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S.Daedalus

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In 1995, Ted Jacobson derived Einstein's equations from thermodynamics across a horizon. Roughly, he showed that if the horizon's entropy is given by the Bekenstein-Hawking formula, then the second law of thermodynamics implies that Einstein's equations hold.

More recently, it's become clear that there is a deep relationship between entanglement and gravity in the AdS/CFT duality---roughly, the entanglement entropy between a patch of the CFT and its surroundings is given by the area of a minimal AdS-surface that has the same boundary as the patch (Ryu-Takayanagi formula). In fact, it can be shown, again with a thermodynamical argument, that for small perturbations around the vacuum of the CFT, the corresponding AdS obeys the Einstein equations to linear order.

Now these two things seem too close not to be related, in particular if you follow the suggestion that the Bekenstein-Hawking entropy might in fact itself be entanglement entropy, as first proposed by Srednicki and Bombelli et al. (This is not without its own problems, as the entanglement entropy, while it does follow an area law, diverges at the horizon, but there may be ways around that.)

So if this can be made sense of, both approaches seem to relate the Einstein equations to entanglement entropy and its dynamics. But what's puzzling me is that they work on different stages, so to speak: Jacobson's approach gets the Einstein equations of 3+1-dim spacetime by considering horizons within that spacetime; the more modern approach gets them by considering entanglement across boundaries that themselves exist on the boundary of an AdS-spacetime, that is, the entangling surface has (at least) one less dimension. Jacobson uses a surface whose area is directly proportional to the entanglement across that surface, while van Raamsdonk et al. use, via the Ryu-Takayanagi prescription, a surface in the boundary CFT whose entanglement is proportional to the area of a surface in the higher-dimensional AdS-spacetime.

So, where does that difference come from? How can you get, apparently, the Einstein equations within the same space from entanglement across a horizon, and in a space of higher dimensionality? Or is my whole outlook on this just muddled, and the two things really aren't related at all?