Black Holes
collaboration of numerical relativity groups is gen- will use their merger observations to test numer- 9. For example, the simulation by Owen described erating a large catalog of waveforms, and a dic- ical relativity’sgeometrodynamicpredictions.This in figure 5 of (4). 10. M. Campanelli, C. O. Lousto, Y. Zlochower, D. Merritt, tionary of the information in the waveforms (13). will be just one of many science payoffs from Phys. Rev. Lett. 98,231102(2007). These waveforms will be observed by the LIGO and its partners, but it is the payoff that 11. J. A. González, M. Hannam, U. Sperhake, B. Brügmann, Laser Interferometer Gravitational Wave Observ- excites me the most. S. Husa, Phys. Rev. Lett. 98,231101(2007). atory (LIGO), with interferometers in Livingston, 12. J. L. Jaramillo, R. P. Macedo, P. Moesta, L. Rezzolla, Phys. Rev. D Part. Fields Gravit. Cosmol. 85,084030 Louisiana, and Hanford, Washington, and by (2012). References and Notes LIGO’sinternationalpartners:theVirgointerferom- 13. H. P. Pfeiffer, Class. Quant. Grav., in press; available at 1. J. A. Wheeler, Geometrodynamics (Academic Press, eter near Pisa, Italy, and (just recently funded) the http://xxx.lanl.gov/abs/1203.5166v2. New York, 1962). 14. www.ligo.caltech.edu/, www.ligo.org/, www.cascina.virgo. Kamioka Gravitational Wave Antenna (KAGRA) 2. B. Carter, S. Chandrasekhar, G. W. Gibbons, in General infn.it, and http://gwcenter.icrr.u-tokyo.ac.jp/en/ in the Kamioka mine in Japan (14). The initial Relativity, An Einstein Centenary Survey,S.W.Hawking, 15. LIGO Scientific Collaboration and Virgo Collaboration, W. Israel, Eds. (Cambridge Univ. Press, Cambridge, 1979), LIGO and Virgo interferometers (with sensitivities “Sensitivity achieved by the LIGO and Virgo gravitational chapters 6, 7, and 13. at which it is plausible but not likely to see waves) wave detectors during LIGO’ssixthandVirgo’ssecond 3. E. Witten, Science 337,538(2012). and third science runs,” http://xxx.lanl.gov/abs/1203.2674. were designed to give the experimental teams 4. R. Owen et al., Phys. Rev. Lett. 106,151101(2011). 16. All published results are available at https://www.lsc-group. enough experience to perfect the techniques and 5. If two freely falling particles have a vector separation phys.uwm.edu/ppcomm/Papers.html. Dx, then (i) the tidal field gives them a relative design for advanced interferometers—that will 17. https://www.advancedligo.mit.edu/ and https://www.cascina. acceleration Da = −E·D x,and(ii)gyroscopescarried have sensitivities at which they are likely to see virgo.infn.it/advirgo/ by the particles precess with respect to each other with lots of waves. angular velocity DW = B·D x. Acknowledgments: For insights into the geometrodynamics of Initial LIGO and Virgo exceeded and reached 6. Each tendex line (or vortex line) is the integral black holes, I thank my research collaborators: J. Brink, curve of an eigenvector field of the tidal field E their design sensitivities, respectively (15), and Y. Chen, J. Kaplan, G. Lovelace, K. Matthews, D. Nichols, (or frame-drag field B), and the line’stendicity(or carried out searches for waves in 2006–2007 and R. Owen, M. Scheel, F. Zhang, and A. Zimmerman. My vorticity) is that eigenvector’seigenvalue.Aknowledge geometrodynamic research is supported by the Sherman 2009–2010. As was expected, no waves have been of the tendex and vortex lines and their tendicities Fairchild Foundation, the Brinson Foundation, and NSF seen, though the data are still being analyzed and vorticities is equivalent to a full knowledge of grant PHY-1068881. and many interesting results have been obtained the Riemann tensor. 7. For pictures and movies from these and other simulations (16). The advanced LIGO and advanced Virgo Supplementary Materials
see, e.g., http://black-holes.org/explore2.html and on September 12, 2012 interferometers are now being installed (17)and www.sciencemag.org/cgi/content/full/337/6094/536/DC1 http://numrel.aei.mpg.de/images. by 2017 should reach sensitivities at which black- 8. G. Lovelace et al., Phys. Rev. D Part. Fields Gravit. Cosmol. Movie S1 hole mergers are observed. The LIGO/Virgo team 82,064031(2010). 10.1126/science.1225474
It seems, then, that black holes are impos- PERSPECTIVE sible in light of quantum mechanics. To learn more, let us consider another physical principle that is also seemingly violated by the existence
Quantum Mechanics of Black Holes of a black hole. This is time-reversal symmetry, www.sciencemag.org which says that if a physical process is possible, Edward Witten then the time-reversed process is also possible. Clearly, black holes seem to violate this as well. The popular conception of black holes reflects the behavior of the massive black holes found Time reversal is a subtle concept, and ele- by astronomers and described by classical general relativity. These objects swallow up whatever mentary particle physicists have made some un- comes near and emit nothing. Physicists who have tried to understand the behavior of black holes expected discoveries about it (2). However, for from a quantum mechanical point of view, however, have arrived at quite a different picture. applications to black holes, the important prob- The difference is analogous to the difference between thermodynamics and statistical mechanics. lem with time reversal is that in everyday life, it Downloaded from The thermodynamic description is a good approximation for a macroscopic system, but statistical simply does not appear to be valid. We can spill mechanics describes what one will see if one looks more closely. acupofwaterontotheground,butthewater never spontaneously jumps up into the cup. nquantummechanics,ifatime-dependent scopic body, perhaps a rock or an astronaut. Fi- The explanation has to do with randomness transition is possible from an initial state |i> nally, let B* be a heavier black hole that can be at the atomic level, usually called entropy. Spill- Ito a final state | f >, then it is also possible to made by combining A and B. General relativity ing the cup of water is an irreversible operation have a transition in the opposite direction from tells us that the reaction A+B→ B* will occur in practice, because it greatly increases the num- | f>to |i>. The most basic reason for this is that whenever A and B get close enough. Quantum ber of states available to the system at the atomic the sum of quantum mechanical probabilities must mechanics tells us, then, that the reverse reaction level, even after one specifies all of the variables— always equal 1. Starting from this fact, one can B* → A+Bcan also happen, with an equiv- such as temperature, pressure, the amount of show that, on an atomic time scale, there are equal alent amplitude. water, the height of the water above the ground, probabilities for a transition in one direction or The reverse reaction, though, is one in which and so on—that are visible macroscopically. The the other (1). the heavier black hole B* spontaneously emits water could jump back up into the cup if the ini- This seems, at first, to contradict the whole the body A,leavingbehindalighterblackhole tial conditions are just right at the atomic level, idea of a black hole. Let B denote a macroscopic B. That reverse reaction is exactly what does not but this is prohibitively unlikely. black hole—perhaps the one in the center of the happen, according to classical general relativ- The second law of thermodynamics says that Milky Way—and let A be some other macro- ity. Indeed, the nonoccurrence of the reverse re- in a macroscopic system, like a cup of water, a action, in which a black hole re-emits whatever process that reduces the randomness or entropy School of Natural Sciences, Institute for Advanced Study, it has absorbed, is often stated as the defining in the universe can never happen. Now suppose Princeton, NJ 08540, USA. E-mail: [email protected] property of a black hole. that we consider what is sometimes called a
538 3AUGUST2012 VOL337 SCIENCE www.sciencemag.org SPECIALSECTION mesoscopic system—much larger than an atom, Although it was a shock at the time, perhaps of a black hole. What sort of theory would this but not really macroscopic. For example, we could in hindsight we should not be surprised that clas- be, and how could one possibly find it? consider 100 water molecules instead of a whole sical general relativity does not describe proper- cupful. Then we should use statistical mechanics, ly the emission of an atom or elementary particle Gauge-Gravity Duality which tells us that a rare fluctuation in which ran- from a black hole. After all, classical general rel- The known forces in nature other than gravity domness appears to diminish can happen, but ativity is not a useful theory of atoms and indi- are all well described in the standard model of very rarely. Finally, at the level of a single particle vidual elementary particles, or quantum mechanics particle physics in terms of quantum field theories or a handful of particles, we should focus on the would never have been needed. that are known as gauge theories. The prototype fundamental dynamical equations: Newton’slaws However, general relativity is a good theory is Maxwell’stheoryofelectromagnetism,inter- and their modern refinements. These fundamen- of macroscopic bodies, and when it tells us that preted in modern times as a gauge theory. tal laws are completely reversible. ablackholecanabsorbamacroscopicbodybut Quantum gauge theory is a subtle yet relatively Since the late 19th century, physicists have cannot emit one, we should listen. well-understood and well-established subject. The understood that thermodynamic irreversibility principles are known, but the equations are hard to arises spontaneously by applying reversible equa- Black Holes and the Rest of Physics solve. On the other hand, quantum gravity is much tions to a macroscopic system, but it has always Is the quantum theory of black holes just a theo- more mysterious. String theory has given some in- been vexingly hard to make this concrete. retical construct, or can we test it? Unfortunately, sight, but the foundations are still largely unknown. the usual astrophysical black holes, formed from In the 1990s, string theorists began to discover Black Hole Entropy and Hawking Radiation stellar collapse or in the centers of galaxies, are that aspects of black hole physics can be modeled Now let us go back to the conflict between black much too big and too far away for their micros- by gauge theory (6, 7). Such insights led to a remark- holes and quantum mechanics. What is really wrong copic details to be relevant. However, one of the able new way to use gauge theory to study black with the reverse reaction B* → A+B,wherein cornerstones of modern cosmology is the study holes and other problems of quantum gravity (8). aheavierblackholeB* decays to a lighter black of the cosmic microwave radiation that was This relied on the fact that string theory has hole B plus some other system A? created in the Big Bang. It has slightly different extended objects known as branes (9), which are Agreatinsightofthe1970s[originatingfrom temperatures in different directions. The theory rather like membranes except that in general they asuggestionbyBekenstein(3)] is that what is of how this came about is in close parallel with are not two-dimensional. In fact, the word “brane” wrong with the reverse reaction involving black the theory of Hawking radiation from black holes, is a riff on “membrane.” on September 12, 2012 holes is just like what is wrong with a time- and its success adds to our confidence that the Branes can be described by gauge theory; on reversed movie in which a puddle of water flies Hawking theory is correct. the other hand, because black holes can be made off the wet ground and into the cup. A black Surprisingly, in the past 15 years, the theory of out of anything at all, they can be made out of hole should be understood as a complex system Hawking radiation and related ideas about quan- branes. When one does this, one finds that the with an entropy that increases as it grows. tum black holes have turned out to be useful for membrane that describes the horizon of the black In a sense, this entropy measures the igno- theoretical physicists working on a variety of more hole is the string theory brane. rance of an outside observer about what there down-to-earth problems. To understand how this Of course, we are cutting corners with this is inside the black hole. When a black hole B happened, we need one more idea from the early very simple explanation. One has to construct a absorbs some other system A in the process A+ period. string theory model with a relatively large neg- www.sciencemag.org B → B*,itsentropyincreases,alongwithits ative cosmological constant (in contrast with the mass, in keeping with the second law of ther- The Membrane Paradigm for Black Holes very small positive one of the real world), and then, modynamics. The reverse reaction B* → A+B The entropy of an ordinary body like Earth or the under appropriate conditions, one gets a gauge diminishes the entropy of the black hole as well Sun is basically a volume integral; to compute the theory description of the black hole horizon. as its mass, so it violates the second law. entropy, one computes the entropy density and in- How can one test this idea? If the irreversibility tegrates it over the interior of Earth or the Sun. Solving the Equations of Gauge Theory found in black hole physics is really the sort of Black holes seem to be different. According to Gauge-gravity duality was discovered with the aim irreversibility found in thermodynamics, then it Bekenstein and Hawking, the entropy of a black of learning about quantum gravity and black holes. Downloaded from should break down if A is not a macroscopic sys- hole is proportional to the surface area of the black One can turn the relationship between these two tem but a single elementary particle. Although a hole, not to its volume. This observation led in the subjects around and ask whether it can help us whole cupful of water never jumps off the floor early days to the membrane paradigm for black better understand gauge theory. and into the cup, a single water molecule certainly holes (5). The idea of the membrane paradigm is Even though gauge theory is the well-established might do this as a result of a lucky fluctuation. that the interactions of a black hole with particles framework for our understanding of much of phys- This is what Hawking found in a celebrated and fields outside the hole can be modeled by ics, this does not mean that it is always well under- calculation (4). A black hole spontaneously emits treatingthe surfaceor horizon of the black hole as a stood. Often, even if one asks a relatively simple elementary particles. The typical energy of these macroscopic membrane. The membrane associ- question, the equations turn out to be intractable. particles is proportional to Planck’sconstant,sothe ated with a black hole horizon is characterized by In the past decade, the gauge theory descrip- effect is purely quantum mechanical in nature, and macroscopic properties rather similar to those that tion of black holes has been useful in at least two the rate of particle emission by a black hole of as- one would use to characterize any ordinary mem- areas of theoretical physics. One involves heavy tronomical size is extraordinarily small, far too small brane. For example, the black hole membrane has ion collisions, studied at the Relativistic Heavy Ion to be detected. Still, Hawking's insight means that a temperature, entropy density, viscosity, and elec- Collider at Brookhaven. The expanding fireball black hole is potentially no different from any other trical conductivity. created in a collision of two heavy nuclei turns out quantum system, with reactions A+B→ B* and In short, there was a satisfactory thermodynam- to be a droplet of nearly ideal fluid. In principle, B* → A+Boccurring in both directions at the ic theory of black hole membranes, but can one go this should all be described by known equations of microscopic level. The irreversibility of classical farther and make a microscopic theory that de- gauge theory—quantum chromodynamics, to be black hole physics is just like the familiar irrevers- scribes these membranes? An optimist, given the precise—but the equations are intractable. It turns ibility of thermodynamics, valid when what is ab- ideas of the 1980s, might hope that some sort of out that by interpreting the gauge theory as a de- sorbed by the black hole is a macroscopic system. quantum field theory would describe the horizon scription of a black hole horizon, and using the
www.sciencemag.org SCIENCE VOL 337 3 AUGUST 2012 539 Black Holes
Einstein equations to describe the black hole, one dissipative behavior in model systems with a de- transition amplitude
REVIEW both were moderately certain it was a black hole, Hawking wanted an insurance policy). In 1990, Hawking conceded the bet, accepting that the Stellar-Mass Black Holes source contained a black hole. Since then, astron- on September 12, 2012 omers have discovered many hundreds of x-ray binaries within the Milky Way and beyond, several and Ultraluminous X-ray Sources tens of which are good candidate BHXRBs (4). Rob Fender1* and Tomaso Belloni2 Over the past decade, repeating empirical pat- terns connecting the x-ray, radio, and infrared emis- We review the likely population, observational properties, and broad implications of stellar-mass sion from these objects have been found and used black holes and ultraluminous x-ray sources. We focus on the clear empirical rules connecting to connect these observations to physical compo- accretion and outflow that have been established for stellar-mass black holes in binary systems in nents of the accretion flow (Fig. 1). It is likely that
the past decade and a half. These patterns of behavior are probably the keys that will allow us some of these empirical patterns of behavior also www.sciencemag.org 5 9 to understand black hole feedback on the largest scales over cosmological time scales. apply to accreting supermassive (10 to 10 M⊙) black holes in the centers of some galaxies, and tellar-mass black holes are the end points galaxy, under the assumption that all stars of ini- that from studying BHXRBs on humanly accessi- of the evolution of the most massive stars. tial mass >10 times that of the Sun met this fate. ble time scales, we may be learning about the forces SThe collapse of an iron core of >3 solar The strongest evidence for the existence of this that shaped the growth of galaxies over the life- masses (M⊙)cannotbestoppedbyeitherelectron population of stellar-mass black holes comes from time of the universe. Between the stellar-mass black or neutron degeneracy pressure (which would other- observations of x-ray binary systems (XRBs). In holes and the supermassive, there could be a popu- wise result in a white dwarf, or neutron star, respec- XRBs, matter is accreted (gravitationally captured lation of intermediate-mass black holes (IMBHs), Downloaded from 2 5 tively). Within the framework of classical general into/onto the accretor), releasing large amounts of with masses in the range of 10 to 10 M⊙.These relativity (GR), the core collapses to a singularity gravitational potential energy in the process. The may be related to the ultra-luminous x-ray sources that is cloaked in an event horizon before it can be efficiency of this process in releasing the gravita- (ULXs), very luminous x-ray sources that have been viewed. Like a giant elementary particle, the result- tional potential energy is determined by the ratio observed in external galaxies. However, the problem ing black hole is then entirely described by three of mass to radius of the accretor. For neutron ofthe nature of these sources is still unsettled, and parameters: mass, spin, and charge (1). Because stars, more than 10% of the rest mass energy can alternative options involving stellar-mass black galaxies are old—the Milky Way is at least 13 bil- be released—aprocessmoreefficientatenergyre- holes are still open. lion years old—and the most massive stars evolve lease than nuclear fusion. For black holes, the ef- quickly (within millions of years or less), there ficiency can be even higher (3), but the presence of Black Hole X-ray Binaries are likely to be a large number of such stellar-mass an event horizon—from within which no signals can There are several different approaches to clas- black holes in our galaxy alone. Shapiro and ever be observed in the outside universe—means sifying BHXRBs and their behavior, each of Teukolsky (2)calculatedthattherewerelikelyto that this accretion power may be lost. which can lead to different physical insights. One be as many as 108 stellar-mass black holes in our In some of these systems, dynamical mea- important approach is to look at the orbital pa- surements of the orbit indicate massive (>3 M⊙) rameters, and the most important of these is the accretors that, independently, show no evidence mass of the donor star because it relates to the 1 Physics and Astronomy, University of Southampton, Southampton for any emission from a solid surface. The first age of the binary. High-mass x-ray binaries have SO17 1BJ, UK. 2Istituto Nazionale di Astrofisica–Osservatorio Astronomico di Brera, Via Emilio Bianchi 46, I-23807 Merate such candidate black hole x-ray binary (BHXRB) OB-type (5)massivedonorsandareyoungsys- (LC), Italy. system detected was Cygnus X-1, which led to a tems, typically with ages less than a million years *To whom correspondence should be addressed. E-mail: bet between Kip Thorne and Stephen Hawking or so. They are clustered close to the midplane of [email protected] as to the nature of the accreting object (although the Galactic disc and associated with star-forming
540 3AUGUST2012 VOL337 SCIENCE www.sciencemag.org