8.6 New interpretations and future directions

Throughout this review, we have focused on classical general relativity as expressed in the Newman–Penrose (or spin-coefficient) formalism; yet despite this highly classical setting, the geometric structures discussed here bear a striking resemblance to ideas from more ambitious areas of theoretical physics. In particular, the relationship between real, twisting asymptotically shear-free NGCs in asymptotically flat spacetime and complex, twist-free NGCs in (complex) ℋ-space appears to form a dual pair. The complexified asymptotic boundary ℑ+ℂ acts as the translator between these two descriptions: from ℑ+ ℂ we determine the complexified congruence in ℋ-space via the angle fields L and &tidle; L or the real congruence (in the real spacetime) via L and -- L. Imposing reality conditions in the former case gives an open world sheet (the ‘ℋ-space ribbon’ discussed in the text) for the NGC’s complex source. In the real case, the twisting congruence’s caustic set in Minkowski space (which is interpreted as its real source) forms a closed loop propagating in real time [7Jump To The Next Citation Point], or a closed world sheet. In the asymptotically flat case, we cannot follow twisting congruences back to their real caustic set, and they must be represented by the dual ‘ribbon’.

In both cases, we see that the congruence’s source is a structure which has a natural interpretation as a classical string, with the boundary ℑ+ ℂ interpolating between the two descriptions. This is suggestive of the so-called ‘holographic principle’, which aims to equate a theory in a d-dimensional compact space with another theory defined on its d − 1-dimensional boundary [77, 17]. In practice, this can allow one to interpolate between a ‘physical’ theory in one space and a ‘dual’ theory living on its boundary (or vice versa). In our case, the ‘physical’ information is the real, twisting NGC in the asymptotically flat spacetime; ℑ+ ℂ acts as a lens into ℋ-space, which serves as the virtual image space where physical data (such as the mass, linear momentum, angular momentum, etc.) is computed by the methods reviewed here. Hence, it is tempting to refer to + ℑ ℂ as the ‘holographic screen’ for some application of the holographic principle to general relativity. The presence of classical string-like structures on both sides of the duality makes such a possibility all the more intriguing.

This should be contrasted with the most well-known instance of the holographic principle: the AdS/CFT correspondence [47, 78, 9]. Here, the AdS-boundary acts as the holographic screen between a type IIB string theory in AdS5 × S5 (the virtual image space) and maximally supersymmetric Yang–Mills theory in real four-dimensional Minkowski spacetime (other versions exist, but all involve some supersymmetry). It is interesting that we appear to be describing an instance of the holographic principle that requires no supersymmetry, although ’t Hooft’s original work relating the planar limit of gauge theories to string-type theories did not use supersymmetry either [76]. In ’t Hooft’s work, an extra dimension for string propagation enters the picture due to anomaly cancellation in the same way that the extra dimensions of AdS5 × S5 allow for anomaly cancellation in a full supersymmetric string theory. In our investigations, one can think of the analytic continuation of ℑ+ to ℑ+ ℂ in the same manner: instead of canceling a (quantum) anomaly, the ‘extra dimensions’ arising from analytic continuation allow us to reinterpret the real twisting NGC in terms of a simpler geometric structure, namely the complex light-cones in ℋ-space.

It is worth noting that this is not the first time that there has been a suggested connection between structures in asymptotically flat spacetimes and the holographic principle. Most prior studies have attempted to understand such a duality in terms of the BMS group, which serves as the symmetry group of the asymptotic boundary [10]. Loosely speaking, these studies take their cue more directly from the AdS/CFT correspondence: by studying fields living on ℑ+ which carry representations of the BMS group, one hopes to reconstruct the full interior of the spacetime ‘holographically’. It would be interesting to see how, if at all, our methodology relates to this program of research.

Additionally, as we have mentioned throughout this review (and further elaborated in Appendix A), the nature of many of the objects studied here is highly twistorial. This is essentially because ℋ-space is a complex vacuum spacetime equipped with an anti-self-dual metric, and hence possesses a curved twistor space by Penrose’s nonlinear graviton construction [32]. It would be interesting to know if our procedure for identifying the complex center of mass (and/or charge) in an asymptotically flat spacetime could be phrased purely in terms of twistor theory. Furthermore, the past several years have seen dramatic progress in using twistor theory to study gauge theories and their scattering amplitudes [1]. These techniques may provide the most promising route for connecting our work with any quantized version of gravity, as illustrated by the recent twistorial derivation of the tree-level MHV graviton scattering amplitude [48].

While the interpretations we have suggested here are far from precise, they suggest a myriad of further directions which research in this area could take. It would be truly fascinating for a topic as old as asymptotically flat spacetimes to make meaningful contact with ambitious new areas of mathematics and physics such as holography or twistorial scattering theory.

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