In this proceedings for the MG14 conference, we discuss the construction of a phenomenology of Planck-scale effects in curved spacetimes, underline a few open issues and describe some perspectives for the future of this research line.
Quantum gravity phenomenology suggests an effective modification of the general relativistic dispersion relation of freely falling point particles caused by an underlying theory of quantum gravity. Here we analyse the consequences of modifications of
the general relativistic dispersion on the geometry of spacetime in the language of Hamilton geometry. The dispersion relation is interpreted as the Hamiltonian which determines the motion of point particles. It is a function on the cotangent bundle of spacetime, i.e. on phase space, and determines the geometry of phase space completely, in a similar way as the metric determines the geometry of spacetime in general relativity. After a review of the general Hamilton geometry of phase space we discuss two examples. The phase space geometry of the metric Hamiltonian $H_g(x,p)=g^{ab}(x)p_ap_b$ and the phase space geometry of the first order q-de Sitter dispersion relation of the form $H_{qDS}(x,p)=g^{ab}(x)p_ap_b + ell G^{abc}(x)p_ap_bp_c$ which is suggested from quantum gravity phenomenology. We will see that for the metric Hamiltonian $H_g$ the geometry of phase space is equivalent to the standard metric spacetime geometry from general relativity. For the q-de Sitter Hamiltonian $H_{qDS}$ the Hamilton equations of motion for point particles do not become autoparallels but contain a force term, the momentum space part of phase space is curved and the curvature of spacetime becomes momentum dependent.
It was recently realized that Planck-scale momentum-space curvature, which is expected in some approaches to the quantum-gravity problem, can produce dual-curvature lensing, a feature which mainly affects the direction of observation of particles emi
tted by very distant sources. Several gray areas remain in our understanding of dual-curvature lensing, including the possibility that it might be just a coordinate artifact and the possibility that it might be in some sense a by product of the better studied dual-curvature redshift. We stress that data reported by the IceCube neutrino telescope should motivate a more vigorous effort of investigation of dual-curvature lensing, and we observe that studies of the recently proposed $rho$-Minkowski noncommutative spacetime could be valuable from this perspective. Through a dedicated $rho$-Minkowski analysis, we show that dual-curvature lensing is not merely a coordinate artifact and that it can be present even in theories without dual-curvature redshift.
We use our previously developed identification of dispersion relations with Hamilton functions on phase space to locally implement the $kappa$-Poincare dispersion relation in the momentum spaces at each point of a generic curved spacetime. We use thi
s general construction to build the most general Hamiltonian compatible with spherical symmetry and the Plank-scale-deformed one such that in the local frame it reproduces the $kappa$-Poincare dispersion relation. Specializing to Planck-scale-deformed Schwarzschild geometry, we find that the photon sphere around a black hole becomes a thick shell since photons of different energy will orbit the black hole on circular orbits at different altitudes. We also compute the redshift of a photon between different observers at rest, finding that there is a Planck-scale correction to the usual redshift only if the observers detecting the photon have different masses.
The covariant understanding of dispersion relations as level sets of Hamilton functions on phase space enables us to derive the most general dispersion relation compatible with homogeneous and isotropic spacetimes. We use this concept to present a Pl
anck-scale deformation of the Hamiltonian of a particle in Friedman-Lema^itre-Robertson-Walker (FLRW) geometry that is locally identical to the $kappa$-Poincare dispersion relation, in the same way as the dispersion relation of point particles in general relativity is locally identical to the one valid in special relativity. Studying the motion of particles subject to such Hamiltonian we derive the redshift and lateshift as observable consequences of the Planck-scale deformed FLRW universe.
Momentum-space curvature, which is expected in some approaches to the quantum-gravity problem, can produce dual redshift, a feature which introduces energy dependence of the travel times of ultrarelativistic particles, and dual lensing, a feature whi
ch mainly affects the direction of observation of particles. In our recent arXiv:1605.00496 we explored the possibility that dual redshift might be relevant in the analysis of IceCube neutrinos, obtaining results which are preliminarily encouraging. Here we explore the possibility that also dual lensing might play a role in the analysis of IceCube neutrinos. In doing so we also investigate issues which are of broader interest, such as the possibility of estimating the contribution by background neutrinos and some noteworthy differences between candidate early neutrinos and candidate late neutrinos.
Two recent publications have reported intriguing analyses, tentatively suggesting that some aspects of IceCube data might be manifestations of quantum-gravity-modified laws of propagation for neutrinos. We here propose a strategy of data analysis whi
ch has the advantage of being applicable to several alternative possibilities for the laws of propagation of neutrinos in a quantum spacetime. In all scenarios here of interest one should find a correlation between the energy of an observed neutrino and the difference between the time of observation of that neutrino and the trigger time of a GRB. We select accordingly some GRB-neutrino candidates among IceCube events, and our data analysis finds a rather strong such correlation. This sort of studies naturally lends itself to the introduction of a false alarm probability, which for our analysis we estimate conservatively to be of 1%. We therefore argue that our findings should motivate a vigorous program of investigation following the strategy here advocated.
We present the first detailed study of the kinematics of free relativistic particles whose symmetries are described by a quantum deformation of the de Sitter algebra, known as $q$-de Sitter Hopf algebra. The quantum deformation parameter is a functio
n of the Planck length $ell$ and the de Sitter radius $H^{-1}$, such that when the Planck length vanishes, the algebra reduces to the de Sitter algebra, while when the de Sitter radius is sent to infinity one recovers the $kappa$-Poincare Hopf algebra. In the first limit the picture is that of a particle with trivial momentum space geometry moving on de Sitter spacetime, in the second one the picture is that of a particle with de Sitter momentum space geometry moving on Minkowski spacetime. When both the Planck length and the inverse of the de Sitter radius are non-zero, effects due to spacetime curvature and non-trivial momentum space geometry are both present and affect each other. The particles motion is then described in a full phase space picture. We find that redshift effects that are usually associated to spacetime curvature become energy-dependent. Also, the energy dependence of particles travel times that is usually associated to momentum space non-trivial properties is modified in a curvature-dependent way.
We describe the Hamilton geometry of the phase space of particles whose motion is characterised by general dispersion relations. In this framework spacetime and momentum space are naturally curved and intertwined, allowing for a simultaneous descript
ion of both spacetime curvature and non-trivial momentum space geometry. We consider as explicit examples two models for Planck-scale modified dispersion relations, inspired from the $q$-de Sitter and $kappa$-Poincare quantum groups. In the first case we find the expressions for the momentum and position dependent curvature of spacetime and momentum space, while for the second case the manifold is flat and only the momentum space possesses a nonzero, momentum dependent curvature. In contrast, for a dispersion relation that is induced by a spacetime metric, as in General Relativity, the Hamilton geometry yields a flat momentum space and the usual curved spacetime geometry with only position dependent geometric objects.
This proceeding is based on a talk prepared for the XIII Marcell Grossmann meeting. We summarise some results of work in progress in collaboration with Giovanni Amelino-Camelia about momentum dependent (Rainbow) metrics in a Relative Locality framewo
rk and we show that this formalism is equivalent to the Hamiltonian formalization of Relative Locality obtained in arXiv:1102.4637.
