# Curved spacetimes with local $\kappa$-Poincar\'e dispersion relation

## TL;DR

This paper develops a framework to incorporate the $
abla$-Poincaré dispersion relation into curved spacetimes, revealing energy-dependent photon orbits and Planck-scale corrections to redshift.

## Contribution

It introduces a general Hamiltonian construction compatible with spherical symmetry and Planck-scale deformation, applied to Schwarzschild geometry.

## Key findings

- Photon sphere becomes a thick shell with energy-dependent orbits
- Planck-scale corrections to redshift occur for observers with different masses
- Framework allows local implementation of deformed dispersion relations in curved spacetime

## Abstract

We use our previously developed identification of dispersion relations with Hamilton functions on phase space to locally implement the $\kappa$-Poincar\'e dispersion relation in the momentum spaces at each point of a generic curved spacetime. We use this 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 $\kappa$-Poincar\'e 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.