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

Understand

We investigate the ghostfree scalar-tensor theory with a timelike scalar field, with derivatives of the scalar field up to the third order and with the Riemann tensor up to the quadratic order.

• We build two types of linear spaces.
• 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 argue that these two types of linear space are isomorphic to each other in the sense of gauge fixing/recovering procedures.