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Entropy and energy are found to be closely tied on our quest for quantumgravity. We point out an interesting connection between the recently proposedouter entropy, a coarse-grained entropy defined for a compact spacetime domainmotivated by the holographic duality, and the Bartnik-Bray quasilocal mass longknown in the mathematics community. In both scenarios, one seeks an optimalspacetime fill-in of a given closed, connected, spacelike, codimension-twoboundary. We show that for an outer-minimizing mean-convex surface, theBartnik-Bray inner mass matches exactly with the irreducible mass correspondingto the outer entropy. The equivalence implies that the area laws derived fromthe outer entropy are mathematically equivalent as the monotonicity property ofthe quasilocal mass. It also gives rise to new bounds between entropy and thegravitational energy, which naturally gives the gravitational counterpart toWall's ant conjecture. We also observe that the equality can be achieved in aconformal flow of metrics, which is structurally similar to the Ceyhan-Faulknerproof of the ant conjecture. We compute the small sphere limit of the outerentropy and it is proportional to the bulk stress tensor as one would expectfor a quasilocal mass. Lastly, we discuss some implications of taking quantummatter into consideration in the semiclassical setting.

Outer entropy = Bartnik-Bray inner mass, and the gravitational ant conjecture