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

TL;DR

This paper presents a unified entropic framework for deriving the Brown-York tensor on null and timelike hypersurfaces, extending to scalar-tensor theories and clarifying boundary-bulk relations in gravity.

Contribution

It introduces a robust entropy-based approach to obtain the Brown-York tensor for different hypersurfaces, including null cases, and extends the analysis to scalar-tensor theories.

Findings

Brown-York tensor naturally arises from entropy functional.

The approach applies to both timelike and null hypersurfaces.

Extension to scalar-tensor theories reproduces expected equations of motion.

Abstract

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.