# Computers, quantum information and computational complexity in Kerr-AdS black hole backgrounds

A Kerr-AdS black hole is eternal, never evaporates and has a Malament-Hogarth metric. Bob, a universally programmable reversible classical computer with a fixed maximum memory who only outputs one bit, orbits around the black hole for a superpolynomial time. Bob aspires to be a universal computer. He accepts any program for a universal computer and tries to compute it, but doesn't always succeed because he might not survive long enough, but is never wrong if he manages to output an answer. Alice selects an $$l$$ bit program of her own choosing, programs Bob with it and initiates the program. She commits a copy of her program to her memory permanently and will never forget it. Then, she falls through the outer event horizon. In the Penrose diagram of the Kerr-AdS black hole, there are actually two different inner event horizons Alice may fall through afterward. There's also another entangled AdS double universe to prevent firewalls from forming. Only one of the two inner horizons has the Malament-Hogarth property, and Alice chooses that one to fall through. Alice has a rocket so she doesn't have to follow a geodesic.

Meanwhile, Bob computes the superpolynomially long reversible program Alice chose. Reversibility means Bob doesn't have to dump a superpolynomial amount of entropy into his environment according to Landauer's principle. Actually, the volume Bob can take up goes exponentially in $$kr$$ where $$r$$ is the radius of Bob's orbit and the number of computational steps per unit coordinate time also grows exponentially. Geodesic orbits no longer exist if $$kr$$ is large enough, but we can let Bob be a spherical shell of radius $$r$$ totally surrounding the black hole instead. As a spherical shell instead of a satellite, Bob doesn't have to worry about spiraling inward after radiating gravitational waves or Brownian motion from the Hawking heat bath. The mass density of Bob then needs to be low enough so that Bob doesn't collapse into a black hole of his own. However, the energy contribution of Bob to the universe can be much greater than the mass of the black hole. Despite being a classical reversible computer, Bob can still broadcast internal information because classical information can be cloned.

At the end of the computation, Bob deputizes Carol, who also has a rocket, to catch up with Alice and inform her about the output bit. So, Alice learns about the output of the program of her choice after experiencing a linear amount of subjective time. Alice doesn't come to an end at a singularity, so black hole complimentarity isn't needed, but why would black hole complimentarity suddenly apply in the limit of the angular velocity going to zero? Can the CFT dual of this future AdS universe perform some superpolynomial computations?

Well, Bob can't compute indefinitely due to being immersed in a Hawking heat bath to equilibrate with, but for how long can he remain computing? If Bob's temperature is much larger than the Hawking temperature, what's the heat conductivity between Bob and the black hole? Or is Bob's temperature the Hawking temperature?

Even without Bob, Alice can make some PP computations with a different program of her choice by making use of closed timelike curves after passing through the Cauchy horizon of the timelike ring singularity. Can this show up in the CFT dual of this future AdS universe?

Actually, blue shifting of generic small perturbations are likely to prevent inner horizons from forming in the first place due to gravitational back-reaction causing a spacelike singularity to develop instead, but is this necessarily true for all possible quantum states of the entangled universe and its double? After all, that's not what we expect if we have locality at inner horizons. Just as we can have a thermodynamic past hypothesis at the big bang, we might also have a thermodynamic future hypothesis leading to a local reversal of the thermodynamic arrow of time. (Speaking of which, if AdS has no past hypothesis, then why can it have a thermodynamic arrow of time?) The Hawking temperature of the inner horizon differs from that of the outer horizon. If there are exceptions, can one such quantum state be described by a polynomial amount of classical information? After all, if such a state exists, I just described it in a few words together with Alice' $$l$$ bit program!

Also, there's a lack of global hyperbolicity right after Alice and Carol pass the inner horizon.

Even if a singularity definitely prevents inner horizons from forming, Carol might still use her rocket to catch up with Alice an inverse superpolynomially small time before Alice hits the singularity. This is much smaller than the Planck time, so Planck scale dynamics might modify this description. Black hole complementarity now has to apply unless superluminal signaling is admitted. So, can the stretched outer horizon compute some superpolynomial computations? What about the combined CFT dual of our universe and its double?

Actually, Alice, Carol and Bob can't be classical because there's no invariant decoherent pointer basis for any of them. So, let Alice carry $$l$$ qubits, Carol carry a qubit and Bob be a quantum computer instead. Prepare Alice' $$l$$ qubit program to be entangled with Bob, and Bob hands over the output qubit to Carol. Actually, quasiclassicality remains possible if Alice, Carol and Bob store a huge number of redundant copies of their information, as an error correction code. If Bob’s program were quantum, there can only be one copy of it throughout all of Bob. If it were classical, there can be numerous copies of it all over Bob.

As Landauer once said, information is physical.

I'm just learning this subject myself, but I'll share some excerpts from a closely-related paper that happened to be on my reading list when you posted this question.

The paper arXiv:1911.12413 studied how instability of the inner horizon(s) prevents any inconsistencies with the AdS/CFT correspondence's realization of the holographic principle.$$^*$$ From the abstract:

we find that strong cosmic censorship holds for all AdS black holes except rotating BTZ.

From the intro:

We develop these ideas further to study the implications of extending the black hole metric beyond the inner horizon on CFT correlation functions. The potential ambiguities in this extension of spacetime ... are fixed if we require analyticity; thus the maximal analytic extension of is unique. ... We use this prescription to conduct two tests of inner horizon stability in AdS black holes. ... We will examine the implications of these results and interpret them as saying that in quantum gravity the black hole spacetime cannot be extended beyond the inner horizon for charged black holes in any dimension, and for rotating black holes in more than three dimensions.

$$^*$$ Caveats: They also show that the situation is different in lower-dimensional spacetimes, and the paper arXiv:2010.03575 seems to propose a different mechanism for preventing inconsistencies with the holographic principle, but I haven't studied either paper carefully enough to really compare them.

• FWIW, inner horizons of the BTZ BHs seems to be destroyed by higher order effects: 2002.02083. Commented Oct 25, 2021 at 15:25
• Scenario 2 on page 5 and possibility 2 on page 11 applies. The inner KMS condition is violated. For almost all quantum states, there's a singularity instead of an inner horizon. However, maybe for some rare very fine tuned quantum states, the inner horizon exists. These are the exceptions I mentioned in the question. The projection of the KMS state onto this rare subspace is exponentially small. So, the contributions of the inner horizon extension are exponentially suppressed in $n$-point functions for low $n$.
– QGR
Commented Oct 26, 2021 at 7:53
• The inner states of Bob the computer are also invisible to low $n$-point functions. We need at least polynomial sized $n$'s to detect Bob's internal states.
– QGR
Commented Oct 26, 2021 at 7:53
• BTZ is a TQFT, which can't simulate universal computers.
– QGR
Commented Oct 26, 2021 at 7:55
• @QGR "...maybe for some rare very fine tuned quantum states, the inner horizon exists." In classical GR, we often think of spacetime as meaningful on its own, because mathematically we can probe it with test-objects that don't modify the spacetime. But in a quantum theory of gravity, I'm not sure that makes sense anymore. If introducing an object to probe the spacetime would perturb it out of that special subset of states that have an inner horizon, then the inner horizon wouldn't really exist in any physically meaningful sense. Is this where your question is coming from? Commented Oct 31, 2021 at 20:11