  When physics is expressed in a way that is independent of local choices of
unit systems, Riemannian geometry is replaced by conformal geometry. Moreover
masses become geometric, appearing as Weyl weights of tractors (conformal
multiplets of fields necessary to keep local unit invariance manifest). The
relationship between these weights and masses is through the scalar curvature.
As a consequence mass terms are spacetime dependent for off-shell gravitational
backgrounds, but happily constant for physical, Einstein manifolds.
Unfortunately this introduces a naturalness problem because the scalar
curvature is proportional to the cosmological constant. By writing down tractor
stress tensors (multiplets built from the standard stress tensor and its first
and second derivatives), we show how back-reaction solves this naturalness
problem. We also show that classical back-reaction generates an interesting
potential for scalar fields. We speculate that a proper description of how
physical systems couple to scale, could improve our understanding of
naturalness problems caused by the disparity between the particle physics and
observed, cosmological constants. We further give some ideas how an ambient
description of tractor calculus could lead to a Ricci-flat/CFT correspondence
which generalizes the AdS side of Maldacena's duality to a Ricci-flat space of
one higher dimension.
