Space-time symmetry breaking on non-geodesic leaves and a new form of matter

We examine the permanent damage caused by the historical breakdown of full diffeomorphism invariance induced by a foliation. We focus on the case where the foliation is allowed to be non-geodesic after the interactions with the foliation switch off. Gravity and other forms of matter recover full diffeomorphism invariance only at the expense of introducing a new matter-like component, carrying the non-vanishing Hamiltonian (and momentum, as it turns out) left over from the violating past interactions. This matter form must be stress-free in the preferred frame; this is the only way a matter action can mimic the evolution of the leftover Hamiltonian (and momentum) driven by the Dirac hypersurface deformation algebra. Hence, if the preferred frame is non-geodesic, the equivalent matter component must have energy and a momentum current in this frame, but still no spatial stresses: an unusual form of "matter". It is equivalent to a fluid with anisotropic stress in some regimes, reducing to dust in others, or even displaying completely new features in extreme situations. Its stress energy tensor is conserved. We provide two examples based on accelerated frames: Rindler space-time and the canonical Schwarzchild frame.