the introduction, the amount of results and developments related to the conformal structure of space-times and, in particular, to conformal infinity is overwhelming. We take the opportunity to refer to several other developments which are not treated here in detail.

There exist several reviews of the subject from different points of view, e.g. by Geroch [66Jump To The Next Citation Point In The Article], by Penrose [110], by Schmidt [133], by Newman and Tod [103], by Ashtekar [6Jump To The Next Citation Point In The Article, 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 tex2html_wrap_inline3847, 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 tex2html_wrap_inline3847, 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 tex2html_wrap_inline3847 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 tex2html_wrap_inline3905 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 tex2html_wrap_inline3683 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 tex2html_wrap_inline3847 between the two cuts.

Furthermore, there exist the somewhat mysterious Newman-Penrose constants [102Jump To The Next Citation Point In The Article], five complex quantities which are also defined by surface integrals over a cut of tex2html_wrap_inline3847 . 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 tex2html_wrap_inline4134, the NP constants turn out to be the value of the gravitational field at tex2html_wrap_inline4134 . If tex2html_wrap_inline4134 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 tex2html_wrap_inline4134 . Other interpretations relate them to certain combinations of multipole moments of the gravitational field [102, 118Jump To The Next Citation Point In The Article]. 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 [58Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline3847 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 tex2html_wrap_inline3847 . 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 tex2html_wrap_inline3699 -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 tex2html_wrap_inline3699 -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]).

3 The Regular Conformal Field 2 General Background2.4 Example: Minkowski space

image Conformal Infinity
Jörg Frauendiener
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