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29.01.2014

Celebrated theoretical physicist Stephen Hawking has opened a can of worms in his 1976 paper on black holes. In a recent article, he is trying to put the worms back into the can. It may prove a little trickier than expected.

Black holes are solutions of Einstein’s equations of general relativity which have the unique property that they possess a closed surface in space which is the ultimate “point of no return.” Even the fastest things that nature allows us – light rays – will not escape if they cross this surface, dubbed the horizon. Despite their strange properties, black holes are not really exotic objects. In the theoretical realm, if one takes enough matter, throws it all to a point and fast forwards using Einstein’s equations, one ends up with a black hole. In the observational realm, there is ample astrophysical evidence that massive stars end their life as black holes, and that there are mega black holes at the center of many galaxies (including our own).

But black holes don’t just gobble things up. Rather they quite effortlessly chew things up – thoroughly and completely – and spit them out in a completely undecipherable form. This is the main point of Hawking’s work from 1976, in which he showed that a black hole in empty space emits precisely thermal (black body) radiation when quantum mechanics is taken into account. In fact it emits thermal radiation until it completely evaporates, and any initial state of the system will end up exactly the same, in the form of thermal radiation.

Very loosely we can understand this as follows: Consider creating an electron and positron pair, in a specific quantum state, just outside the horizon. Each of these particles can have either spin up or spin down so there are a total of four states. Suppose the black hole now gobbles up the positron, never to be seen again, and that the electron makes it back to our lab. The electron has only two states. So we started with a system which had four states and ended with a system that has two states – we lost information! Equivalently we can say that quantum mechanics is not unitary (reversible) in the presence of black holes, or in more technical terms we can say that the emitted electron is in a density matrix and not a pure state, just the same as exactly thermal radiation.

Hawking’s computation is extremely elegant and robust – it only uses 1) quantum field theory on 2) curved space. The former is well tested and verified in just about any high energy physics experiment, and the latter is just Einstein’s general relativity (as a classical theory). Furthermore, a very similar set of computations is successful in the context of generating the structures in the universe from primordial quantum fluctuations after the big bang. Yet around the black hole, the synthesis of these two sets of ideas leads to a bewildering result, since any high-enough energy experiment will create a black hole, and end in information-free thermal radiation.The universe just can’t help losing the information of where the keys are. This is unlike any other quantum mechanical system whose time evolution does not lose any information.

Interestingly, the surprising prediction for a flux of thermal radiation from a black hole fits very nicely with other properties of the black hole. Shortly before Hawking’s article, Jacob Bekenstein suggested that black holes have entropy. Bekenstein’s entropy, Hawking’s temperature and the mass of the black hole, which is the same as its energy, satisfy the ordinary laws of thermodynamics.

The synthesis of quantum mechanics and general relativity has been an outstanding problem for quite some time. Using string theory, and more specifically Maldacena’s AdS/CFT correspondence, it was finally established that evolution of black holes is unitary and that we do not lose any information, since we can embed black holes in standard quantum theories which we know are completely unitary. In these theories, black holes seem no different than lumps of coal that burn. The issue remains, however: Where exactly does the synthesis of field theory and classical general relativity fail, and which of their well tested properties are we forced to modify?

There has been a renewed interest in this question in recent years. An elegant argument from Almheiri, Marolf, Polchinski and Sully suggested that one needs to quantum-mechanically modify the horizon of a black hole into a hot membrane (whose nature is not clear). This solution has been called the “firewall” solution. In this solution, one gives up some aspects of Einstein’s equivalence principle, as well as parts of the black hole solution in classical general relativity, where it naively seems that quantum effects should be small.

In another solution, suggested a few years ago by Mathur, one replaces the black hole by a large set of horizon-free solutions of string theory – this is the “fuzzball” solution. This solution is quite attractive, but so far no one has been able to construct enough “fuzzballs” to account for the black hole entropy. Other solutions suggest some non-locality in space-time, which allows information to be transported from the interior of the black hole to its exterior, or replacing space-time itself by an algebraic construction, keeping only quantum mechanics. Hawking conceded already 10 years ago that black holes do not really lose information, and his recent paper provides evidence for the “fuzzball” proposal for the description of black holes.

This topic is one of the topics of research of the String theory group at the Weizmann Institute, Profs. Ofer Aharony, Micha Berkooz, Zohar Komargodski and Adam Schwimmer, who hosted a workshop on “Black Holes and Quantum Information” earlier this month. The workshop explored the role of entanglement entropy and quantum information theory in the resolution of the black hole information paradox, and in the very emergence of space-time as a derived concept, which seems to appear in a way similar to how thermodynamics is derived from statistical physics.

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