  A general framework of the novel matter coupling in the Einstein gravity is
introduced. We firstly prove that a class of theories whose Hamiltonian
constraint is given by an arbitrary function $f(H_g)$, where $H_g$ is the
Hamiltonian constraint of general relativity (GR), is equivalent to GR in the
vaccum. A novel Jordan frame is defined when GR is rewritte in terms of one of
its equivalents in this class. The transformation between the novel Jordan
frame and the Eintein frame is a redifinition of lapse. We discuss two types of
consistency condition for matter to couple to Einstein gravity in the novel
Jordan frame. The Type I consistency condition is found by demanding all
constraints to be first class; the type II consistency condition is found by
demanding the algebra is closed when matter minimally couples to gravity in the
novel Jordan frame, which an additional gauge condition is required. We discuss
the cosmological implications from these two types of matter coupling.
