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 interpolating between the two descriptions. This is suggestive of
the so-called ‘holographic principle’, which aims to equate a theory in a
-dimensional compact space
with another theory defined on its
-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;
acts as a lens into
-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
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 (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
allow for anomaly
cancellation in a full supersymmetric string theory. In our investigations, one can think of the
analytic continuation of
to
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
-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 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 -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.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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