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.
