Black Holes And The Information Paradox
Black Holes andthe Information ParadoxWhat happens to the information in matter destroyedby a black hole? Searching for that answer, physicistsare groping toward a quantum theory of gravityby Leonard SusskindSomewhere in outer space, Professor Windbag’s time capsule hasbeen sabotaged by his arch rival,Professor Goulash. The capsule containsthe only copy of a vital mathematicalformula, to be used by future generations. But Goulash’s diabolical schemeto plant a hydrogen bomb on board thecapsule has succeeded. Bang! The formula is vaporized into a cloud of electrons, nucleons, photons and an occasional neutrino. Windbag is distraught.He has no record of the formula andcannot remember its derivation.Later, in court, Windbag charges thatGoulash has sinned irrevocably: “Whatthat fool has done is irreversible. Why,the fiend has destroyed my formula andmust pay. Off with his tenure!”“Nonsense,” says an unflustered Goulash. “Information can never be destroyed. It’s just your laziness, Windbag. Although it’s true that I’ve scrambled things a bit, all you have to do isgo and find each particle in the debrisand reverse its motion. The laws of nature are time symmetric, so on reversing everything, your stupid formula willbe reassembled. That proves, beyond ashadow of a doubt, that I could neverhave destroyed your precious information.” Goulash wins the case.Windbag’s revenge is equally diabolical. While Goulash is out of town, hiscomputer is burglarized, along with allhis files, including his culinary recipes.Just to make sure that Goulash will never again enjoy his famous Matelote d’anguilles with truffles, Windbag launches the computer into outer space andstraight into a nearby black hole.At Windbag’s trial, Goulash is besidehimself. “You’ve gone too far this time,Windbag. There’s no way to get my filesout. They’re inside the black hole, andif I go in to get them I’m doomed to becrushed. You’ve truly destroyed information, and you’ll pay.”“Objection, Your Honor!” Windbag jumpsup. “Everyone knows that black holeseventually evaporate. Wait long enough,and the black hole will radiate away allits mass and turn into outgoing photonsand other particles. True, it may take1070 years, but it’s the principle thatcounts. It’s really no different from thebomb. All Goulash has to do is reversethe paths of the debris, and his computer will come flying back out ofthe black hole.”“Not so!” cries Goulash.Copyright 1997 Scientific American, Inc.Black Holes and the Information ParadoxBLACK HOLE’S SURFACE looks toWindbag (in the spaceship) like a spherical membrane, called the horizon. Windbag sees Goulash, who is falling into theblack hole, as being slowed down andflattened at the horizon; according tostring theory, Goulash also seems to bespread all over it. Thus, Windbag, whorepresents the outside observer, sees theinformation contained in everything thatfalls into the black hole as stopping at thesurface. But Goulash finds himself fallingright through the horizon to the blackhole’s center, where he becomes crushed.52Scientific American April 1997
zon, black holes are a fundamental newsource of irreversibility in nature. Windbag really did destroy information.”Her Honor turns to Windbag: “Whatdo you have to say to that?” Windbagcalls on Professor Gerard ’t Hooft ofUtrecht University.“Hawking is wrong,” ’t Hooft begins.“I believe black holes must not lead toviolation of the usual laws of quantummechanics. Otherwise the theory wouldbe out of control. You cannot undermine microscopic reversibility withoutdestroying energy conservation. IfHawking were right, the universe wouldheat up to a temperature of 1031 degreesin a tiny fraction of a second. Becausethis has not happened, there must besome way out of this problem.”Twenty more famous theoretical physicists are called to the stand. All thatbecomes clear is that they cannot agree.The Information ParadoxWindbag and Goulash are, ofcourse, fictitious. Not so Hawking and ’t Hooft, nor the controversy ofwhat happens to information that fallsinto a black hole. Hawking’s claim thata black hole consumes information hasdrawn attention to a potentially seriousconflict between quantum mechanicsand the general theory of relativity. Theproblem is known as the informationparadox.When something falls into a blackhole, one cannot expect it ever to comeflying back out. The information codedin the properties of its constituent atomsis, according to Hawking, impossible toretrieve. Albert Einstein once rejectedquantum mechanics with the protest:“God does not play dice.” But Hawkingstates that “God not only plays dice, Hesometimes throws the dice where theycannot be seen”—into a black hole.The problem, ’t Hooft points out, isthat if the information is truly lost,quantum mechanics breaks down. Despite its famed indeterminacy, quantummechanics controls the behavior of particles in a very specific way: it is reversible. When one particle interacts withanother, it may be absorbed or reflectedor may even break up into other particles. But one can always reconstruct theinitial configurations of the particlesfrom the final products.If this rule is broken by black holes, en-YAN NASCIMBENE“This is different. My recipe was lostbehind the black hole’s boundary, itshorizon. Once something crosses thehorizon, it can never get back out without exceeding the speed of light. AndEinstein taught us that nothing can everdo that. There is no way the evaporation products, which come from outside the horizon, can contain my lostrecipes even in scrambled form. He’sguilty, Your Honor.”Her Honor is confused. “We needsome expert witnesses. Professor Hawking, what do you say?”Stephen W. Hawking of the University of Cambridge comes to the stand.“Goulash is right. In most situations,information is scrambled and in a practical sense is lost. For example, if a newdeck of cards is tossed in the air, theoriginal order of the cards vanishes. Butin principle, if we know the exact detailsof how the cards are thrown, the original order can be reconstructed. This iscalled microreversibility. But in my 1976paper I showed that the principle of microreversibility, which has always heldin classical and quantum physics, is violated by black holes. Because information cannot escape from behind the hori-Black Holes and the Information ParadoxCopyright 1997 Scientific American, Inc.Scientific American April 199753
HORIZON:“POINT OF NO RETURN”INVISIBLE HORIZON is represented in this analogy as a line in a river. To the left ofit, the water flows faster than a “lightfish” can swim. So if a lightfish happens to driftbeyond this line, it can never get back upstream; it is doomed to be crushed in the falls.But the fish notices nothing special at the line. Likewise, a light ray or person who is inside the horizon can never get back out; the object inevitably falls into the singularity atthe black hole’s center, but without noticing anything special about the horizon.SINGULARITYRISING PULLOF GRAVITYBRYAN CHRISTIERISING PULLOF GRAVITYergy may be created or destroyed, threatening one of the most essential underpinnings of physics. The conservation ofenergy is ensured by the mathematicalstructure of quantum mechanics, whichalso guarantees reversibility; losing onemeans losing the other. As ThomasBanks, Michael Peskin and I showed in1980 at Stanford University, information loss in a black hole leads to enormous amounts of energy being generated. For such reasons, ’t Hooft and I believe the information that falls into ablack hole must somehow become available to the outside world.Some physicists feel the question ofwhat happens in a black hole is academic or even theological, like counting angels on pinheads. But it is not so at all:at stake are the future rules of physics.Processes inside a black hole are merelyextreme examples of interactions between elementary particles. At the energies that particles can acquire in today’slargest accelerators (about 1012 electronvolts), the gravitational attraction between them is negligible. But if the particles have a “Planck energy” of about1028 electron volts, so much energy—and therefore mass—becomes concentrated in a tiny volume that gravitational forces outweigh all others. The resulting collisions involve quantummechanics and the general theory ofrelativity in equal measure.It is to Planckian accelerators that wewould nominally look for guidance inbuilding future theories of physics. Alas,Shmuel Nussinov of Tel Aviv Universityconcludes that such an accelerator wouldhave to be at least as big as the entireknown universe.Nevertheless, the physics at Planck energies may be revealed by the knownproperties of matter. Elementary particles have a variety of attributes that leadphysicists to suspect they are not so ele54Scientific American April 1997mentary after all: they must actuallyhave a good deal of undiscovered internal machinery, which is determined bythe physics at Planck energies. We willrecognize the right confluence of general relativity and quantum physics—orquantum gravity—by its ability to explain the measurable properties of electrons, photons, quarks or neutrinos.Very little is known with absolute certainty about collisions at energies beyondthe Planck scale, but there is a good educated guess. Head-on collisions at theseenergies involve so much mass concentrated in a tiny volume that a black holewill form and subsequently evaporate.So figuring out whether black holes violate the rules of quantum mechanics ornot is essential to unraveling the ultimate structure of particles.A black hole is born when so muchmass or energy gathers in a small volume that gravitational forces overwhelmall others and everything collapses under its own weight. The material squeezes into an unimaginably small regioncalled a singularity, the density inside ofwhich is essentially infinite. But it is notthe singularity itself that will interest us.Surrounding the singularity is an imaginary surface called the horizon. For ablack hole with the mass of a galaxy, thehorizon is 1011 kilometers from the center—as far as the outermost reaches ofthe solar system are from the sun. For ablack hole of solar mass, the horizon isroughly a kilometer away; for a blackhole with the mass of a small mountain,the horizon is 10 –13 centimeter away,roughly the size of a proton.Copyright 1997 Scientific American, Inc.Black Holes and the Information Paradox
YAN NASCIMBENEThe horizon separates space into tworegions that we can think of as the interior and exterior of the black hole. Suppose that Goulash, who is scouting forhis computer near the black hole, shootsa particle away from the center. If he isnot too close and the particle has a highvelocity, then it may overcome the gravitational pull of the black hole and flyaway. It will be most likely to escape ifit is shot with the maximum velocity—that of light. If, however, Goulash is tooclose to the singularity, the gravitational force will be so great that even a lightray will be sucked in. The horizon is theplace with the (virtual) warning sign:Point of No Return. No particle or signal of any kind can cross it from the inside to the outside.At the HorizonAn analogy inspired by William G.Unruh of the University of BritishColumbia, one of the pioneers in blackhole quantum mechanics, helps to explain the relevance of the horizon. Imagine a river that gets swifter downstream.Among the fish that live in it, the fastestswimmers are the “lightfish.” But atsome point, the river flows at the fish’smaximum speed; clearly, any lightfishthat drifts past this point can never getback up. It is doomed to be crushed onthe rocks below Singularity Falls, locatedfarther downstream. To the unsuspecting lightfish, though, passing the pointof no return is a nonevent. No currentsBlack Holes and the Information Paradoxor shock waves warn it of the crossing.What happens to Goulash, who in acareless moment gets too close to theblack hole’s horizon? Like the freelydrifting fish, he senses nothing special:no great forces, no jerks or flashinglights. He checks his pulse with his wristwatch—normal. His breathing rate—normal. To him the horizon is just like anyother place.But Windbag, watching Goulash froma spaceship safely outside the horizon,sees Goulash acting in a bizarre way.Windbag has lowered to the horizon acable equipped with a camcorder andother probes, to better keep an eye onGoulash. As Goulash falls toward theblack hole, his speed increases until itapproaches that of light. Einstein foundthat if two persons are moving fast relative to each other, each sees the other’sclock slow down; in addition, a clockthat is near a massive object will runslowly compared with one in emptyspace. Windbag sees a strangely lethargic Goulash. As he falls, the latter shakeshis fist at Windbag. But he appears to bemoving ever more slowly; at the horizon, Windbag sees Goulash’s motionsslow to a halt. Although Goulash fallsthrough the horizon, Windbag neverquite sees him get there.In fact, not only does Goulash seem toslow down, but his body looks as if it isbeing squashed into a thin layer. Einstein also showed that if two personsmove fast with respect to each other,each will see the other as being flattenedin the direction of motion. More strangeCopyright 1997 Scientific American, Inc.ly, Windbag should also see all the material that ever fell into the black hole,including the original matter that madeit up—and Goulash’s computer—similarly flattened and frozen at the horizon.With respect to an outside observer, allof that matter suffers a relativistic timedilation. To Windbag, the black holeconsists of an immense junkyard of flattened matter at its horizon. But Goulashsees nothing unusual until much later,when he reaches the singularity, thereto be crushed by ferocious forces.Black hole theorists have discoveredover the years that from the outside, theproperties of a black hole can be described in terms of a mathematical membrane above the horizon. This layer hasmany physical qualities, such as electrical conductivity and viscosity. Perhapsthe most surprising of its properties waspostulated in the early 1970s by Hawking, Unruh and Jacob D. Bekenstein ofthe Hebrew University in Israel. Theyfound that as a consequence of quantum mechanics, a black hole—in particular, its horizon—behaves as though itcontains heat. The horizon is a layer ofhot material of some kind.The temperature of the horizon depends on just where it is measured. Suppose one of the probes that Windbag hasattached to his cable is a thermometer.Far from the horizon he finds that thetemperature is inversely proportional tothe black hole’s mass. For a black holeof solar mass, this “Hawking temperature” is about 10–8 degree—far colderthan intergalactic space. As Windbag’sthermometer approaches the horizon,Scientific American April 199755
Copyright 1997 Scientific American, Inc.Black Holes and the Information ParadoxLIGHTSOURCEDISTANCE FROM SINGULARITYBRYAN CHRISTIETIMEHORIZONs it possible that Goulash and Windbag are in a sense both correct? Can itbe that Windbag’s observations are indeed consistent with the hypothesis thatGoulash and his computer are thermalized and radiated back into space beforeever reaching the horizon, even thoughGoulash discovers nothing unusual un-til long after, when he encounters thesingularity? The idea that these are notcontradictory but complementary scenarios was first put forward as the principle of black hole complementarity byLárus Thorlacius, John Uglum and meat Stanford. Very similar ideas are alsofound in ’t Hooft’s work. Black holecomplementarity is a new principle ofrelativity. In the special theory of relativity we find that although differentobservers disagree about the lengths oftime and space intervals, events takeplace at definite space-time locations.Black hole complementarity does awaywith even that.How this principle actually comesinto play is clearer when applied to thestructure of subatomic particles. Suppose that Windbag, whose cable is alsoequipped with a powerful microscope,watches an atom fall toward the horizon. At first he sees the atom as a nucleus surrounded by a cloud of negativecharge. The electrons in the cloud moveso rapidly they form a blur. But as theatom gets closer to the black hole, its internal motions seem to slow down, andthe electrons become visible. The protons and neutrons in the nucleus stillmove so fast that its structure is obscure.But a little later the electrons freeze, andthe protons and neutrons start to showup. Later yet, the quarks making up theseparticles are revealed. (Goulash, whofalls with the atom, sees no changes.)Many physicists believe elementaryparticles are made of even smaller constituents. Although there is no definitivetheory for this machinery, one candidatestands out as being the most promising—namely, string theory. In this theory, an elementary particle does not resemble a point; rather it is like a tinyrubber band that can vibrate in manymodes. The fundamental mode has thelowest frequency; then there are higherharmonics, which can be superimposedon top of one another. There are an infinite number of such modes, each ofwhich corresponds to a different elementary particle.Here another analogy helps. One cannot see the wings of a hovering hummingbird, because its wings flutter toofast. But in a photograph taken with afast shutter speed, one can see thewings—so the bird looks bigger. If a hummer falls into the black hole, Windbagwill see its wings take form as the birdapproaches the horizon and the vibrations appear to slow down; it seems togrow. Now suppose that the wings haveLIGHT CONES describe the path of light rays emanating from a point. Outside thehorizon the light cones point upward—that is, forward in time. But inside, the lightcones tip so that light falls straight into the black hole’s center.however, it registers higher temperatures. At a distance of a centimeter, itmeasures about a thousandth of a degree; at a nuclear diameter it records 10billion degrees. The temperature ultimately becomes so high that no imaginable thermometer can measure it.Hot objects also possess an intrinsicdisorder called entropy, which is relatedto the amount of information a systemcan hold. Think of a crystal lattice withN sites; each site can house one atom ornone at all. Thus, every site holds one“bit” of information, corresponding towhether an atom is there or not; the total lattice has N such bits and can contain N units of information. Becausethere are two choices for each site andN ways of combining these choices, thetotal system can be in any one of 2Nstates (each of which corresponds to adifferent pattern of atoms). The entropy(or disorder) is defined as the logarithmof the number of possible states. It isroughly equal to N—the same numberthat quantifies the capacity of the system for holding information.Bekenstein found that the entropy ofa black hole is proportional to the areaof its horizon. The precise formula, derived by Hawking, predicts an entropyof 3.2 1064 per square centimeter ofhorizon area. Whatever physical systemcarries the bits of information at the horizon must be extremely small and densely distributed: their linear dimensionshave to be 1/1020 the size of a proton’s.They must also be very special forGoulash to completely miss them as hepasses through.The discovery of entropy and otherthermodynamic properties of black holesled Hawking to a very interesting conclusion. Like other hot bodies, a blackhole must radiate energy and particlesinto the surrounding space. The radia56Scientific American April 1997tion comes from the region of the horizon and does not violate the rule thatnothing can escape from within. But itcauses the black hole to lose energy andmass. In the course of time an isolatedblack hole radiates away all its massand vanishes.All of the above, though peculiar, hasbeen known to relativists for some decades. The true controversies arise when,following H
52 Scientific American April 1997 Black Holes and the Information Paradox BLACK HOLE’S SURFACE looks to Windbag (in the spaceship) like a spheri-cal membrane, called the horizon. Wind-bag sees Goulash, who is falling into the black hole, as being slowed down and flattened at the horizon; according to string theory, Goulash also seems to be spread all over it. Thus, Windbag, who represents .
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