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In this paper, we propose an approch to derive three dimensional geometriesfrom the dual CFT$_2$ entanglement entropies. We only employ the holographicprinciple and do not use any assumption about the AdS/CFT correspondence andbulk geometry. Our strategy is to use both UV and IR-like CFT$_2$ entanglemententropies to fix the bulk geodesics. By a simple trick, then the metric can beextracted from the geodesics. As examples, we derive the pure AdS$_3$ metricfrom the entanglement entropies of pure CFT$_2$, and more importantly the BTZblack hole from the entanglement entropies of finite temperature CFT$_2$.Consequently, CFT$_2$ with finite size or topological defects can be worked outby simple transformations. Following the same steps, in principle, it ispossible to derive all three dimensional (topologically distinct) holographicclassical geometries from the dual CFT$_2$ entanglement entropies.

Derive three dimensional geometries from entanglement entropies of CFT$_2$