Carrollian Conformal Dynamics and Flat-Space Holography

Abstract: The conformal boundary of asymptotically flat spacetime is null and therefore carries a Carrollian rather than Lorentzian geometry. This changes the relation between symmetry, variational principles and evolution equations in an essential way: unlike a Lorentzian Levi-Civita connection, a torsion-free Carrollian connection compatible with the conformal structure contains genuine independent data. We develop a metric-affine formulation of conformal dynamics that keeps track of these connection degrees of freedom through hypermomenta. Local Carroll boosts, rotations, Weyl transformations and diffeomorphisms then imply a set of covariant evolution equations. We apply this framework to Einstein gravity in four-dimensional asymptotically flat spacetimes. Penrose's conformal compactification identifies null infinity as a conformal Carrollian manifold; the undetermined transverse symmetric part of its compatible connection is the asymptotic Bondi shear. After fixing a Bondi-van der Burg-Metzner-Sachs (BMS) frame and identifying the Carrollian momenta and hypermomenta with the Bondi data, the general Carrollian evolution equations reproduce the standard evolution equations for the mass and angular-momentum aspects. The result provides a boundary-intrinsic derivation of the gravitational evolution equations and a geometric realization of radiative degrees of freedom in flat-space holography. This is a proceedings contribution to the "Athens Workshop in Theoretical Physics: 10th Anniversary", held at the National and Kapodistrian University of Athens on December 17-19 2025.