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Holographically gluing an AdS space out of Rindler wedges

In some previous articles, I discussed Mark van Raamsdonk's clever efforts to explain how the spacetime in quantum gravity may be glued out of pieces whose individual microstates are being entangled.

Today, he and three collaborators in Vancouver apply these ideas to Rindler wedges in an AdS space:

Rindler Quantum Gravity
Bartlomiej Czech, Joanna L. Karczmarek, Fernando Nogueira, and Mark Van Raamsdonk (I know two of them and the Czech guy isn't among them, greetings to them!) consider an X-shaped wedge inside the anti de Sitter spacetime.

In the opposite spatially separated quarters of the X-separated spacetime, there are uniformly accelerating observers. So the regions accessible to them look like Rindler wedges. A funny thing is that one may give a holographic description to these causal wedges, in terms of the boundary CFT.

Normally, the boundary CFT would be defined on \(S^{d}\times \RR\). But here, the sphere is replaced by \(H^d\), the hyperbolic "Lobachevsky" space, so we deal with two copies of a holographic CFT on \(H^d\times\RR\). Each of them has lots of microstates and they may be entangled into the 2-CFT state\[

\ket{0}_{\rm global} = \frac{1}{Z} \sum_i e^{-\pi R_H E_i} \ket{E_i^L} \otimes \ket{E_i^R}

\] where the sum goes over the corresponding pairs of microstates (it may be helpful to label them more than just with energy) and \(R\) is the curvature radius of the hyperbolic space. If you trace over one of the two CFTs, you get a thermal density matrix for the other – note that the exponential gets squared so \(-\pi R_H E_i\) gets doubled to \(-2\pi R_H E_i=-\beta E_i\) which depends on the inverse temperature – as expected for Rindler observers.

I've been thinking about similar entanglement issues as well and I actually think that the formula for the state above may be proved. Just consider the space in the boundary theories. The hyperbolic space is topologically a ball which is the same thing as a hemisphere and the bilinear expression above is exactly what is needed to glue these two hemispheres into a sphere. The "half-thermal" exponential plays the role of the evolution by an imaginary amount of time which is needed to map the boundary of one of the hemispheres to the other (essentially a rotation by \(\pi\) radians and yes, that's how you can get the Southern Hemisphere out of the Northern one, if I neglect the detail that you must also shoot the Australians and replace them by the Canadians).

The Vancouver authors point out that all the microstates of the hyperbolic-space-based CFTs may be interpreted as a nearly empty AdS wedge that only differs from the truly empty space by the behavior near the horizon where the microstate's world is terminated by a particular singularity. The precise information in this singularity may also be interpreted as an infinite-horizon-size generalization or limit of a Samir Mathur's fuzzball.

This is a cute picture. For quite some time, I've been convinced that the Rindler space is the right "toy setup" in which black hole thermodynamics and quantum gravity should be studied. After all, every large enough black hole has a nearly flat event horizon which is made out of nearly flat patches. In the strictly flat limit, things should simplify. And the authors show that they simplify, indeed.

I think it's a neat paper – and sequence of papers. By gluing the sphere (or more general spaces) out of the open patches, one gets a more specific idea about the right description of the numerous microstates that are responsible for the large entropy of event horizons.

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