Gamma ray burst delay times probe the geometry of momentum space
  We study the application of the recently proposed framework of relative locality to the problem of energy dependent delays of arrival times of photons that are produced simultaneously in distant events such as gamma ray bursts. Within this framework, possible modifications of special relativity are coded in the geometry of momentum space. The metric of momentum space codes modifications in the energy momentum relation, while the connection on momentum space describes possible non-linear modifications in the laws of conservation of energy and momentum. In this paper, we study effects of first order in the inverse Planck scale, which are coded in the torsion and non-metricity of momentum space. We find that time delays of order Distance * Energies/m_p are coded in the non-metricity of momentum space. Current experimental bounds on such time delays hence bound the components of this tensor of order 1/m_p. We also find a new effect, whereby photons from distant sources can appear to arrive from angles slightly off the direction to the sources, which we call gravitational lensing. This is found to be coded into the torsion of momentum space. 
Laurent Freidel and Lee Smolin
hep-th gr-qc