Sekino and Susskind have argued that black holes scramble information faster than any quantum field theory in this paper. What is the mechanism for such scrambling?


2 Answers 2


The mechanism by Charging Bull is interesting, but deeply flawed. It is true at any given moment, there are about $A/\ell_P^2$ entangled Hawking pairs across the horizon, and that the curvature of the black hole serves to pull them apart. The particle on the inside will of course hit the singularity. But also note that unless the direction of of the outgoing Hawking particle points nearly exactly outward in direction, it will only move out from the horizon a bit before following a curved trajectory in Eddington-Finkelstein coordinates causing it to fall back through the horizon and also hit the singularity, but at a different moment. Only an average of an order unity of the $A/\ell_P^2$ many entangled pairs lead to an outgoing Hawking particle escaping for good. Those which escape are mostly s-waves, with a few p-waves and d-waves delocalized over all directions. What we get instead are $A/\ell_P^2$ entangled pairs of ingoing Hawking particles hitting the singularity at different moments. Also, it's also true there are $A/\lambda^2$ entangled pairs of wavelength $\lambda$ for the vacuum, but they are really redshifted versions of some of the $A/\ell_P^2$ entangled pairs from an earlier moment due to the redshifting mechanism of Eddington-Finkelstein warping. They are not independent pairs. The other entangled pairs also show up as $\lambda$ wavelength entanglements later after redshifting, but of the nonvacuum sort, a squeezed state sort.

In fact, if we're only interested in pairs of infalling Hawking particles hitting the singularity at different moments, it's not even necessary for them to be produced across the horizon. That's only necessary for entangled pairs with one escaping for good and the other hitting the singularity. The location of pair production of both eventually infalling Hawking particles can be outside the horizon, across it, or inside it.

On the average, we only get an order of unity of outgoing Hawking qubits per unit Schwarzschild time. So, over a time period of $R\ln(R/\ell_P)$, only somelike like $c\ln(R/\ell_P)$ qubits could have escaped where $c$ is of order unity. It's possible for the spacecraft to dump a lot more information than that, and we have Holevo bounds. The problem with the Hayden-Preskill mechanism is, maybe Alice's top secret diary contains $k$ qubits of information. But that doesn't necessarily mean collecting only $ck\ln(R/\ell_P)$ qubits of outgoing radiation will suffice to snoop into her top secret confidential diary. What they overlooked is the microstates of the craft contain a lot more than k qubits worth of info. To suggest that only collecting that few qubits is enough to crack the k chosen diary qubits out of the that many more craft microstate qubits is to violate information causality.

But this suggests another very interesting mechanism. If the craft hits the singularity at time $t$, it hits simultaneously with $A/\ell_P^2$ ingoing Hawking particles. Actually, probably more than that, maybe $R^3/\ell_P^3$ because the location of pair production can be far inside the hole. Retrocausal influences still exist, but most of them relate to pairs of entangled infalling Hawking radiation, both infalling. Via these entanglements, this retrocausal effect zigzags back forward causally again to the singularity at a different moment. This zigzag mechanism can continue for many many iterations in both directions in Eddington-Finkelstein time, which really becomes space inside the black hole. A tiny fraction of the entangled Hawking pairs really have an escaping outgoing Hawking particle. These carry scrambled information about the craft. This information is spread out over a long time both far into the future and far into the past of the moment when the craft plunges into the hole. A very long zigzag of alternating causality and retrocausality carried over many entangled pairs of infalling Hawking particles hitting the singularity at different moments.


Fast scrambling presupposes black hole complementarity, which is why it's so mysterious.

An alternative explanation is retrocausality. Black hole singularities exert retrocausal influences. Suppose you have a black hole with a radius of one light-second. A spacecraft carrying some quantum information passes through the event horizon and hits the singularity. Work in Eddington-Finkelstein coordinates. Ingoing light rays take one second to go from the horizon to the singularity. The spacecraft moves slower than that, so it might take two or so seconds to hit the singularity. What hits the singularity at that moment? It is the spacecraft and the quantum information it carries. But also, pair entangled Hawking particle pairs of massless particles are constantly produced across the event horizon. One of them lies outside and escapes the black holes gravitational pull. The other falls in and hits the singularity. Infalling Hawking radiation also hits the singularity at the same instant. Retrocausal influences from the singularity shows up as a fixing of the form of entanglement between everything which will hit the singularity at the same instant, i.e. the internal microstates of the spacecraft and the infalling Hawking radiation. This can be described in the two-state formalism as postselection. However, the infalling Hawking radiation is also entangled with the outgoing Hawking radiation. So, information on the microstates of the spacecraft are transmitted retrocausally to the outgoing Hawking radiation via this mechanism. In flat spacetime, "virtual" particle pair entangled states don't increase in spatial separation. In the warped geometry of a black hole, a pair of "not-so-virtual" massless particles both travelling locally outward at the speed of light across the event horizon will see the spatial distance between them increasing with time. If their initial wavelength is $\lambda$, there will be around $A/\lambda^2$ many of such pairs at any given instant. The smallest wavelengths are cutoff around the Planck scale, and at that scale, there are about $A/\ell_P^2$ of them. Such pairs dominate the counting, but longer wavelengths also contribute. The infalling Hawking particle is travelling outward locally, but is still dragged inward by coordinate dragging. Initially, it is about $\lambda$ inward from the boundary. So, it would take about $R\ln(R/\lambda)$ to hit the singularity. There's a cutoff about the Planck wavelength, so the longest time it would take is around $R\ln(R/\ell_P)$ which is about 100 seconds or so. So, all you need is to collect all the outgoing Hawking radiation starting 100 seconds or so before the craft passes the horizon until one second after (two seconds minus one second) to extract the infalling quantum information. Any earlier Hawking radiation is not needed. Because infalling Hawking radiation hitting the singularity can come in from all directions, outgoing Hawking radition has to be collected from all directions.

The Hayden-Presskill mechanism is all wrong because it assumes collecting all info starting from the moment the craft hits the horizon for about 100 seconds or so after that, with the entire history of all the outgoing hawking radiation since the formation of the black hole needed as context.

If there’s a possibility that once craft is just inside the horizon, it hits something which causes it to recoil locally outward near the speed of light, then it can take longer than two seconds to hit the singularity. In that case, we might still need to collect all outgoing Hawking radiation up till 100 secs later “just in case”.

The black hole information paradox is only so mysterious because most people have an automatic mental block against retrocausality.


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