  Boundary term and Brown-York (BY) formalism, which is based on the
Hamilton-Jacobi principle, are complimentary of each other as the gravitational
actions are not, usually, well-posed. In scalar-tensor theory, which is an
important alternative to GR, it has been shown that this complementarity
becomes even more crucial in establishing the equivalence of the BY quasi-local
parameters in the two frames which are conformally connected. Furthermore,
Brown-York tensor and the corresponding quasi-local parameters are important
from two important yet different aspects of gravitational theories: black hole
thermodynamics and fluid-gravity correspondence. The investigation suggests
that while the two frames are equivalent from the thermodynamic viewpoints,
they are not equivalent from the perspective of fluid-gravity analogy or the
membrane paradigm. In addition, the null boundary term and null Brown-York
formalism are the recent developments (so far obtained only for GR) which is
non-trivial owing to the degeneracy of the null surface. In the present
analysis these are extended for scalar-tensor theory. The present analysis also
suggests that, regarding the equivalence (or inequivalence) of the two frame,
the null formalism draws the same inferences as of the timelike case, which in
turn establishes the consistency of the newly developed null Brown-York
formalism.
