Holographic Entanglement and Emergent Gravity

Abstract

We investigate the emergence of bulk geometry from boundary entanglement entropy in the context of holographic duality. Considering a thermal $1+1$-dimensional conformal field theory dual to the BTZ black hole, we compute four distinct holographic entanglement measures: the standard spacelike HEE, its complementary spacelike counterpart HEE$^s$, the timelike HTEE and its complementary timelike counterpart HEE$^t$. These measures probe both the exterior and interior regions of the black hole horizon. Remarkably, by taking appropriate derivatives of these entanglement entropies with respect to the boundary intervals and the turning point of the extremal surface, we reconstruct the full BTZ metric, including the radial component $g_{zz}$, without imposing the Einstein equations. The reconstruction yields the correct metric components in all regions of the geometry. We further show that in the zero-temperature limit, our results consistently reduce to the pure AdS$_3$ metric and the familiar vacuum CFT$_2$ entanglement entropies. Our findings demonstrate that entanglement entropy data alone contains sufficient information to uniquely determine the dual gravitational background, offering a concrete realization of holographic emergence where gravity arises from the quantum information structure of the boundary theory.