Higher derivative scalar-tensor theory from the spatially covariant gravity: a linear algebraic analysis

Abstract

We investigate the ghostfree scalar-tensor theory with a timelike scalar field, with the derivatives of the scalar field up to the third order and with the Riemann curvature tensor up to the quadratic order. There are two kinds of linear space that are isomorphic to each other in the sense of gauge fixing/recovering procedures. One is the set of linearly independent generally covariant scalar-tensor monomials, the other is the set of linearly independent spatially covariant gravity monomials. We concentrate on the subspaces of the spatially covariant gravity, which are spanned by the linearly independent monomials built of the extrinsic curvature, the intrinsic curvature, the lapse function as well as their spatial derivatives. The vectors in these subspaces, i.e., spatially covariant gravity polynomials, automatically propagate at most three degrees of freedom. As a result, their images under the gauge recovering mappings are automatically the subspaces of the generally covariant scalar-tensor theory that propagate up to three degrees of freedom as long as the scalar field is timelike, although the higher derivatives of the scalar field are present generally. We also derive the explicit expressions for the projection matrices, which encode the mappings from the subspaces of the spatially covariant gravity to the subspaces of the generally covariant scalar-tensor theory, up to the fourth order in the total number of derivatives in the spatially covariant gravity. Our formalism and results will be useful in deriving the generally covariant higher derivative scalar-tensor theory without ghost(s).