The Brown-York quasilocal energy is traditionally obtained from the Hamilton-Jacobi formulation of general relativity as the boundary contribution to the gravitational Hamiltonian. In the present work we investigate whether the same boundary geometry also appears in the dynamics of freely falling particles. Starting from the autoparallel equation, we derive the normal projection of the geodesic equation with respect to the Brown-York boundary foliation. The resulting expression naturally separates into a geometric contribution involving the combination $k-\mathrm{\Theta }$, a kinematical term associated with the particle motion, and a mixed observer/particle contribution determined by the foliation geometry. The combination $k-\mathrm{\Theta }$ is free of the reference-subtraction ambiguity inherent in integrated quasilocal energies. For a time-symmetric foliation $K_{ij}=0$, and for geodesic configurations stationary with respect to the observer congruence, ${\pounds }_u\beta =0$, the mixed observer/particle term vanishes. In Schwarzschild spacetime the remaining equation reproduces the Newtonian inverse-square law in the weak-field limit. The result may be interpreted geometrically as arising from the mismatch between the spatial embedding of a closed two-surface and the embedding of its timelike evolution in spacetime. We further derive the normal variation of the referenced Brown-York energy and show that, for Schwarzschild spheres, it satisfies the exact relation $-d{E}_{\mathrm{BY}}/ds=E_{\mathrm{BY}}N(1+N{)}^{-1}(k-\mathrm{\Theta })$, thereby relating the acceleration term entering the projected geodesic equation directly to the radial variation of the integrated quasilocal energy. The analysis does not derive the geodesic equation from the Brown-York energy functional. Rather, it demonstrates that the same boundary geometry underlying Brown-York quasilocal energy also governs the normal component of free-fall dynamics, thereby establishing a direct geometrical connection between Hamiltonian boundary geometry and geodesic motion.

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• General relativity equations & solutions
• General relativity formalism