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~ Achievements an] Opportunities in ~ravitationat Physics I~ G~L wins Key Questions It is a tenet of Einstein's 1905 special relativity that no information can be transmitted or carried in any way at a speed faster than that of light, an idea prefigured in Maxwell's earlier theory of electromagnetic waves. When general relativity was worked out by Einstein using special relativity as a base, it was natural that it should predict that moving masses would communicate their changed gravitational fields at the speed of light, through the propagation of gravitational waves. Gravitational waves are a quintessential relativistic strong- gravitational-field phenomenon one that is completely absent in Newtonian gravitational theory. In the years since general relativity was proposed, many of its predictions have been spectacularly verified, but a few key features still remain unconfirmed. (See Section IV of this chapter.) Remarkably, one idea yet to be fully checked is the feature most closely related to the principle of relativity: gravitational waves. Just as interestingly, the eventual detection of gravitational waves will probably provide the best possible way to verify the other most spectacular of the unveri- fied predictions of general relativity: the existence of black holes. There are practical reasons that the earliest relativistic idea about gravity might be among the last to be verified. Compared to electric or magnetic forces, 32

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 33 gravity is extremely weak. This means that it is much harder to construct a practical receiver for gravitational waves than it is to construct an electromag- netic (e.g., radio) receiver. Worse, construction of a gravitational wave generator (or transmitter) from laboratory-scale components is hopeless. Einstein himself thought gravitational waves might never be detected. As is described below, new technology just coming on line is expected to be able to detect gravitational waves generated by the rapid motion of astronomical bodies whose masses are comparable to those of stars. In a radio receiver, the reception of an electromagnetic wave begins with the acceleration of electrons in the receiver's antenna by the electric field component of the electromagnetic wave. Similarly, a gravitational wave will cause motions among a set of masses that are free to move. Only relative motions are meaning- ful, though, because all objects are required to have the same free fall motion by the principle of equivalence. (For more on this principle, see Box 3.3 in Section IV of this chapter.) The measurable effect of a gravitational wave is a distortion in the distances between a set of free masses, characterized by the fractional change in distances AL/L. (It is traditional to refer to the wave amplitude h _ 2AL/L.) The strongest waves arriving regularly at Earth (say, several times per year) are expected to cause fractional length changes AL/L between pairs of detector masses separated by a distance L of no larger than about 1 part in 102~. It is this fact that scales the technological challenge of detecting gravitational waves. Still, the reception and study of gravitational waves can help answer many key ques- tions in basic physics and astrophysics: Do waves such as those predicted by Einstein propagate away from dy- namic massive objects, and do they interact with test bodies in the way described by general relativity? . Do gravitational waves propagate at the speed of light, and do they have the polarization that general relativity predicts? What is the nature of gravity in the strong-field regime where general relativity makes its most dramatic predictions? . Do black holes exist? What are the properties of the highly relativistic spacetime just outside their horizons? Can we use neutron star and white dwarf binary systems to study gravita- tional physics? Are massive black hole binaries present in galactic centers? What is the state of matter inside neutron stars or in the collapsing cores of supernovae? What is the origin of gamma-ray bursts? Are there gravitational waves left over from the very early universe?

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34 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ED TIME Achievements The Binary Pulsar and the Emission of Gravitational Waves For some years after Einstein' s initial prediction, even the existence of gravi- tational waves was in doubt, at least for some. That is no longer the case, thanks to the remarkable work of radio astronomers Joseph Taylor and Russell Hulse. In 1974, they discovered the pulsar PSR1913+16. Its frequency varied with a period of 7 3/4 hours, revealing it to be a member of a binary system. (For this reason, it was called the Binary Pulsar; now about 50 other binary pulsars are known. For background on the Binary Pulsar see Chapter 2.) Over the succeed- ing years, careful measurement of the arrival times of the pulses of radio emission from the pulsar revealed the shape of the pulsar's orbit in unprecedented detail. By recognizing a variety of relativistic effects (including orbit precession, gravi- tational redshift, and the special-relativistic time-dilation), Taylor and his co- workers were able to show that both the pulsar and its companion were neutron stars with precisely measured masses around 1.4 times the mass of the Sun. The most exciting result of these studies was the recognition that the motion of the pulsar around its companion could not be understood unless the dissipative reaction force associated with gravitational wave production was included. The two neutron stars, by virtue of their motion about one another, execute precisely the sort of motion that generates gravitational waves. Those waves carry away energy. Thus, the two stars must gradually fall closer to each other, with the result that their orbit steadily speeds up. The motion of the Binary Pulsar has shown that this orbital speedup is occurring in accordance with the rate predicted by general relativity, to a precision of a third of a percent. (See Figure 3.1.) For the discovery of this remarkable object, Hulse and Taylor were awarded the Nobel Prize in physics in 1993. Experimental Searches, Ongoing and New A gravitational wave interacts with matter by producing differential forces and thus relative motions between sets of masses. Experiments to detect gravita- tional waves involve setting up systems of a few test masses, then looking in as sensitive a way as possible for relative motions between them. Resonant detec- tors, based on the original idea pioneered by Joseph Weber in the 1960s, use a large single extended body such as a solid cylinder, whose ends may be thought of as separate masses being pulled apart or pushed together by the wave as it passes. The newer interferometric detectors use three or more small masses that are widely separated; a propagating laser beam is used to monitor their separa- tions, which will be perturbed when a gravitational wave passes.

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 35 Ll 1 1 1 1 I T 7 1 1 7 1 1 7 1 1 1 1 1 7 1 1 7 1 1 1 11 o En a) . _ o a) Q -5 -10 0 - 15 En al > . _ -20 -25 -30 `.. be; General Relativity Prediction' / \ / t-l I I I I I I I I I I I I I I I I I I I I I I I mu 1 975 1 980 1 985 1 990 1 995 2000 Year FIGURE 3.1 The orbital period of any body around another decreases because of the energy lost to gravitational radiation. That effect is strongest in highly relativistic sys- tems such as the binary pulsar PSR1913+16. One measure of this decrease in orbital period is the steady shift over time of the time of the pulsar's closest approach (perias- tron) to its companion star. The figure above shows the cumulative value of this shift measured by J. Taylor and J. Weisberg at the Arecibo radio telescope in Puerto Rico over several decades. The points are their data points. The solid line is the shift predicted by general relativity. The agreement is better than a third of a percent. (Courtesy of J.H. Taylor and J.M. Weisberg; to be published.)

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36 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ED TIME Resonant Detectors. The first detector to reach astrophysically interesting sensi- tivity was the ultracold resonant bar at Stanford University, operating at 4 K (degrees above absolute zero). In 1980, it operated with a sensitivity to short bursts with strain amplitudes (AL/L) of around 1 part in 10~. The first observa- tions looking for coincident events in widely separated detectors were carried out in 1986. High-sensitivity coincidence observations were performed between an Italian detector and the Allegro detector at Louisiana State University (Figure 3.2) in 1991. This run determined the strongest upper limit yet on the flux of gravitational waves. A new generation of resonant detectors has now begun operation, using dilution refrigerators to bring their several-ton resonators to temperatures of around 50 mK (5/100 degree above zero). The pioneers of this class are the Nautilus and Auriga detectors in Italy. They should eventually attain a sensitivity of about 1 part in 102. Designs have been produced for resonant detectors that could reach sensi- tivities of parts in 10-2~. These detectors would extend the technology already developed for aluminum cylinders to spheres as much as 10 times more massive. Interferometr~c Detectors. For many years, work on interferometers, directed at kilometer-scale devices that could achieve astrophysically motivated sensitivi- ties, was devoted mainly to proof-of-principle devices and engineering tests. Finally, in the early 1990s large interferometer construction projects were ap- proved in several countries around the world. The U.S. entry, the Laser Interfer- ometer Gravitational-Wave Observatory (LIGO), will consist of two facilities- one in Hanford, Washington, and the other in Livingston, Louisiana, each of which will contain a Michelson interferometer of arm length 4 kilometers (Figure 3.3~. (The Hanford site will also carry an interferometer of half that length for additional coincidence measurements.) When LIGO becomes operational in 2002, it is expected to be able to make unambiguous detections of waves with strains AL/L around 1 part in 102~. Similar results are expected from the 3-kilometer VIRGO interferometer (a French-Italian project located near Pisa). The British-German GEO 600-meter interferometer near Hannover has the handi- cap of shorter arm length, but early application of advanced interferometer tech- nology will allow it to be competitive in some frequency ranges, at least for a while. There is also a 300-meter interferometer called TAMA under construction near Tokyo, and an Australian project in the planning stage called ACIGA. Theoretical Studies of Gravitational Wave Sources During the last decade, the theoretical prediction of gravitational wave sources reached new levels of sophistication and promise. This effort was driven by progress in gravitational wave detectors and made possible by advances in numerical and analytic techniques for solving Einstein' s equations. The ability to

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 37 FIGURE 3.2 A view of the Allegro resonant bar gravitational wave detector at Louisiana State University. The photo shows the bar nestled in its cryogenic dewar, shortly before it was closed up. The end of the bar is visible as the circular structure in the lower half of the dewar. The rest of the internal components are structures for cooling the bar to 4.2 K, for bringing out the signal from the transducer on the bar's end, and for isolating the whole system from external disturbances. Since 1991 Allegro has functioned as the most sensitive continuously operating gravitational wave detector in the world. (Courtesy of Bill Hamilton, Louisiana State University Physics and Astronomy.)

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38 GRAVITATIONS PHYSICS: E~LOHNG THE STRUCTURE OF SPACE ED TIME FIGURE 3.3 An aerial photograph of the Laser Interferometer Gravitational-Wave Ob- servatory (LIGO) facility at Hanford, Washington, in 1998. The large building in the foreground holds the vertex of the interferometers, along with the lasers, input test mass- es, and the input and output optics. The building also contains the control room, experi- ment staging areas, laboratory space, and offices for the observatory staff. One of the 4-kilometer arms disappears out of the right-hand side of the photo. The other is seen stretching to the upper left. At the 2-kilometer point can be seen a smaller building holding the end mass of the half-length interferometer that will run in parallel to the 4-kilometer interferometer, providing a local check on the observations. (Courtesy of the LIGO Laboratory.) carry out numerical simulations of gravitational collapse in three spatial dimen- sions (i.e., without restrictive symmetries), together with improvements in inte- grating realistic microphysics into the description of the collapsing stellar matter, gave results that demonstrated a remarkable sensitivity of the gravitational wave output from a supernova to the details of neutrino physics, hydrodynamics, and thermal physics. Similarly, large-scale numerical simulations of the merger and coalescence of double black hole or double neutron star systems are on the verge of achieving reliable results. (See the discussion under "Computational General Relativity" in Section II of this chapter.)

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 39 The theory of gravitational waves from the inspiral phase of double star systems was advanced using analytic tools based on the post-Newtonian tech- nique, a method for approximating solutions of Einstein's equations by succes- sive improvements on its first-order, Newtonian approximation. The results, carried to remarkably high order in successive steps, provided very accurate gravitational wave "templates," which will play a role in analysis of signals detected by LIGO-type gravitational wave detectors. The theory of small perturbations of stars and black holes, initiated in the early 1960s, was taken to new levels of development. One result was the discov- ery of entirely unsuspected unstable modes of oscillation of rotating stars, which could be promising sources of gravitational radiation and could explain the rapid spin-down of newly formed neutron stars. Another was the development of a nearly complete description of the gravitational wave emission from a small mass orbiting a massive black hole and of the "ringing" modes of distorted black holes. Finally, over the last 5 years a new method was developed to study the properties of gravitational waves emitted in the very final stages of black hole mergers. The method, called the close limit approximation, combines analytic and numerical approximation methods and has yielded fresh insights into the diverse physical processes responsible for various features of the emitted radia- tion. It paves the way for gravitational wave phenomenology along the lines of the standard quantum mechanical perturbation theory used in analyzing spectra in atomic physics. Opportunities Ground-based Reception of Gravitational Waves The decade just ending saw the National Science Foundation make a sub- stantial investment in the construction of the research facilities of the LIGO project (see Figure 3.3~. The great opportunity of the coming decade is to exploit those facilities by operating receivers of sufficient sensitivity to detect the gravi- tational waves emitted by astronomical bodies. The expectation that gravita- tional waves will be detected during the coming decade represents one of the most exciting research opportunities of gravitational physics. The spectrum of gravitational waves expected from known sources is shown in Figure 3.4. Detec- tion and study of those waves can address many of the key questions listed above. Key Questions Addressed Do waves such as those predicted by Einstein propagate away from dy- namic massive objects, and do they interact with test bodies in the way described by general relativity?

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40 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ED TIME 10 . E 10 1 id . _ CD 10 -22 1 0-24 "'"'' / ResolVE3d ~ Gil BlnarleS "''' ~ \ :~ . Ad-, / -4 -2 10 10 o 10 Frequency (Hz) 10 10 4 FIGURE 3.4 A schematic view of the gravitational wave spectrum, showing the project- ed sensitivity of the advanced version of LIGO and of a proposed space-based interferom- eter. LIGO and the space-based detector will each be able to look for gravitational wave sources in a band a decade or two wide. LIGO will have its best sensitivity near 100 Hz, while an instrument in space should be most sensitive near 10 mHz. The high-frequency window accessible to LIGO is best suited for studying the signals from the coalescence of neutron star (NS) binaries, and from binaries consisting of black holes (BH) with masses around 10 times that of the Sun. Binaries of massive (106 solar masses) black holes, such as are found in galactic nuclei, will be a primary target of a space-based interferometer. Each detector should also be capable of finding the signals from a variety of other astro- nomical objects, as described in the text. For example, a space-based detector would be able to record the signals of many known binary star systems. SN, supernova. (Courtesy of the Jet Propulsion Laboratory, California Institute of Technology.) The work of Taylor and his collaborators tracking the orbit of the binary pulsar PSR1913+16 established dramatically that gravitational waves were being emitted by the binary neutron star system, with a rate of energy loss in agreement with the predictions of general relativity. In a very real sense, that measurement can be said to have "detected" the emission of gravitational waves. But physicists' paradigm of establishing the existence of a wave phenom- enon is the set of l9th-century experiments on electromagnetic waves performed by Heinrich Hertz (1857-1894), which demonstrated not only energy loss in the transmitter, but also (1) propagation across spatial intervals large compared to the

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 4 wavelength and (2) excitation of test bodies in a manner consistent with the field carried by the wave. We may never have the capability to construct and manipu- late a gravitational wave transmitter, but can always rely on the existence of natural ones, as did the Hulse-Taylor binary pulsar experiment. However, receiv- ers can be built that establish that a wave propagated across the space from the transmitter and that this wave interacted with test particles in the expected way. This is the fundamental role LIGO and the other gravitational wave detectors can be expected to play, when they successfully detect the arrival of gravitational waves of astrophysical origin. Do gravitational waves propagate at the speed of light, and do they have the polarization that general relativity predicts? The speed of gravitational waves is unambiguously predicted by general relativity to be equal to the speed of light in vacuum. Similarly, general relativity states that the polarization of the waves should be strictly quadrupolar, although other gravitation theories predict some admixture of other polarizations in addi- tion. The relativistic predictions are equivalent to the statements that the quan- tum of gravitation analogous to the photon the graviton is massless and has spin 2. (However, no detector is foreseen as being sensitive enough to detect individual gravitons.) These features of gravitational waves can be checked once the waves are detected. The polarization is most directly measured by verifying that the signal strengths detected by receivers at different locations on Earth (hence with differ- ent orientations) agree with those predicted from the source's position on the sky (as determined by time delays). The propagation speed can be checked against that of light whenever the gravitational wave emission is accompanied by some electromagnetic counterpart; examples might include the optical flash of a super- nova, or perhaps a gamma-ray burst. What is the nature of gravity in the strong-field regime where general relativity makes its most dramatic predictions? Gravitational waves are emitted most strongly when large masses move at relativistic speeds in close proximity to one another, especially as those masses approach the degree of compactness of black holes (as in neutron stars or black holes themselves). These are inherently strong-field situations, with dynamics dramatically different than Newtonian theory would predict. The dynamics of the ultimate strong-field sources, black holes, are even more distinctive. Do black holes exist? What are the properties of the highly relativistic spacetime just outside their horizons?

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42 GRAVITATIONAL PHYSICS: EXPLORING THE STRUCTURE OF SPACE AND TIME Black holes are relativistic strong gravitational phenomena, and they show the most dramatic effects of strong gravity. The many aspects of the study of black holes are discussed in Section II of this chapter. The "cleanest" test of the existence of black holes (and of the predictions of theory regarding strong-field gravity in general) would be the measurement of the gravitational waves emitted when a black hole is disturbed, or when it forms in a gravitational collapse or merger. These waves provide a direct probe of the dynamics of the region just outside the black hole's horizon. The spectrum of the quasi-normal modes (the "ringing" of a disturbed black hole) has been exten- sively studied theoretically and should be easily recognizable in a gravitational waveform. Its most distinctive feature is that all of the modes are highly damped. The damping mechanism is the gravitational wave emission process itself. The frequency of the signal is inversely proportional to the mass; a 10-solar-mass black hole has its fundamental resonance near 1 kHz. Detection of waves from the merger phase will also probe strong-field gravity effects. Interpretation of such signals will require numerical integration of the full Einstein equations. (See "Advances in Computational General Relativity" in Section II of this chapter.) What can we learn from the study of coalescing neutron star binaries? The gravitational wave source whose signal is most securely predicted is the coalescence of neutron star binaries, of which the Hulse-Taylor binary pulsar is a prototype. The shrinkage of the orbit of such a system is driven by gravitational radiation until the two stars coalesce, through a process that can be calculated in precise detail until nearly the final moments. The gravitational luminosity of the system grows as the orbit shrinks, with the final signal at frequencies of several hundred hertz. It is a key aim of the LIGO project to develop receivers capable of detecting several of these events per year, which means being able to detect them to dis- tances of several hundred megaparsecs (Mpc). This is the goal of the planned upgrades to the initial LIGO interferometers, scheduled to start in the middle of the coming decade. (See Chapter 1, Table 1.1, which is based on the analysis described in the addendum to this section.) When the signals from coalescing neutron star binaries are detected, we will probe strongly post-Newtonian orbital dynamics. Just as interesting will be what we will learn about the properties of nuclear matter, which will strongly influence the final phases of the orbit and ring-down of the coalesced star. It is a difficult problem to calculate the signals researchers should expect from the end of a coalescence event. This led to the formation of the NASA Neutron Star Grand Challenge effort in computational physics. (See the discussion on neutron stars under "Advances in Computational General Relativity" in Section II of this chap- ter for more details.)

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88 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ED TIME agree with general relativity to parts in a thousand. Another test of nonlinearities could be provided by a space experiment in which atomic clocks travel close to the Sun. In such an experiment, the shift in frequency between satellite and Earth clocks, or between satellite clocks of different physical structure, contains not only the first-order, or linear contribution, which could be tested to a part in a billion, but also a nonlinear term, which could be tested to a part in a thousand. Lunar laser ranging has been one of the most cost-effective and scientifically productive projects arising from the space program. It has yielded new informa- tion on the orbit and rotational motion of the Moon and on crustal motions on Earth. It has also verified the equivalence principle for Earth and the Moon and has helped set bounds on a temporal variation of the Newtonian gravitational constant. Improved lasers, better modeling of the lunar motion, and the contin- ued accumulation of data will provide further improvements in all of these areas. For example, the measurement of the equality of acceleration could be improved by an order of magnitude. Binary and Millisecond Pulsars To date, only about 1000 out of a predicted 105 pulsars in our Galaxy have been discovered; of these, 100 could be in binary systems. While continued searches for and observations of millisecond and binary pulsars will be driven by independent astrophysical considerations, they will provide opportunities for fur- ther tests of relativity in the radiative and strong-field regimes. For example, the fortuitous discovery of binary pulsars of the right characteristics, such as systems containing both a pulsar and a black hole, could result in a 10-fold improvement in accuracy in the test of gravitational radiation damping, provide a high-preci- sion measurement of a companion black hole mass, detect the precession of the spin of a neutron star, contribute to a determination of the distribution of neutron star masses, and help sharpen the event rate of inspiraling and coalescing neutron star binaries. The Newtonian Gravitational Constant The Newtonian constant G. which governs the strength of all gravitational interactions, was historically the first "fundamental constant" in physics. Yet today, with an official uncertainty of about a part in 104, it is the least precisely determined of any of these constants. Recently, measurements of G at several laboratories have cast doubt on the accepted value of G and especially on its uncertainty. The next decade will see the completion of G measurements using a wide variety of techniques and devices, such as torsion balances, fountains of ultracold atoms, or gravimeters that see a modulated field, possibly reaching a level of a part in 106. Consistent results from several groups will be needed to give confidence that the systematic errors in the measurements are understood.

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 89 Improved bounds on any temporal variation of G of cosmic origin could help constrain alternative gravitational and cosmological models, such as those arising from string theory. Key Questions There have been three major crises in theoretical physics in this century. In each case, two well-established theories were found to be incompatible, either because they were based on contradictory assumptions about the workings of the physical world, or because they led to physically untenable conclusions. The first was the conflict between Newtonian gravity and special relativity, which was resolved by Einstein's theory of general relativity. The second arose from ten- sion between thermodynamics and electromagnetism, which led to the develop- ment of quantum theory. In both of these cases, the crisis was resolved not by a small modification of one of the theories, but rather by an entirely new funda- mental theory that introduced a new framework and made some old concepts obsolete. Over time, both of these theories passed stringent experimental tests and now form the cornerstones of modern physics. Unfortunately, however, general relativity and quantum theory are themselves mutually incompatible, presenting physicists with a third crisis. It is likely that the resolution of this crises will be as profound as the previous two, yielding a major change in our view of nature. The resulting theory, quantum gravity, is expected to be crucial for under- standing the strongest gravitational fields in the universe. It is possible to make a rough estimate of when the effects of quantum gravity should become impor- tant. This is because, as mentioned in Chapter 2, there is a unique combination of the fundamental constants of general relativity and quantum theory which has dimensions of length, up = (hGIc3~/2 ~ 1 0-33 cm, called the Planck length. When the gravitational field is so strong that space is curved on this scale, then quantum gravity is indispensable. Such extreme conditions were present in the early universe. Our current cosmological models can in principle be extrapolated back to t = up I c ~ 10~3 seconds after the big bang, but then they completely break down. Quantum gravity should provide a description of the first moments after the big bang, and perhaps of the big bang itself. In certain situations, effects of quantum gravity can be important even when the gravitational fields are significantly weaker than the above estimate. In particular this is the case around black holes. It turns out that, because of the unusual properties of space and time near a black hole, when quantum effects are included black holes are not really black; they radiate via a quantum tunneling process, losing their mass and becoming hotter. This radiation can be quite significant. For example, if black holes with mass of the order of 10~5 grams

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90 GRAVITATIONS PHYSICS: E~LOHNG THE STRUCTURE OF SPACE ED TIME were formed in the early universe, they would appear today to be white hot and would explode. Some of the key questions in quantum gravity are the following: . How can the principles of general relativity and quantum theory be uni fled into one consistent framework? . What are the quantum properties of black holes? What is the nature of space and time at the smallest possible scales? Is this smallest-scale structure responsible for curing the short-distance infinities of quantum theories of other fundamental forces? What was the universe like 10~3 seconds after the big bang? As the COP describes below, there has been enormous progress over the past decade in trying to answer these questions, especially the first three. This is due partly to a new formulation of the problem, and partly to the influx of ideas from high-energy physics. As the search for a unified theory of all forces expanded to encompass gravity, particle physicists were naturally led to seek a quantum theory of gravity. The next decade promises to be a very exciting one, in which ideas from different approaches may fuse together, providing deeper insights and per- haps complete answers to these fundamental questions about nature. Achievements The usual formulations of quantum theory require a fixed notion of time since, e.g., quantum states are specified at an instant of time. But in quantum gravity, physicists expect the spacetime geometry to fluctuate. This raises deep conceptual problems, especially in the cosmological context where there are no external observers with clocks. To address such problems, a new route was developed in the l990s; quantum theory was generalized to accommodate quan- tum spacetime geometry. Usual quantum mechanics is recovered in those epochs and situations in which the spacetime geometry is approximately classical. This conceptual advance resulted from a fruitful interchange of ideas between experts working on quantum gravity, quantum theory of closed systems, and foundations of quantum mechanics. In spite of this progress in constructing a framework, however, the task of actually constructing a quantum theory of gravity remains formidable. Straight- forward quantized general relativity is "perturbatively non-renormalizable" it yields infinite answers to physical questions. Therefore, it is natural to start with approximation schemes. Perhaps the simplest among them is quantum field theory in curved spacetimes, where the gravitational field is treated as a classical, passive entity and analyzes the effects of the curved spacetime geometry on quantized matter. At first, this appears to be an oversimplification of the prob- lem. However, this approach led to some key insights in the mid-1970s. The

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 9 most striking among them is the Hawking effect: black holes radiate as though they are blackbodies. This discovery brought together general relativity, quan- tum field theory, and thermodynamics and provided powerful hints for quantum gravity. In particular, it became physically meaningful to assign thermodynamic parameters such as temperature and entropy to black holes in terms of their geometric properties. This posed a concrete challenge to any candidate quantum theory of gravity: Explain the origin of these thermodynamic properties in terms of microscopic degrees of freedom as is done for ideal gases and other everyday systems. As described below, there has been considerable recent progress on this challenge. Over the past decade, quantum field theory in curved spacetimes has evolved considerably, especially through the development of powerful algebraic methods, and has now become a mature branch of mathematical physics. On the physical side, to gain insight into the role of very high frequencies in the Hawking effect, quantum physics around "dumb holes" the acoustic analogs of black holes- has been studied. These model systems also exhibit Hawking radiation (see Box 3.5) and the radiation spectrum turns out to be quite robust, surprisingly insensi- tive to changes in the assumed properties of the extreme high frequency modes. Substantial progress occurred also in another approximation method, that of ef- fective field theories. Here, theorists do treat the gravitational field quantum mechanically, but they focus only on the implications of these quantum effects on length scales large compared to up. Thus, although quantum general relativity is "perturbatively non-renormalizable," theorists can nonetheless extract from it meaningful information to describe physics in the situation where the curvature of spacetime is small compared to that at the Planck scale. This is an interesting development, especially because in the 1970s, such non-renormalizable quantum theories were widely regarded as being devoid of physical content. A number of approaches have been developed to go beyond such approxima- tions. Notable among them are Euclidean quantum gravity, the dynamical trian- gulation method, Regge calculus, asymptotic quantization, reformulation of gen- eral relativity as a dynamical theory of null hyper-surfaces, twister theory, and non-commutative geometry. However, in terms of providing answers to the key questions, two directions stand out: string theory and quantum theory of geom- etry. In both of these approaches, considerable progress has already been made on the first three key questions listed above. String Theory During the past decade, string theory has emerged as a leading candidate for a quantum theory of gravity. In addition, this theory appears to achieve another long-standing goal of theoretical physics: It may provide a unified theory of all known forces and particles. The starting point is remarkably simple. One as- sumes that elementary particles are not point-like, but rather actually different

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92 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE^D TIME BOX 3.5 Black Holes and Quantum Physics In the early 1 970s, the properties of black holes were studied and formulated in terms of the laws of black hole mechanics. It was noticed that there was a striking similarity between these laws and the ordinary laws of thermodynamics. The an- alog of the temperature T was a multiple of the surface gravity of the black hole K, which is similar to g on Earth's surface. The analog of the entropy S was a multiple of the area of the event horizon A. More precisely, there were the following corre- spondences: Black Hole Mechanics K iS constant. aM= KDA/8~G. aA 2 0. Thermodynamics Tis constant. aE= 7OS. aside. It was believed at the time that the above relationships were only an analogy, since the defining property of black holes was that nothing can escape their grav- itational pull. In particular, they did not emit the radiation characteristic of an object with non-zero temperature. However, a few years later it was shown that when quantum effects are included, this analogy becomes exact. Because of an analog of the tunneling process in elementary quantum mechanics called the Hawking effect, particles can escape from black holes. Thus, black holes are not really black. They radiate just like hot objects with temperature Tbh = (~/2~c). This implies that black hole entropy is given by Sbh = A14~2p, where Ip is the Planck length. Therefore, in a fundamental statistical mechanical description, a macro- scopic black hole of area A should have eSbh microstates a huge number. It has taken 25 years to obtain a fundamental description of these microstates. In the past few years this fundamental description has finally been achieved in both of the main approaches to quantum gravity. However, important issues still remain, and there should be further exciting developments in this area over the next decade. excitations of a one-dimensional extended object the string. When an ordinary violin string is plucked, it vibrates at certain characteristic frequencies producing the usual notes. Fundamental strings are much smaller (of order up in size), but when they are excited they also vibrate at certain frequencies. Different modes of vibration are seen as electrons, quarks, photons, and so on. This provides a strikingly simple unified picture. The basic interaction between strings is through a simple splitting and join- ing process. Remarkably, the description of this process automatically incorpo- rates the known interactions between the elementary particles. The relation between string theory and general relativity can be seen in two ways. First, one mode of the string describes a graviton (a small fluctuation of the gravitational field), and the classical scattering of strings in this mode reproduces the

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 93 perturbative expansion of general relativity. More importantly, when strings in curved spacetime are studied, a consistency condition on the spacetime geometry emerges which is Einstein's equation of general relativity, augmented by correc- tions that are only important at the Planck scale. In fact, if general relativity were not already known, string theory would predict it as the description of gravity for distances much larger than the size of the string. Until recently, quantum effects in string theory were mostly discussed in the context of perturbation theory. A background spacetime geometry is assumed (satisfying the above consistency condition) and small fluctuations about it are quantized. This is not sufficient to answer the important questions about the big bang or what happens deep inside a black hole. These issues appear to require a complete non-perturbative formulation of string theory which is not yet available. However, in the past few years, a number of non-perturbative facts about the theory have nonetheless been found. This is possible largely because string theory incorporates supersymmetry, a powerful symmetry first discussed in the 1970s. Supersymmetry implies that certain results computed perturbatively are, in fact, exact. String theory has had a number of achievements over the past decade. Here the COP focuses on those achievements that relate directly to gravity and the structure of space and time. The apparently obvious three-dimensional nature of space need not be correct. It is possible that we experience three dimensions because the extra dimensions are curled up into a very small ball. Our measure- ments of space so far might simply be too crude to detect such extra dimensions. In string theory, space and time are no longer fundamental, but instead are de- rived concepts. One early result is that space must have more than three dimen- sions. (The perturbative formulation of string theory predicts nine dimensions, but recent non-perturbative arguments suggest that in the full theory, the number is ten.) T-duality. Suppose one direction in space is curled up into a circle of radius R. In addition to the usual string states, there are now extra states corresponding to strings winding around this circle. The net effect of these extra states is that the spectrum of the string is exactly the same as if the circle had radius 1/R. Further- more, the interactions between string states are also invariant under changing the radius of a circle from R to 1/R. This means that very small circles are indistin- guishable from large circles in string theory. Singularities. As discussed elsewhere in this report, general relativity predicts the existence of places in the universe where the spacetime curvature is infinite. General relativity breaks down at these "singularities." Since string theory modi- fies general relativity even classically, it is important to know whether singu- larities exist in string theory as well. It has been shown that several spacetimes that are singular in general relativity are completely nonsingular when embedded

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94 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE^D TIME in string theory. However, it has also been shown that there are some singular solutions even in string theory. These examples are not very physical, and so it is not yet known whether these singularities are likely to arise in nature. Topology Change. The topology of space is a measure of how the space is connected e.g., whether it has holes like the surface of a donut. A long-standing question is whether the topology of space in our universe can change over time. According to general relativity, the answer is no: Topology change always results in singularities. In string theory the situation is different. It has been shown that the topology of space can change in a way that is non-singular in string theory. Quantum Properties of Black Holes. The most important achievement has undoubtedly been the successful description of black hole entropy in terms of quantum string states. As discussed above, this has been a challenge for more than 20 years. For certain large black holes with electric charge near the maxi- mum allowed value, the number of string states with the same charge and mass turns out to be precisely the number predicted from black hole thermodynamics. Even more importantly, the interactions between these states turn out to precisely reproduce the Hawking spectrum of radiation. This is a remarkable achievement. For more general black holes, theorists can identify a class of string states associ- ated with a black hole which scale in the expected way with the mass and charge, but numerical coefficients in the entropy formulas have not yet been checked. Quantum Theory of Geometry In general relativity, spacetime geometry is a dynamical entity that interacts with matter and has degrees of freedom of its own. Therefore, to unify the principles of general relativity and quantum theory, it is natural to take this physical role of geometry to be fundamental and probe its quantum nature from first principles. Over the past decade, a detailed theory has been developed starting from this viewpoint which in turn has provided some key insights on the nature of quantum gravity effects. This approach is "non-perturbative" in the sense that a classical spacetime is not the starting point to which quantum fluctuations to its geometry are then added. There is no background spacetime; everything, including geometry, is dynamical and quantum mechanical. Indeed, the strategy is just the opposite of that followed in perturbative treatments: Rather than starting with quantum matter on classical spacetimes, one first quantizes geometry and then incorpo- rates matter. This procedure is motivated by two considerations. The first comes from general relativity in which some of the simplest and yet most interesting physical systems black holes and gravitational waves consist of "pure geom- etry." The second comes from quantum theory where the occurrence of infinities

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 95 at short distances suggests that it may be physically incorrect to quantize matter assuming that spacetime can be regarded as a smooth continuum at arbitrarily small scales. The detailed implementation of these ideas requires a new mathematical and conceptual framework, since the standard methods used in quantum theories of non-gravitational forces are tied to the availability of a background spacetime that is now absent. An important recent advance was the systematic construction of the appropriate substitutes. The resulting mathematical framework is now sufficiently rigorous to ensure that there are no hidden infinities or other internal . . . Inconsistencies. The key results obtained from this framework can be summarized as follows. Fundamental Discreteness. Over the past 5 years, this approach has led to a detailed quantum theory of geometry. The framework shares several basic con- cepts with gauge theories, thereby bringing gravity closer to other fundamental forces. In particular, the fundamental excitations of geometry are coded in the gravitational Wilson loops. They are one-dimensional so that quantum geometry resembles a polymer. However, when densely packed in appropriate configura- tions, these excitations can approximate the three-dimensional spatial continuum. Quantum analogs of observable geometrical quantities such as areas of surfaces and volumes of regions called geometric operators are well-defined. They have the striking property that their values are quantized, that is, can change only in discrete steps. Thus, at the Planck scale, the continuum picture breaks down and geometry becomes "polymer-like." Several properties of the geometric operators have been worked out. For instance, the allowed discrete values of area crowd rapidly as area increases; the difference between them, called the level spacing, goes to zero exponentially quickly, making the continuum picture an excellent approximation in laboratory physics. However, since the Planck length up is so small, can such details ever bear on the macroscopic world? What if, for example, the level spacing were uniform, like that in a harmonic oscillator, with steps of the order of {p? Could such alternatives be physically distinguished? Surprisingly, using quantum field theory in black hole spacetimes, one can. While the actual level spacing of area is consistent with the blackbody spectrum of the Hawking effect, the uniform level spacing is not. Thus, there are checks on predictions. Black Hole Thermodynamics. Since a black hole in general relativity is "pure geometry," it is natural to use quantum geometry to unravel its microscopic degrees of freedom. Recently, this task was carried out for nonrotating black holes, possibly with charges. For large black holes, the number of microstates grows exponentially with area, showing that the entropy is proportional to area. From this perspective, the mechanism underlying black hole evaporation is strik- ingly simple: Quanta of area are converted to quanta of matter. This ongoing

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96 GRAVITATIONS PHYSICS: E~LOHNG THE STRUCTURE OF SPACE ED TIME work provides a quantum mechanical manifestation of the idea that geometry is a physical entity. Quantum Dynamics. Quantum geometry provides a mathematical language to formulate a wide variety of quantum gravity theories, just as differential geom- etry does in the case of classical gravity. However, so far quantum dynamics has been explored only in general relativity, possibly coupled to matter, and super- gravity, the extension of general relativity which incorporates supersymmetry. The central question is, Can these quantum theories admit an exact, mathemati- cally consistent formulation even though they are "perturbatively nonrenormal- izable?" In two (rather than three) space dimensions, the answer is in the affirma- tive even with certain types of matter. Although this lower-dimensional theory is not of direct physical interest, it faces most of the conceptual difficulties of the three-dimensional theory, and the application of perturbative methods had led to a general belief that a consistent quantum theory would not exist. Not only does a satisfactory theory exist but the non-perturbative formulation also provides useful hints for higher-dimensional theories. Over the last 3 years, there has also been considerable work in four dimen- sions which has provided an example of how quantum Einstein equations can be formulated rigorously. This is an interesting development in mathematical phys- ics. However, it is not yet clear whether this formulation can successfully answer those physical questions that are motivated by semi-classical considerations. Another approach to quantum dynamics leads to an unexpected interplay with a branch of topology, namely, the theory of knots. In particular, some of the well- known "knot invariants" automatically solve quantum Einstein equations, and there are indications of deeper relations between quantum general relativity and knot theory. Opportunities The coming decade is likely to see substantial further progress in quantum gravity. With an eye toward the key questions listed above, the COP briefly describes some of the opportunities that await us. The pace of progress in string theory has been extremely rapid during the past few years. Indeed, within the past year, a proposal for a non-perturbative formulation of the theory has been made, which is applicable when the cosmo- logical constant is negative. While this is not believed to be the case in nature, this formulation can still be used as a model to study quantum gravitational processes such as the evaporation of black holes. This proposal incorporates a novel "holographic" view of space and time, in which our usual notions of local- ity and causality hold only approximately. Much effort will be devoted in com- ing years to establish in detail that this proposal is correct and to extend it in such

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ACHIEVEMENTS AND OPPORTUNITIES IN GRAVITATIONAL PHYSICS 97 a way that the cosmological constant need not be specified a priori. If these steps can be completed, we may finally have a workable quantum theory of gravity. Significant advances are also expected in the approach based on quantum geometry. In this framework, quantum dynamics was initially discussed using Hamiltonian methods. Spacetime formulations of the required theory are now being pursued vigorously in which quantum geometry serves to unify results from apparently three distinct areas of research, pursued independently by rela- tivists, quantum field theorists, and mathematicians. These methods provide a new avenue to formulate and discuss quantum Einstein equations and are better suited for semi-classical considerations. Physical ramifications of the quantum nature of geometry will also be explored further. In particular, it is likely that quantum field theories on quantum geometries will be studied. Since the funda- mental geometric excitations are one-dimensional, the effective spacetime di- mension is now reduced. The key question then is whether this feature will free quantum theories of non-gravitational interactions from the usual short-distance infinities. If so, the old and cherished hope that quantum gravity may cure quantum field theories will be realized. So far, string theory has been developed largely by high-energy theorists and quantum geometry by relativists. This is reflected in the choice of issues that are emphasized in the two approaches. However, there are tantalizing similarities such as the importance of one-dimensional objects. Furthermore, as the results on black hole thermodynamics indicate, both approaches are now addressing the same physical problems. Their strengths are complementary. One enables quan- tum physics to be done without a background spacetime but provides no guidance on how various physical fields couple to gravity and to one another. The other has an in-built powerful principle that dictates all couplings but has yet to com- pletely free itself from reliance on a background geometry. Much progress would occur if there were more interaction between the two communities. It is clear that recent results on the quantum properties of black holes will be extended using both string theory and quantum geometry. In this area, there are several exciting challenges. The two approaches have led to rather different physical pictures of a quantum black hole, one based on extended objects in higher-dimensional spacetimes, and the other, on the polymer-like excitations of geometry of ordinary space. Are these two pictures "complementary" in a suit- able sense? More generally, what is the relation between them? Another chal- lenge is to derive the laws of black hole thermodynamics from quantum gravity, in full generality, allowing for departures from thermal equilibrium. An even more important open question is whether information thrown into a black hole is lost forever, or is ultimately recovered in the evaporation process. If it is indeed lost forever, as suggested by the original semi-classical calculations in the 1970s, then some of the basic principles of quantum theory would have to be modified. However, the recent string calculations indicate that information is not lost. It is

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98 GRAVITATIONAL PHYSICS: EXPLORING THE STRUCTURE OF SPACE AND TIME possible that this long-standing "black hole information puzzle" will be resolved in the near future. Considerable progress can also be expected in applying these developing theories of quantum gravity to cosmology. Indeed it could be argued that while most effects of quantum gravity are not observable, the effects of the quantum fluctuations in geometry and matter near the big bang are all around us. We see them in anisotropies in the cosmic background radiation and in the large-scale distributions of galaxies. The objective of quantum cosmology is to understand the earliest moments after the big bang. Quantum gravity is central to this task, and cosmology is one of quantum gravity's most important applications. (These issues are further discussed in Section III of this chapter.) It is quite possible that qualitatively new and unexpected effects will be discovered in the coming years, the quantum gravity analogs of E = mc2 of special relativity. For example, it was recently found that in the two- and three- dimensional exactly soluble models, unexpectedly large quantum fluctuations can arise in the spacetime geometry because the coupling between general rela- tivistic gravity and matter can magnify small quantum uncertainties in matter sources of gravitation into huge uncertainties in the gravitational field. In the coming years, these results are likely to be extended to four dimensions and may then have experimental consequences. Similarly, there are directions in which the current ideas in string theory could be confronted with experiments. For example, discovery of supersymmetry in particle accelerators would lend support to an important ingredient of string theory. A second example arises from the fact that one mode of the string, the dilator, has interactions like gravity but couples to matter in a different way. (See Section IV of this chapter.) This might produce violations of the equivalence principle at a detectable level. More direct tests may also be possible in spite of the fact that the energy scale of quantum gravity is very high, 10~9 GeV. In the 1980s, for example, experiments were performed to directly test the predictions of grand unified theories on proton decay. The processes responsible for this phenomenon are only a few orders of magnitude below the quantum gravity scale. Yet, it was possible to test these theories without having to accelerate particles to such high energies; the experiments involved confining a very large number of protons and waiting sufficiently long to see if any of them decayed. In the same spirit ~ ~ ~ .# .# , attempts have recently been made to put limits on certain quantum gravity effects using observations of TeV gamma-ray flares. The idea is that the tiny effects on the propagation of gamma-rays due to the Planck-scale fluctuations in the spacetime geometry can accumulate during their long flight over cosmological distances and lead to an observable dispersion. It is likely that these ideas will be refined over the next decade and enable researchers to experimentally distinguish between possible quantum gravity scenarios.