Hints from non-commutative geometry
By Marco de Cesare, Mairi Sakellariadou, and Patrizia Vitale
It is often argued that modifications of general relativity can potentially explain the properties of the gravitational field on large scales without the need to postulate a (so far unobserved) dark sector. However, the theory space seems to be virtually unconstrained. One may then legitimately ask whether there is any guiding principle —such as symmetry— that can be invoked to build such a modified gravity theory and ground it in fundamental physics. We also know that the classical picture of spacetime as a Riemannian manifold must be abandoned at the Planck scale. The question then arises as to what kind of geometric structures may replace it, and if there are any novel gravitational degrees of freedom that they bring along. Importantly, one may ask whether there are any potentially observable effects away from the experimentally inaccessible Planck regime. These questions are crucial both from the point of view of quantum gravity and for model building in cosmology; trying to answer them will help us in the attempt to bridge the gap between the two fields, and could have far-reaching implications for our understanding of the quantum structure of spacetime.
In our recent work, we consider a modification of (tetrad) general relativity based on spacetime non-commutativity. In fact, the existence of a fundamental length scale implies that there is an intrinsic limitation to the possibility of localising spacetime events with extreme precision, which justifies modelling quantum spacetime as a non-commutative manifold. The particular type of non-commutative structure that we consider is obtained via a ‘twist’ deformation of the differential geometry (roughly speaking, this can be regarded as a tool generalising the notion of a ★-product to tensor fields). The non-commutative deformation has non-trivial consequences: it demands an extension of the gravitational sector by simple arguments of mathematical consistency of the theory. In fact, by insisting on the gauge principle, one is naturally lead to enlarge the local Lorentz symmetry and double the tetrad degrees of freedom. Thus, the geometric framework is necessarily much richer than Riemmanian geometry. In particular, the theory is bimetric and it also entails torsion and (Weyl) non-metricity.
There are essentially two independent sources of new physical effects in this framework. In first place, the ‘twist’ introduces corrections to general relativity that become more and more relevant as the non-commutativity scale is approached. In second place, the commutative limit yields a modified gravity theory, which extends general relativity with novel gravitational degrees of freedom and interactions. In particular, we show in our paper that the underlying non-commutative structure only allows interactions of some specific form between the two tetrads; remarkably, in the commutative limit such interactions turn out to correspond to partially massless bigravity. Moreover, there is an emergent conformal symmetry in the commutative limit, and a related gauge field.
Besides the usual gauge symmetries of general relativity —suitably generalised to the non-commutative setting— our model displays new duality symmetries, stemming from the doubling of the tetrad degrees of freedom. Supplementing the action by the Holst term, the dynamics is also invariant under a generalisation of the standard Hodge-duality. The introduction of the Holst term is particularly relevant since it allows for a great simplification of the dynamics when a specific choice is made for the value of the Barbero-Immirzi parameter, corresponding to self-dual variables.
Going beyond the geometric setting of general relativity seems to be necessary in order to gain a better understanding of gravity at a fundamental level. From this point of view, non-commutative geometry represents (in its various incarnations) a very general framework that is able to capture non-classical properties of spacetime, which are expected to play a dominant role in extreme regimes, such as the very early Universe. Moreover, as exemplified in our paper, spacetime non-commutativity can motivate, or even require, an extended gravitational sector. This is promising of many developments and gives us hints of a fruitful interplay between quantum gravity and cosmological model building.
 M. de Cesare, M. Sakellariadou and P. Vitale, “Noncommutative gravity with self-dual variables” , Class. Quantum Grav. 35 215009 (2018)
 P. Aschieri and L. Castellani, “Noncommutative D=4 gravity coupled to fermions” , JHEP 0906 (2009) 086
 S. Doplicher, K. Fredenhagen and J. E. Roberts, “The Quantum structure of space-time at the Planck scale and quantum fields” , Commun. Math. Phys. 172 (1995) 187-220
Read the full article in Classical and Quantum Gravity:
Noncommutative gravity with self-dual variables
Marco de Cesare et al 2018 Class. Quantum Grav. 35 215009
About the authors:
Marco de Cesare is a postdoctoral research fellow at the University of New Brunswick and part of the UNB gravity group. His main research interests are in quantum gravity and cosmology.
Mairi Sakellariadou is a professor of theoretical physics at King’s College London and a member of the LIGO Scientific Collaboration. She is also Chair of the Gravitational Physics Division of the European Physical Society. Link to institutional webpage.
Patrizia Vitale is associate professor of Theoretical Physics at the University of Naples Federico II and research associate at INFN. Link to institutional web page.
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