Entropic route to Brown-York tensor: A unified framework for null and timelike hypersurfaces

Building on Padmanabhan's entropy functional, originally introduced to derive Einstein's equations and highlight the emergent nature of gravity, we demonstrate its robustness in a broader context. Using the same entropy density, we show that the Brown-York (BY) tensor arises naturally as the projection of the canonical momentum conjugate to the normal vectors on the relevant hypersurface, thereby providing a common construction applicable to both timelike and null hypersurfaces. This perspective also offers insight into the structural differences of the null BY tensor, including its non-symmetric character. We further extend the analysis to scalar-tensor theories, showing that the entropy-based formulation reproduces the expected equations of motion along with the corresponding BY tensor, and, clarifies its non-conservation in the presence of additional scalar field which is non-minimally coupled. Our results provide a coherent variational interpretation of quasi-local gravitational quantities and reveal a common underlying structure linking bulk dynamics and boundary momentum.