Carrollian geometry is the c → 0 limit of relativity. Where Galilean physics takes c → ∞ (infinite speed of light, no causal barrier), Carrollian physics takes the opposite extreme: the speed of light goes to zero, and causal structure collapses to a point. Nothing can propagate. Every event is causally disconnected from every other. The geometry is named after Lewis Carroll — in this world, you run as fast as you can to stay in the same place.
This sounds like a mathematical curiosity, but Carrollian geometry appears naturally at null surfaces — black hole horizons, null infinity, cosmological horizons — where one lightlike direction contracts to zero. Understanding field theory on Carrollian backgrounds is relevant for flat-space holography and BMS symmetry.
Andrew James Bruce (arXiv:2603.07081, March 2026) asks a concrete question: can you build a propagating scalar field theory on the Carrollian plane? Not as a limit of a Lorentzian theory (which inherits propagation from its relativistic ancestor) but intrinsically — starting from Carrollian geometry and building upward.
The answer is no. The extended Carrollian symmetry group includes supertranslations — an infinite-dimensional family of transformations that generalize ordinary translations. Invariance under supertranslations (which include boosts as a special case) forces the energy density to be static and the momentum density to vanish. On shell, the field cannot propagate. The symmetry kills motion.
This is not a technical obstruction or a failure of imagination. The proof is that any Lagrangian minimally coupled to the Carrollian connection, invariant under the full symmetry group, produces frozen dynamics. The class of allowed Lagrangians is large — any function independent of spacetime coordinates works. But the symmetry constraint is stronger than the Lagrangian freedom. No choice of dynamics can overcome it.
To get propagation in Carrollian physics, you must either break the supertranslation symmetry, add multiple fields, or couple non-minimally to the geometry. The single, minimally coupled scalar is the simplest possible field theory, and it's exactly the one that the symmetry forbids from moving. The geometry permits existence but not dynamics. The field can be. It cannot go.
Bruce, "Frozen Motion: Why Single Carrollian Scalars Cannot Propagate," arXiv:2603.07081 (March 2026).