collaboration of numerical relativity groups is gen-erating a large catalog of waveforms, and a dic-tionary of the information in the waveforms (13).

These waveforms will be observed by theLaser Interferometer Gravitational Wave Observ-atory (LIGO), with interferometers in Livingston,Louisiana, and Hanford, Washington, and byLIGOs international partners: theVirgo interferom-eter near Pisa, Italy, and (just recently funded) theKamioka GravitationalWave Antenna (KAGRA)in the Kamioka mine in Japan (14). The initialLIGO and Virgo interferometers (with sensitivitiesat which it is plausible but not likely to see waves)were designed to give the experimental teamsenough experience to perfect the techniques anddesign for advanced interferometersthat willhave sensitivities at which they are likely to seelots of waves.

Initial LIGO and Virgo exceeded and reachedtheir design sensitivities, respectively (15), andcarried out searches for waves in 20062007 and20092010. As was expected, no waves have beenseen, though the data are still being analyzedand many interesting results have been obtained(16). The advanced LIGO and advanced Virgointerferometers are now being installed (17) andby 2017 should reach sensitivities at which black-hole mergers are observed. The LIGO/Virgo team

will use their merger observations to test numer-ical relativitys geometrodynamic predictions. Thiswill be just one of many science payoffs fromLIGO and its partners, but it is the payoff thatexcites me the most.

References and Notes1. J. A. Wheeler, Geometrodynamics (Academic Press,

New York, 1962).2. B. Carter, S. Chandrasekhar, G. W. Gibbons, in General

Relativity, An Einstein Centenary Survey, S. W. Hawking,W. Israel, Eds. (Cambridge Univ. Press, Cambridge, 1979),chapters 6, 7, and 13.

3. E. Witten, Science 337, 538 (2012).4. R. Owen et al., Phys. Rev. Lett. 106, 151101 (2011).5. If two freely falling particles have a vector separation

Dx, then (i) the tidal field gives them a relativeacceleration Da = ED x, and (ii) gyroscopes carriedby the particles precess with respect to each other withangular velocity DW = BD x.

6. Each tendex line (or vortex line) is the integralcurve of an eigenvector field of the tidal field E(or frame-drag field B), and the lines tendicity (orvorticity) is that eigenvectors eigenvalue. A knowledgeof the tendex and vortex lines and their tendicitiesand vorticities is equivalent to a full knowledge ofthe Riemann tensor.

7. For pictures and movies from these and other simulationssee, e.g., http://black-holes.org/explore2.html andhttp://numrel.aei.mpg.de/images.

8. G. Lovelace et al., Phys. Rev. D Part. Fields Gravit. Cosmol.82, 064031 (2010).

9. For example, the simulation by Owen describedin figure 5 of (4).

10. M. Campanelli, C. O. Lousto, Y. Zlochower, D. Merritt,Phys. Rev. Lett. 98, 231102 (2007).

11. J. A. Gonzlez, M. Hannam, U. Sperhake, B. Brgmann,S. Husa, Phys. Rev. Lett. 98, 231101 (2007).

12. J. L. Jaramillo, R. P. Macedo, P. Moesta, L. Rezzolla,Phys. Rev. D Part. Fields Gravit. Cosmol. 85, 084030(2012).

13. H. P. Pfeiffer, Class. Quant. Grav., in press; available athttp://xxx.lanl.gov/abs/1203.5166v2.

14. www.ligo.caltech.edu/, www.ligo.org/, www.cascina.virgo.infn.it, and http://gwcenter.icrr.u-tokyo.ac.jp/en/

15. LIGO Scientific Collaboration and Virgo Collaboration,Sensitivity achieved by the LIGO and Virgo gravitationalwave detectors during LIGOs sixth and Virgos secondand third science runs, http://xxx.lanl.gov/abs/1203.2674.

16. All published results are available at https://www.lsc-group.phys.uwm.edu/ppcomm/Papers.html.

17. https://www.advancedligo.mit.edu/ and https://www.cascina.virgo.infn.it/advirgo/

Acknowledgments: For insights into the geometrodynamics ofblack holes, I thank my research collaborators: J. Brink,Y. Chen, J. Kaplan, G. Lovelace, K. Matthews, D. Nichols,R. Owen, M. Scheel, F. Zhang, and A. Zimmerman. Mygeometrodynamic research is supported by the ShermanFairchild Foundation, the Brinson Foundation, and NSFgrant PHY-1068881.

Supplementary Materialswww.sciencemag.org/cgi/content/full/337/6094/536/DC1Movie S1

10.1126/science.1225474

PERSPECTIVE

Quantum Mechanics of Black HolesEdward Witten

The popular conception of black holes reflects the behavior of the massive black holes foundby astronomers and described by classical general relativity. These objects swallow up whatevercomes near and emit nothing. Physicists who have tried to understand the behavior of black holesfrom a quantum mechanical point of view, however, have arrived at quite a different picture.The difference is analogous to the difference between thermodynamics and statistical mechanics.The thermodynamic description is a good approximation for a macroscopic system, but statisticalmechanics describes what one will see if one looks more closely.

In quantum mechanics, if a time-dependenttransition is possible from an initial state |i>to a final state | f >, then it is also possible tohave a transition in the opposite direction from| f > to |i>. The most basic reason for this is thatthe sum of quantum mechanical probabilities mustalways equal 1. Starting from this fact, one canshow that, on an atomic time scale, there are equalprobabilities for a transition in one direction orthe other (1).

This seems, at first, to contradict the wholeidea of a black hole. Let B denote a macroscopicblack holeperhaps the one in the center of theMilky Wayand let A be some other macro-

scopic body, perhaps a rock or an astronaut. Fi-nally, let B* be a heavier black hole that can bemade by combining A and B. General relativitytells us that the reaction A + B B* will occurwhenever A and B get close enough. Quantummechanics tells us, then, that the reverse reactionB* A + B can also happen, with an equiv-alent amplitude.

The reverse reaction, though, is one in whichthe heavier black hole B* spontaneously emitsthe body A, leaving behind a lighter black holeB. That reverse reaction is exactly what does nothappen, according to classical general relativ-ity. Indeed, the nonoccurrence of the reverse re-action, in which a black hole re-emits whateverit has absorbed, is often stated as the definingproperty of a black hole.

It seems, then, that black holes are impos-sible in light of quantum mechanics. To learnmore, let us consider another physical principlethat is also seemingly violated by the existenceof a black hole. This is time-reversal symmetry,which says that if a physical process is possible,then the time-reversed process is also possible.Clearly, black holes seem to violate this as well.

Time reversal is a subtle concept, and ele-mentary particle physicists have made some un-expected discoveries about it (2). However, forapplications to black holes, the important prob-lem with time reversal is that in everyday life, itsimply does not appear to be valid. We can spilla cup of water onto the ground, but the waternever spontaneously jumps up into the cup.

The explanation has to do with randomnessat the atomic level, usually called entropy. Spill-ing the cup of water is an irreversible operationin practice, because it greatly increases the num-ber of states available to the system at the atomiclevel, even after one specifies all of the variablessuch as temperature, pressure, the amount ofwater, the height of the water above the ground,and so onthat are visible macroscopically. Thewater could jump back up into the cup if the ini-tial conditions are just right at the atomic level,but this is prohibitively unlikely.

The second law of thermodynamics says thatin a macroscopic system, like a cup of water, aprocess that reduces the randomness or entropyin the universe can never happen. Now supposethat we consider what is sometimes called a

School of Natural Sciences, Institute for Advanced Study,Princeton, NJ 08540, USA. E-mail: witten@ias.edu

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mesoscopic systemmuch larger than an atom,but not really macroscopic. For example, we couldconsider 100 water molecules instead of a wholecupful. Then we should use statistical mechanics,which tells us that a rare fluctuation in which ran-domness appears to diminish can happen, butvery rarely. Finally, at the level of a single particleor a handful of particles, we should focus on thefundamental dynamical equations: Newtons lawsand their modern refinements. These fundamen-tal laws are completely reversible.

Since the late 19th century, physicists haveunderstood that thermodynamic irreversibilityarises spontaneously by applying reversible equa-tions to a macroscopic system, but it has alwaysbeen vexingly hard to make this concrete.

Black Hole Entropy and Hawking RadiationNow let us go back to the conflict between blackholes and quantummechanics. What is really wrongwith the reverse reaction B* A + B, whereina heavier black hole B* decays to a lighter blackhole B plus some other system A?

A great insight of the 1970s [originating froma suggestion by Bekenstein (3)] is that what iswrong with the reverse reaction involving blackholes is just like what is wrong with a time-reversed movie in which a puddle of water fliesoff the wet ground and into the cup. A blackhole should be understood as a complex systemwith an entropy that increases as it grows.

In a sense, this entropy measures the igno-rance of an outside observer about what thereis inside the black hole. When a black hole Babsorbs some other system A in the process A +B B*, its entropy increases, along with itsmass, in keeping with the second law of ther-modynamics. The reverse reaction B* A + Bdiminishes the entropy of the black hole as wellas its mass, so it violates the second law.

How can one test this idea? If the irreversibilityfound in black hole physics is really the sort ofirreversibility found in thermodynamics, then itshould break down if A is not a macroscopic sys-tem but a single elementary particle. Although awhole cupful of water never jumps off the floorand into the cup, a single water molecule certainlymight do this as a result of a lucky fluctuation.

This is what Hawking found in a celebratedcalculation (4). A black hole spontaneously emitselementary particles. The typical energy of theseparticles is proportional to Plancks constant, so theeffect is purely quantum mechanical in nature, andthe rate of particle emission by a black hole of as-tronomical size is extraordinarily small, far too smallto be detected. Still, Hawking's insight means that ablack hole is potentially no different from any otherquantum system, with reactions A + B B* andB* A + B occurring in both directions at themicroscopic level. The irreversibility of classicalblack hole physics is just like the familiar irrevers-ibility of thermodynamics, valid when what is ab-sorbed by the black hole is a macroscopic system.

Although it was a shock at the time, perhapsin hindsight we should not be surprised that clas-sical general relativity does not describe proper-ly the emission of an atom or elementary particlefrom a black hole. After all, classical general rel-ativity is not a useful theory of atoms and indi-vidual elementary particles, or quantum mechanicswould never have been needed.

However, general relativity is a good theoryof macroscopic bodies, and when it tells us thata black hole can absorb a macroscopic body butcannot emit one, we should listen.

Black Holes and the Rest of PhysicsIs the quantum theory of black holes just a theo-retical construct, or can we test it? Unfortunately,the usual astrophysical black holes, formed fromstellar collapse or in the centers of galaxies, aremuch too big and too far away for their micros-copic details to be relevant. However, one of thecornerstones of modern cosmology is the studyof the cosmic microwave radiation that wascreated in the Big Bang. It has slightly differenttemperatures in different directions. The theoryof how this came about is in close parallel withthe theory of Hawking radiation from black holes,and its success adds to our confidence that theHawking theory is correct.

Surprisingly, in the past 15 years, the theory ofHawking radiation and related ideas about quan-tum black holes have turned out to be useful fortheoretical physicists working on a variety of moredown-to-earth problems. To understand how thishappened, we need one more idea from the earlyperiod.

The Membrane Paradigm for Black HolesThe entropy of an ordinary body like Earth or theSun is basically a volume integral; to compute theentropy, one computes the entropy density and in-tegrates it over the interior of Earth or the Sun.

Black holes seem to be different. According toBekenstein and Hawking, the entropy of a blackhole is proportional to the surface area of the blackhole, not to its volume. This observation led in theearly days to the membrane paradigm for blackholes (5). The idea of the membrane paradigm isthat the interactions of a black hole with particlesand fields outside the hole can be modeled bytreating the surface or horizon of the black hole as amacroscopic membrane. The membrane associ-ated with a black hole horizon is characterized bymacroscopic properties rather similar to those thatone would use to characterize any ordinary mem-brane. For example, the black holemembrane hastemperature, entropy density, viscosity, and elec-trical conductivity.

In short, there was a satisfactory thermodynam-ic theory of black hole membranes, but can one gofarther and make a microscopic theory that de-scribes these membranes? An optimist, given theideas of the 1980s, might hope that some sort ofquantum field theory would describe the horizon

of a black hole. What sort of theory would thisbe, and how could one possibly find it?

Gauge-Gravity DualityThe known forces in nature other than gravityare all well described in the standard model ofparticle physics in terms of quantum field theoriesthat are known as gauge theories. The prototypeis Maxwells theory of electromagnetism, inter-preted in modern times as a gauge theory.

Quantum gauge theory is a subtle yet relativelywell-understood and well-established subject. Theprinciples are known, but the equations are hard tosolve. On the other hand, quantum gravity is muchmore mysterious. String theory has given some in-sight, but the foundations are still largely unknown.

In the 1990s, string theorists began to discoverthat aspects of black hole physics can be modeledby gauge theory (6, 7). Such insights led to a remark-able new way to use gauge theory to study blackholes and other problems of quantum gravity (8).

This relied on the fact that string theory hasextended objects known as branes (9), which arerather like membranes except that in general theyare not two-dimensional. In fact, theword braneis a riff on membrane.

Branes can be described by gauge theory; onthe other hand, because black holes can be madeout of anything at all, they can be made out ofbranes. When one does this, one finds that themembrane that describes the horizon of the blackhole is the string theory brane.

Of course, we are cutting corners with thisvery simple explanation. One has to construct astring theory model with a relatively large neg-ative cosmological constant (in contrast with thevery small positive one of the real world), and then,under appropriate conditions, one gets a gaugetheory description of the black hole horizon.

Solving the Equations of Gauge TheoryGauge-gravity duality was discovered with the aimof learning about quantum gravity and black holes.One can turn the relationship between these twosubjects around and ask whether it can help usbetter understand gauge theory.

Even thoughgauge theory is thewell-establishedframework for our understanding of much of phys-ics, this does not mean that it is always well under-stood. Often, even if one asks a relatively simplequestion, the equations turn out to be intractable.

In the past decade, the gauge theory descrip-tion of black holes has been useful in at least twoareas of theoretical physics. One involves heavyion collisions, studied at the Relativistic Heavy IonCollider at Brookhaven. The expanding fireballcreated in a collision of two heavy nuclei turns outto be a droplet of nearly ideal fluid. In principle,this should all be described by known equations ofgauge theoryquantum chromodynamics, to beprecisebut the equations are intractable. It turnsout that by interpreting the gauge theory as a de-scription of a black hole horizon, and using the

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Einstein equations to describe the black hole, onecan get striking insight about a quantum almost-ideal fluid (10). This has become an importanttechnique in modeling heavy ion collisions.

Condensed matter physics is described in prin-ciple by the Schrdinger equation of electrons andnuclei, but for most systems, a full understandingbased on the Schrdinger equation is way out ofreach. Nowadays, there is great interest in under-standing quantum critical behavior in quasitwo-dimensional systems such as high temperaturesuperconductors. These systems are studied by awide variety of methods, and no one approach islikely to be a panacea. Still, it has turned out to bevery interesting to study two-dimensional quantumcritical systems bymapping them to the horizon ofa black hole (11). With this approach, one can per-form calculations that are usually out of reach.

Among other things, this method has beenused to analyze the crossover from quantum to

dissipative behavior in model systems with a de-gree of detail that is not usually possible. In asense, this brings our story full circle. The storybegan nearly 40 years ago with the initial insightthat the irreversibility of black hole physics is anal-ogous to the irreversibility described by the sec-ond law of thermodynamics. In general, to reconcilethis irreversibility with the reversible nature of thefundamental equations is tricky, and explicit cal-culations are not easy to come by. The link be-tween ordinary physics and black hole physics thatis given by gauge-gravity duality has given physi-cists a powerful way to do precisely this. This givesus confidence that we are on the right track inunderstanding quantum black holes, and it alsoexhibits the unity of physics in a most pleasing way.

References and Notes1. The precise mathematical argument uses the fact that

the Hamiltonian operator H is hermitian, so that the

transition amplitude is the complex conjugateof the transition amplitude in the oppositedirection.

2. The precise time-reversal symmetry of nature alsoincludes reflection symmetry and charge conjugation.

3. J. Bekenstein, Phys. Rev. D Part. Fields 7, 2333(1973).

4. S. W. Hawking, Nature 248, 30 (1974).5. K. S. Thorne, D. A. MacDonald, R. H. Price, Eds.,

Black Holes: The Membrane Paradigm (Yale Univ.Press, New Haven, CT 1986).

6. A. Strominger, C. Vafa, Phys. Lett. B 379, 99 (1996).7. S. S. Gubser, I. R. Klebanov, A. W. Peet, Phys. Rev. D

Part. Fields 54, 3915 (1996).8. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).9. M. J. Duff, R. R. Khuri, J.-X. Lu, Phys. Rep. 259, 213

(1995).10. P. K. Kovtun, D. T. Son, A. O. Starinets, Phys. Rev. Lett.

94, 111601 (2005).11. S. Sachdev, Annu. Rev. Cond. Matt. Phys. 3, 9 (2012).

Acknowledgments: This research was supported in partby NSF grant PHY-096944.

10.1126/science.1221693

REVIEW

Stellar-Mass Black Holesand Ultraluminous X-ray SourcesRob Fender1* and Tomaso Belloni2

We review the likely population, observational properties, and broad implications of stellar-massblack holes and ultraluminous x-ray sources. We focus on the clear empirical rules connectingaccretion and outflow that have been established for stellar-mass black holes in binary systems inthe past decade and a half. These patterns of behavior are probably the keys that will allow usto understand black hole feedback on the largest scales over cosmological time scales.

Stellar-mass black holes are the end pointsof the evolution of the most massive stars.The collapse of an iron core of >3 solarmasses (M) cannot be stopped by either electronor neutron degeneracy pressure (whichwould other-wise result in a white dwarf, or neutron star, respec-tively). Within the framework of classical generalrelativity (GR), the core collapses to a singularitythat is cloaked in an event horizon before it can beviewed. Like a giant elementary particle, the result-ing black hole is then entirely described by threeparameters: mass, spin, and charge (1). Becausegalaxies are oldthe Milky Way is at least 13 bil-lion years oldand the most massive stars evolvequickly (within millions of years or less), thereare likely to be a large number of such stellar-massblack holes in our galaxy alone. Shapiro andTeukolsky (2) calculated that there were likely tobe as many as 108 stellar-mass black holes in our

galaxy, under the assumption that all stars of ini-tial mass >10 times that of the Sun met this fate.

The strongest evidence for the existence of thispopulation of stellar-mass black holes comes fromobservations of x-ray binary systems (XRBs). InXRBs, matter is accreted (gravitationally capturedinto/onto the accretor), releasing large amounts ofgravitational potential energy in the process. Theefficiency of this process in releasing the gravita-tional potential energy is determined by the ratioof mass to radius of the accretor. For neutronstars, more than 10% of the rest mass energy canbe releaseda process more efficient at energy re-lease than nuclear fusion. For black holes, the ef-ficiency can be even higher (3), but the presence ofan event horizonfromwithinwhich no signals canever be observed in the outside universemeansthat this accretion power may be lost.

In some of these systems, dynamical mea-surements of the orbit indicate massive (>3 M)accretors that, independently, show no evidencefor any emission from a solid surface. The firstsuch candidate black hole x-ray binary (BHXRB)system detected was Cygnus X-1, which led to abet between Kip Thorne and Stephen Hawkingas to the nature of the accreting object (although

both were moderately certain it was a black hole,Hawking wanted an insurance policy). In 1990,Hawking conceded the bet, accepting that thesource contained a black hole. Since then, astron-omers have discovered many hundreds of x-raybinaries within the Milky Way and beyond, severaltens of which are good candidate BHXRBs (4).

Over the past decade, repeating empirical pat-terns connecting the x-ray, radio, and infrared emis-sion from these objects have been found and usedto connect these observations to physical compo-nents of the accretion flow (Fig. 1). It is likely thatsome of these empirical patterns of behavior alsoapply to accreting supermassive (105 to 109 M)black holes in the centers of some galaxies, andthat from studying BHXRBs on humanly accessi-ble time scales,wemaybe learning about the forcesthat shaped the growth of galaxies over the life-time of the universe. Between the stellar-mass blackholes and the supermassive, there could be a popu-lation of intermediate-mass black holes (IMBHs),with masses in the range of 102 to 105 M. Thesemay be related to the ultra-luminous x-ray sources(ULXs), very luminous x-ray sources that have beenobserved in external galaxies.However, the problemof the nature of these sources is still unsettled, andalternative options involving stellar-mass blackholes are still open.

Black Hole X-ray BinariesThere are several different approaches to clas-sifying BHXRBs and their behavior, each ofwhich can lead to different physical insights. Oneimportant approach is to look at the orbital pa-rameters, and the most important of these is themass of the donor star because it relates to theage of the binary. High-mass x-ray binaries haveOB-type (5) massive donors and are young sys-tems, typically with ages less than a million yearsor so. They are clustered close to the midplane ofthe Galactic disc and associated with star-forming

1Physics and Astronomy, University of Southampton, SouthamptonSO17 1BJ, UK. 2Istituto Nazionale di AstrofisicaOsservatorioAstronomico di Brera, Via Emilio Bianchi 46, I-23807 Merate(LC), Italy.

*To whom correspondence should be addressed. E-mail:r.fender@soton.ac.uk

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