A conformal approach to matter coupled Aristotelian gravity

A bstract We show how to take the first step in the conformal program for constructing general matter couplings to Aristotelian gravity with arbitrary p -brane foliation. For this purpose we extend the p -brane Aristotelian algebra to the direct sum of two conformal algebras: one with Minkowski signature for the longitudinal directions and a second one with Euclidean signature for the transverse directions. For some cases, it is sufficient to work with a subalgebra of this conformal extension that, instead of two dilatations that are isotropic in either the longitudinal or transverse directions, contains a single dilatation that acts on the longitudinal and transverse directions in an an-isotropic way. Using this conformal extension we show how different electric and magnetic versions of Aristotelian gravity can be constructed that all have the distinguishing property that they are not invariant under any (Galilean or Carrollian) boost symmetry. We next consider several matter couplings both for quadratic-derivative models as well as for some higher-derivative models that have recently been considered in connection with studies on fractons.