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Fortschritte der Physik

Computational complexity and black hole horizons

Leonard Susskind

Published Jan 1, 2016• 767 citations
Publisher

Abstract

Computational complexity is essential to understanding the properties of black hole horizons. The problem of Alice creating a firewall behind the horizon of Bob's black hole is a problem of computational complexity. In general we find that while creating firewalls is possible, it is extremely difficult and probably impossible for black holes that form in sudden collapse, and then evaporate. On the other hand if the radiation is bottled up then after an exponentially long period of time firewalls may be common. It is possible that gravity will provide tools to study problems of complexity; especially the range of complexity between scrambling and exponential complexity.

DOI
10.1002/prop.201500092

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Computational complexity is essential to understanding the properties of black hole horizons. The problem of Alice creating a firewall behind the horizon of Bob's black hole is a problem of computational complexity. In general we find that while creating firewalls is possible, it is extremely difficult and probably impossible for black holes that form in sudden collapse, and then evaporate. On the other hand if the radiation is bottled up then after an exponentially long period of time firewalls may be common. It is possible that gravity will provide tools to study problems of complexity; especially the range of complexity between scrambling and exponential complexity.
Computational complexity is essential to understanding the properties of black hole horizons. The problem of Alice creating a firewall behind the horizon of Bob's black hole is a problem of computational complexity. In general we find that while creating firewalls is possible, it is extremely difficult and probably impossible for black holes that form in sudden collapse, and then evaporate. On the other hand if the radiation is bottled up then after an exponentially long period of time firewalls may be common. It is possible that gravity will provide tools to study problems of complexity; especially the range of complexity between scrambling and exponential complexity.
The concept of transparent and opaque horizons is defined. One example of opaqueness is the presence of a firewall. Two apparently contradictory statements are reconciled: The overwhelming number of black hole states have opaque horizons; and: All black holes formed by natural processes have transparent horizons. A diagnostic is proposed for transparency, namely that the computational complexity of the state be increasing with time. It is shown that opaque horizons are extremely unstable and that the slightest perturbation will make them transparent within a scrambling time.
Abstract Firewalls in black holes are easiest to understand by imposing time reversal invariance, together with a unitary evolution law. The best approach seems to be to split up the time span of a black hole into short periods, during which no firewalls can be detected by any observer. Then, gluing together subsequent time periods, firewalls seem to appear, but they can always be transformed away. At all times we need a Hilbert space of a finite dimension, as long as particles far separated from the black hole are ignored. Our conclusion contradicts other findings, as these assume that information that entered into a black hole, cannot re-emerge. But re-emergence of that information is exactly what our version of firewalls is supposed to ensure. Indeed, the firewall transformation removes the problems caused by entanglement between very early and very late in- and out-particles, in a far from trivial way.
We argue that the problem of calculating retention time scales in young black holes is a problem of relative state complexity. In particular, we suggest that Alice’s ability to estimate the time scale for a perturbed black hole to release the extra n qubits comes down to her decoding the Hilbert space of the Hawking radiation. We then demonstrate the decoding task Alice faces is very difficult, and in order to calculate the relative state complexity she would either need to act with an exponentially complex unitary operator or apply an extremely fine-tuned future precursor operator to the perturbed state in S U ( 2 K ) .
We give a quantum circuit interpretation of evaporating black hole geometry. We make an analogy between the appearance of island for evaporating black hole and the transition from two-sided to one-sided black hole in the familiar example of perturbed thermofield double. If Alice perturbs thermofield double and waits for scrambling time, she will have a one-sided black hole with interior of her own. We argue that by similar mechanism the radiation gets access to the interior (island forms) after Page time. The growth of the island happens as a result of the constant transitions from two-sided to one-sided black holes.
Firewalls in black holes are easiest to understand by imposing time reversal invariance, together with a unitary evolution law. The best approach seems to be to split up the time span of a black hole into short periods, during which no firewalls can be detected by any observer. Then, gluing together subsequent time periods, firewalls seem to appear, but they can always be transformed away. At all times we need a Hilbert space of a finite dimension, as long as particles far separated from the black hole are ignored. Our conclusion contradicts other findings, particularly a recent paper by Strauss and Whiting. Indeed, the firewall transformation removes the entanglement between very early and very late in- and out-particles, in a far-from-trivial way.