Scaling Symmetry and Carrollian Gravity

We formulate matter-coupled scaling-Carroll gravity as a gauge theory and analyze its associated gravity multiplet. After fixing the scaling symmetry, the theory is governed by the trace of the extrinsic curvature, the Carroll boost symmetry, and a vector field descending from dilatation. We show that appropriate gauge choices and geometric constraints lead to distinct regimes, including dynamical Carroll gravity, Aristotelian gravity, and a fracton gauge theory coupled to Aristotelian geometry. In the fracton phase, the Carroll boost parameter plays the role of a vector-charge gauge symmetry.