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In the framework of spatially covariant gravity, it is natural to extend a gravitational theory by putting the lapse function $N$ and the spatial metric $h_{ij}$ on an equal footing. We find two sufficient and necessary conditions for ensuring two physical degrees of freedom (DoF) for the theory with the lapse function being dynamical by Hamiltonian analysis. A class of quadratic actions with only two DoF is constructed. In the case that the coupling functions depend on $N$ only, we find that the spatial curvature term cannot enter the Lagrangian and thus this theory possesses no wave solution and cannot recover general relativity (GR). In the case that the coupling functions depend on the spatial derivatives of $N$, we perform a spatially conformal transformation on a class of quadratic actions with nondynamical lapse function to obtain a class of quadratic actions with $dot{N}$. We confirm this theory has two DoF by checking the two sufficient and necessary conditions. Besides, we find that a class of quadratic actions with two DoF can be transformed from GR by disformal transformation.
We investigate the correspondence between generally covariant higher derivative scalar-tensor theory and spatially covariant gravity theory. The building blocks are the scalar field and spacetime curvature tensor together with their generally covaria
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 l
We revisit the problem of building the Lagrangian of a large class of metric theories that respect spatial covariance, which propagate at most two degrees of freedom and in particular no scalar mode. The Lagrangians are polynomials built of the spati
We study the entropy of the black hole with torsion using the covariant form of the partition function. The regularization of infinities appearing in the semiclassical calculation is shown to be consistent with the grand canonical boundary conditions
The one-loop partition function of the $f(R,R_{mu u}R^{mu u})$ gravity theory is obtained around AdS$_4$ background. After suitable choice of the gauge condition and computation of the ghost determinant, we obtain the one-loop partition function of t