Spatially Covariant Theories of a Transverse, Traceless Graviton, Part I: Formalism

General relativity is a covariant theory of two transverse, traceless graviton degrees of freedom. According to a theorem of Hojman, Kuchar, and Teitelboim, modifications of general relativity must either introduce new degrees of freedom or violate the principle of general covariance. In this paper, we explore modifications of general relativity that retain the same number of gravitational degrees of freedom, and therefore explicitly break general covariance. Motivated by cosmology, the modifications of interest maintain spatial covariance. Demanding consistency of the theory forces the physical Hamiltonian density to obey an analogue of the renormalization group equation. In this context, the equation encodes the invariance of the theory under flow through the space of conformally equivalent spatial metrics. This paper is dedicated to setting up the formalism of our approach and applying it to a realistic class of theories. Forthcoming work will apply the formalism more generally.