aa r X i v : . [ phy s i c s . pop - ph ] J a n Black holes and quantum entanglement
Ro Jefferson
Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am M¨uhlenberg 1, D-14476 Potsdam-Golm, Germany
E-mail: [email protected]
Abstract:
The black hole information paradox is a contradiction between fundamentalprinciples which has puzzled physicists for over forty years. The crux of the problem lies inan assumption about the structure of entanglement across the event horizon, namely, that theHilbert space factorizes. While valid in quantum mechanics, this fails drastically in quantumfield theory, and hence a deeper understanding of entanglement is required if further progressis to be made. Recently, ideas from algebraic quantum field theory have provided new insightinto this issue, and show promise for elucidating the connection between entropy and horizonsthat underlies black hole thermodynamics.
Essay written for the Black Hole Initiative’s 2018 paper competition. n 1973, Jacob Bekenstein observed [1] that black holes must be endowed with an entropyin order to preserve the second law of thermodynamics; otherwise, one could decrease theentropy of the universe by simply throwing subsystems with high entropy into a black hole.At face value, this is an intuitive proposal: since the information about the degrees of freedomthat comprise the hypothetical subsystem would then be hidden behind the event horizon, itmakes sense to count them among the microstates of the black hole.The unintuitive twist comes from the realization that this na¨ıve bookkeeping is not atall how black holes operate. The entropy of familiar systems scales with the volume thereof,which is consistent with simply counting the constituent (e.g., particulate) degrees of freedom.Black hole entropy, in stark contrast, scales with the area of the event horizon. Bekenstein’soriginal motivation for this claim hinged largely on Stephen Hawking’s 1971 result [2] thatthe surface area of a black hole cannot decrease in any classical process (the so-called “areatheorem”). This lead Bekenstein to propose an analogy between black holes and statisticalthermodynamics, which has since been enshrined in the laws of black hole thermodynamics .Classically however, black holes do not radiate (hence the name), and therefore havezero temperature and consequently zero thermodynamic entropy. For this reason, black holethermodynamics initially appeared purely epistemic—a reflection of our knowledge of reality,rather than the real (or ontic) nature thereof. But this metaphysical state of affairs soonchanged when Hawking showed [3] that, quantum mechanically, black holes do radiate, withcharacteristic temperature T = 18 πM , (0.1)and entropy S = A , (0.2)where M is the mass of the black hole, and A is the area of its event horizon. The emissionof this Hawking radiation implies that black holes can evaporate, and thus their surface area A , and by extension their entropy S , can in fact decrease when quantum effects are takeninto account. Fortuitously, including the entropy of the Hawking radiation in the total morethan suffices to compensate for this decrease: the second law of thermodynamics is saved.With Hawking’s discovery that black holes are not completely black after all, black holethermodynamics went from epistemic to ontic in one fell swoop—and brought shatteringimplications in its wake. By far the most notorious is the information paradox [4], whichrefers to the apparent loss of information that results from a black hole which forms from apure state, but upon evaporating leaves nothing but a mixed state of thermal radiation. Thisrepresents a grievous violation of quantum mechanics—specifically, the principle of unitarity,which ensures that probabilities are conserved. In a nutshell, quantum mechanics says thatinformation can never be truly lost. Black holes appear to disagree—strenuously.This is a deeply disturbing result: it implies a contradiction between the fundamentalpillars on which the whole edifice of modern physics stands. And despite nearly 45 yearsof intensive effort, the solution continues to elude us. Indeed, the enduring severity of the– 1 –onflict is aptly illustrated by the recent firewall controversy, which was ignited by a 2013paper [5] by Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully (AMPS).In it, the authors honed the long-standing information paradox to razor sharpness, and arguedthat giving up Einstein’s equivalence principle – which underlies general relativity – is themost conservative resolution. But in addition to noticeable violations of low-energy physics,this would imply that an intrepid explorer, rather than falling smoothly through the eventhorizon as Einstein predicts, instead gets incinerated by a “firewall” of ultra high-energyquanta. Clearly, general relativity cannot be so easily tampered with!It is illuminating to contrast black hole evaporation with the apparently pure-to-thermalevolution of normal matter upon incineration, say a burning lump of coal. Supposing this tobe in an initially pure state, the final state again involves a thermal bath of radiation, with theapparent loss of information that implies. But physicists are not concerned about unitarity-violating barbecues. The reason is that subtle correlations between early and late radiationconspire to preserve the purity of the total system. It is only due to our coarse-grained,inevitably imperfect measurements that we perceive a thermal state. In other words, while itmay be impossible to actually recover the information in practice, in principle a sufficientlypowerful computer could do it. The laws of quantum mechanics survive unscathed.The crucial difference between the coal and the black hole is that the former has noevent horizon. Early quanta are entangled with quanta inside the coal, which can – viatheir interactions with other interior quanta – imprint information on the late radiation. Incontrast, the presence of a horizon imposes a very specific, pairwise entanglement structureon the Hawking quanta across it, which forbids them from sharing their entanglement as inthe lump of coal. It is this quantum structure at the horizon that ultimately underlies the blackhole information paradox . Indeed, this entanglement structure was used explicitly by AMPSwhen they kindled the firewall, and has been implicitly assumed in nearly every discussion ofHawking radiation to date.However, while the decomposition of the vacuum that underlies this pairwise entangle-ment is perfectly valid in quantum mechanics, it fails drastically in quantum field theory.The problem is that Hilbert space – the abstract space where quantum states live – doesnot factorize into subspaces (in this case, the inside and outside of the black hole). Thismanifests in the fact that the entanglement entropy between two regions in quantum fieldtheory is formally infinite. In other words, the vacuum is an infinitely entangled state, andthis fact has consequences which we’re only beginning to understand.Perhaps the most famous example of this is the Reeh-Schlieder theorem, which uses theinfinite entanglement of the vacuum to shake our notion of locality to the core. Na¨ıvely,one might suppose that a localized subregion contains no information about the universebeyond. Reeh-Shlieder proves otherwise: there exist operators confined to such regions, whichnonetheless allow one to reconstruct any other state in the global Hilbert space with arbitraryaccuracy. To illustrate how shocking this is to theorists, it means that by acting with suchan operator in the room in which you’re reading this, you could create the Moon—or thesupermassive black hole at the center of our galaxy!– 2 –his is more than merely a mathematical curiosity. Rather, it hints at an incredible rich-ness to which the quantum mechanical approach to entanglement entropy, with its simplisticassumption about Hilbert space factorization, is blind. And the lesson from black holes isthat we must mine this deeper wealth in order to escape from the paradoxes in which four andhalf decades of inquiry from faulty premises have mired us. Black holes force us to rethinkeverything we think we know about entanglement in quantum field theory, and to developmathematical tools capable of casting light where our meagre intuition cannot tread.In fact, there exists a mathematical framework which is particularly well-adapted to foun-dational questions in both physics and the philosophy thereof, known as algebraic quantumfield theory (AQFT). This was developed starting in the 1950’s as a mathematically rigor-ous approach to quantum field theory, but has historically been a relatively esoteric pursuitof mathematical physicists, of little practical use for calculating anything one might wishto measure in a particle accelerator. But this abstract formalism also enjoys powerful ad-vantages; in particular, the pathological divergences that plague the standard approach toentanglement above simply do not arise. For this reason, ideas from AQFT have now begunto surface in the theoretical physics community as promising new tools to deepen our un-derstanding of entanglement [6]. For example, they have been used to shed light on aspectsof locality and causality in the context of the holographic principle (itself inspired by blackhole thermodynamics), thereby further elucidating the emergent relationship between entan-glement and spacetime geometry [7, 8]. More pertinently, this framework has recently beenapplied to black hole interiors [9, 10], in an effort to understand how this information mightbe represented by an external observer.The nascent application of AQFT to the study of black holes represents a novel andinterdisciplinary approach, which draws on the combined strength of physics, mathematics,and philosophy to extract insights about quantum entanglement and the structure of space-time itself. At the very least, it hints at a way beyond the impasse created by the superficialtreatment of entanglement above, and may elucidate the deeper connection between entropyand horizons that underlies black hole thermodynamics. And while it is too soon to tellwhether this approach will ultimately succeed in finally extinguishing the firewall paradox, itis already clear that there are many fascinating insights about black holes yet to be gained.– 3 – eferences [1] J. D. Bekenstein, “Black holes and entropy,”
Phys. Rev. D (Apr, 1973) 2333–2346.[2] S. W. Hawking, “Gravitational radiation from colliding black holes,” Phys. Rev. Lett. (May, 1971) 1344–1346.[3] S. W. Hawking, “Particle creation by black holes,” Comm. Math. Phys. no. 3, (1975) 199–220.[4] S. W. Hawking, “Breakdown of predictability in gravitational collapse,” Phys. Rev. D (Nov, 1976) 2460–2473.[5] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, “Black Holes: Complementarity orFirewalls?,” JHEP (2013) 062, arXiv:1207.3123 [hep-th] .[6] E. Witten, “Notes on Some Entanglement Properties of Quantum Field Theory,” arXiv:1803.04993 [hep-th] .[7] T. Faulkner, M. Guica, T. Hartman, R. C. Myers, and M. Van Raamsdonk, “Gravitation fromEntanglement in Holographic CFTs,” JHEP (2014) 051, arXiv:1312.7856 [hep-th] .[8] T. Faulkner and A. Lewkowycz, “Bulk locality from modular flow,” JHEP (2017) 151, arXiv:1704.05464 [hep-th] .[9] K. Papadodimas, “A class of non-equilibrium states and the black hole interior,” arXiv:1708.06328 [hep-th] .[10] R. Jefferson, “Comments on black hole interiors and modular inclusions,” arXiv:1811.08900 [hep-th] ..