Linearized 3D gravity with dust

Three-dimensional gravity coupled to pressureless dust is a field theory with 1 local degree of freedom. In the canonical framework, the dust-time gauge encodes this field in the metric. We find that its dynamics, up to diffeomorphism flow, is independent of spatial derivatives and is therefore ultralocal. We study this feature further by analyzing the linearized equations of motion about flat and (anti-)de Sitter backgrounds, and show that this field may be viewed as either a traceless or a transverse mode.