Covariant 3+1 correspondence of the spatially covariant gravity and the degeneracy conditions

Abstract

A necessary condition for a generally covariant scalar-tensor theory to be ghostfree is that it contains no extra degrees of freedom in the unitary gauge, in which the Lagrangian corresponds to the spatially covariant gravity. Comparing with analysing the scalar-tensor theory directly, it is simpler to map the spatially covariant gravity to the generally covariant scalar-tensor theory using the gauge recovering procedures. In order to ensure the resulting scalar-tensor theory to be ghostfree absolutely, i.e., no matter if the unitary gauge is accessible, a further covariant degeneracy/constraint analysis is required. We develop a method of covariant 3+1 correspondence, which map the spatially covariant gravity to the scalar-tensor theory in 3+1 decomposed form without fixing any coordinates. Then the degeneracy conditions to remove the extra degrees of freedom can be found easily. As an illustration of this approach, we show how the Horndeski theory is recovered from the spatially covariant gravity. This approach can be used to find more general ghostfree scalar-tensor theory.

Higher derivative scalar tensor theory in unitary gauge

“…In principle, SCG theories can also generate GST theories. The correspondence among GST, SCG and finally also with Lagrangian's 3+1 decomposed form recently has been explored in [40][41][42]. In [41] studying the ghost-free scalar-tensor theory, considering a time-like scalar field as well as derivatives and Riemann tensors of up to quadratic order and also in [42] derived a method in which SGT can be decomposed into covariant 3+1 form without requiring any coordinates and by this method, the Horndeski theory can be recovered from the spatially covariant gravity.…”

Ghost free theory in unitary gauge: a new candidate

“…Recently, effective field theory (EFT) of perturbations on an arbitrary background metric with a time-like scalar profile has been studied in [35,35,36]. Another theory studied under the unitary gauge is the spatially covariant theory of gravity (SCG) [37][38][39][40][41][42][43][44][45][46]. Some examples of SCGs include Horava Lifshitz gravity [47,48], Cuscuton theory [49,50], and its generalization [51,52].…”

Polarized gravitational waves in the parity violating scalar-nonmetricity theory

“…The CS modified gravity can be extended by including higher order derivatives of the scalar field [37], which is proved to be ghostfree on a cosmological background. While on the cosmological background or generally when the scalar field is timelike, the scalar-tensor theory is equivalent to a metric theory respecting only the spatial covariance, which we refer to as the spatially covariant gravity (SCG) [38][39][40][41][42]. The well-studied effective field theory (EFT) of inflation [43,44] and as well as the non-projectable version of Hořava gravity [45,46] can be viewed as subclasses of the SCG theories.…”