Nonpropagation of scalar in the deformed Hořava-Lifshitz gravity

We study the propagation of a scalar, the trace of hij in the deformed Hořava-Lifshitz gravity with coupling constant λ. It turns out that this scalar is not a propagating mode in the Minkowski spacetime background. In this work, we do not choose a gauge-fixing to identify the physical degrees of freedom and instead, make it possible by substituting the constraints into the quadratic Lagrangian.