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8 Gravitation Theory: Highlights NEUTRON STARS The discovery of radio pulsars in 1967, and their subsequent identi- fication as rotating neutron stars, exhibited for the first time objects in the universe with gravity so strong that the effects of general relativity must be important in their structure. Subsequently, neutron stars were also discovered in x-ray binary systems, in which gas from a normal star is accreting onto the neutron star and releasing gravitational binding energy as x rays. The discovery of the binary radio pulsar PSR 1913 + 16 has provided an impressive example of relativistic ejects in its orbital motion (see earlier sections on Perihelion Advance, Einstein's Only Handle in Chapter 2 and on Binary Pulsar in Chapter 5~. However, it has been difficult to discover direct evidence for general relativistic effects in observations of neutron stars themselves. One likely case is the observation of a spectral line in hard x-ray observations of gamma-ray bursters, which can be understood as the positron annihilation line emitted at the surface of a neutron star, redshifted about 10 percent by the general relativistic gravitational redshift. The theory of stellar structure and stellar pulsation, which was originally developed for normal stars, has now been successfully extended to neutron stars, taking full account of general relativity. The most important new effects are the emission of gravitational waves and 61

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62 GRA VITA TION the existence of relativistic instabilities (not present in Newtonian theory), which limit the physical range of stellar possibilities. Detailed results are available for the frequencies and damping rates for nonra- dial oscillations of neutron stars. These oscillations generate gravita- tional-wave emission and might be excited during the birth of a neutron star in stellar collapse. Surprising new effects have been uncovered for gravitational-wave emission by rotating stars. A general theorem has been proved that says roughly that all perfect fluid, rotating stars are unstable to the emission of gravitational radiation via a secular instability. The inclusion of viscosity, always present to some degree in nature, allows the instability to exist only above some critical threshold in stellar rotation rate. The recent discovery of a pulsar with a rotational period of only 1.5 ms has shown the relevance of these results, and it now seems quite possible that there exists a class of fast pulsars with rotational rates at or near the instability threshold, which could be sources of periodic gravitational radiation. GRAVITATIONAL COLLAPSE AND BLACK HOLES There is a maximum mass limit for neutron stars. The exact limit depends on the equation of state for nuclear matter, which is not well known. Nevertheless, the upper mass limit is certainly less than about 5 solar masses and seems likely to be in the range of 1.5-2.5 solar masses. Any stellar core with a mass exceeding the upper limit that undergoes gravitational collapse must collapse to indefinitely high central density to form a singularity. It is generally believed among theorists that in such circumstances a black hole will always form so that the final singularity will be hidden from external observers. A black hole is a region of space-time where the gravitational field is so strong that not even light can escape. Inside a black hole, there exists a space-time singularity, a place where the space-time curvature becomes infinite and all the known laws of physics may break down. The hypothesis that space-time singularities always remain hidden from observers is called the cosmic censorship hypothesis. It remains unproved. A singularity that is visible to external observers, in vio- lation of the hypothesis, would be naked singularity. Naked singulari- ties have been the object of a considerable amount of study and speculation, but most relativists believe that they do not exist, with the exception of the big bang itself. The evidence for existence of black holes is impressive but not conclusive. At least one binary stellar x-ray source, Cygnus X-1, has a mass greater than the upper mass limit for neutron stars and must be of

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GRA VITA TION THEOR Y.: HIGHLIGHTS 63 compact size as evidenced by millisecond variability of its emission. Alternative models are possible but implausible. Other x-ray sources may well contain black holes; a strong possibility is LMC-X3 in the Large Magellanic Cloud. Supermassive black holes of a million to a billion solar masses might be able to form in the nuclei of galaxies by stellar coalescence and accretion. Direct evidence for such black holes remains weak. A1- though some galactic nuclei are found to possess a large accumulation of dark matter at the core, this mass has not been shown to be so compact that it must be a black hole. A popular model of quasars and active galactic nuclei postulates the existence of a supermassive black hole undergoing accretion of surrounding matter, with enormous amounts of gravitational energy released in thermal and nonthermal radiation coming from an accretion disk or a chaotic accretion region around the black hole. Accretion may produce electrodynamic effects or jets of outflowing matter. A black hole may be born in a more or less excited state, depending on the degree of disorder in the gravitational collapse of its progenitor star, but it quickly relaxes to a stationary state by the emission of gravitational waves. The stationary states of black holes are remark- ably simple, according to the uniqueness theorems, which state that a stationary black hole must belong to the three-parameter family of black-hole solutions of the Einstein equations, the Schwarzschild- Kerr-Newman solutions. The three parameters are the mass, the total angular momentum, and the electric charge. The mass, angular mo- mentum, and charge of a black hole cannot disappear because these quantities are conserved charges coupled to long-range fields. In an astrophysical environment, any electric charge on a black hole will be quickly and almost completely neutralized through the conduc- tivity of the surrounding plasma. Since the electromagnetic interaction is so much stronger than gravity, it only takes a tiny amount of charged plasma to neutralize even a maximally charged black hole. On the other hand, the angular momentum of a black hole will persist, and a black hole may remain rotating for hundreds of millions of years or more. A rotating black hole has free energy that can be tapped externally. The energy can be tapped by immersing the black hole in a suitable configuration of conductors and magnetic fields, in which case it acts as a kind of electrical generator, or it can be tapped mechanically by suitable arrangements of particles traversing a certain region near the black hole, called the ergosphere. When all the free energy is removed from a rotating black hole, it is reduced to a nonrotating state. The energetics of black holes are governed by remarkably simple

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64 GRA VI TA TI ON laws, the four laws of black-hole dynamics, which in essence are the four laws of thermodynamics as applied to black holes. The role of entropy is played by a purely geometric quantity, the surface area of the black hole. The area theorem states that, in classical physics, the surface area of a black hole never decreases; this theorem is known as the second law of black-hole dynamics from its parallelism to the Second Law of Thermodynamics, which asserts that entropy never decreases for an isolated system. Small disturbances of stationary black holes, for instance due to small particles falling in or to impinging electromagnetic or gravita- tional waves, can be worked out in linear perturbation theory. This theory reduces to the solution of certain remarkably simple wave equations and is in essence complete. QUANTUM PARTICLE CREATION BY BLACK HOLES A major advance in fundamental understanding of the laws of physics was achieved in the discovery that, when quantum effects are considered, a black hole emits quanta of radiation just as if it were a blackbody at a finite temperature. The temperature is inversely pro- portional to the mass of the black hole. The temperature of a black hole is its key feature that makes possible the identification of the laws of black-hole dynamics with the laws of thermodynamics. The emission of quanta by black holes, known as the Hawking process, causes black holes to become gradually smaller and finally to decay away entirely. This effect seems unobservable for stellar mass black holes, for which the temperature is less than a microkelvin and whose lifetime is greater than 1070 years. On the other hand, if small black holes, with a mass of about 10'5 g, were created in the big bang, they could be observable today. Their lifetime would be roughly 20 billion years, the age of the universe, and their temperatures would become high just before their final decay, so that they would emit a burst of hard electromagnetic radiation. Such bursts have not been found to date. QUANTUM EFFECTS IN THE EARLY UNIVERSE Gravity becomes comparable in strength with the other fundamental forces of nature only at the Planck energy, about 10'9 GeV. The only known places in the present universe where energies reach this level are at space-time singularities. Those inside black holes are thought to be invisible to us, according to the cosmic censorship hypothesis. The initial singularity of the big bang is in principle observable to us,

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GRA VI TA TION THEOR Y.: HIGHLIGHTS 65 although it is shrouded in the hot dense matter of the primeval fireball. The ejects of quantum gravity, imprinted on the universe at times so early that the temperature exceeded the Planck temperature, could have affected the present universe in important ways. An important effect could have been the damping of initial anisotropies, by quantum particle creation, to leave the almost perfectly isotropic universe that we see today. On the other hand, residual anisotropies in the cosmic background radiation could have been created by quantum effects at somewhat later times, for instance during an inflationary era in cosmology (see the section on The Inflationary Universe in Chapter 121. ALTERNATIVE THEORIES The tremendous advances in experimental tests of relativity have changed the theoretical scene greatly in the last decade. Theories that were viable and indeed admirable have now been stringently con- strained or even ruled out by solar-system tests and by observations of the binary pulsar PSR 1913+16. This progress has increased the confidence of most gravitation theorists that general relativity is indeed the correct classical theory of gravity, at least in the long-distance, low-energy domain, despite the fact that many of its most important effects, such as detection of gravitational radiation and magnetic gravity, remain to be demonstrated. Thus, although some work con- tinues on alternative theories, most ongoing theoretical work is based on general relativity. EXACT SOLUTIONS OF THE EINSTEIN EQUATIONS The Einstein equations are a nonlinear set of coupled partial differ- ential equations, and their complete solution is unknown. The discov- ery of exact particular solutions has played an important role in the progress of relativity; for instance, the Kerr solution, which is now known to be the unique solution for a rotating, uncharged, stationary black hole, was first found in a systematic search for certain exact solutions known as algebraically special. Great progress has been made on solution of the Einstein equations in the more general case of a stationary, axisymmetric, vacuum space-time, which is now known to be completely soluble in principle. Soliton methods from mathematical physics have also been applied to this problem. There has even been reason to hope for the complete and general

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66 GRAVITATION solution of the Einstein equations. A set of ideas called twister theory has been developed in a new approach to the issues both of classical and of quantum general relativity. Twistor theory has close connec- tions to modern mathematics, specifically to algebraic topology and algebraic geometry. Twistor theory has already produced new exact solutions for non-Abelian gauge theories in field theory (some of the instanton solutions) and has also produced large new classes of complex valued solutions to the Einstein equations. There has been progress toward a general solution by twister techniques, though as Yet ~ _ _ ~ _ ~ _ ~ _ , it has not been achieved. The initial-value problem for the Einstein evolution equations is itself a deep problem, on which good progress has been made. Known exact solutions for the initial-value constraint equations are few, but constructive methods are now available that give, in principle, the general solution of the Cauchy (spacelike) initial-value problem from freely specifiable initial data for the gravitational field and matter fields. Characteristic (lightlike) initial-value surfaces are likewise often useful, especially in the study of gravitational radiation. ASYMPTOTIC PROPERTIES OF SPACE-TIME An isolated system in general relativity is represented by a space- time that becomes asymptotically flat (Minkowskian) at infinity. Phys- ics should become simple at infinity; and mass, angular momentum, and gravitational waves should become easily measurable there. How- ever, the nonlinearities in the Einstein equations make the study of 7 infinity a subtle one. In a general, asymptotically flat space-time in which gravitational waves are propagating toward infinity, the Riemann curvature tensor falls off only as 1/r in the directions (called null infinity) in which both t and r become large. This slow falloff of the curvature causes many difficulties in principle for the measurement of the properties of isolated systems by distant observers, for instance in the definition of angular momentum. It has been found that if one generalizes the space-time manifold into a four-complex-dimensional manifold by allowing the four space-time coordinates to take complex values instead of just real ones, then a reference system of remarkable simplicity exists at infinity. In it the most troublesome asymptotic terms in the geometry vanish. The new complex space that arises at null infinity is called H-space or the nonlinear graviton. Asymptotic properties of space- time at spacelike infinity (t fixed, r large) also reveal subtleties. The correct definition for the angular momentum of an isolated system as

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GRA VI TA TION THEOR Y.: HIGHLIGHTS 67 measured at spacelike infinity is a problem that has only recently been resolved. NUMERICAL RELATIVITY In the absence of analytic techniques for the general solution of the Einstein equations, relativists have turned to large-scale numerical techniques to solve important problems, such as the collapse of stellar cores or the collision of black holes. The inclusion of general relativity in spherically symmetric computations of stellar collapse is now routinely done when necessary. Nonspherical systems, which unlike spherical ones admit gravitational radiation, are much more difficult to simulate numerically and require both state-of-the-art numerical tech- niques and the largest computers. These computations also require state-of-the-art theoretical analyses of the Cauchy and characteristic initial-value problems of general relativity. Finally, great care is needed for the numerical treatment of hydrodynamics in these simulations. The most ambitious numerical calculation carried out to date in pure general relativity, without any matter present, is the head-on collision of two identical nonrotating black holes. The numerical results show that the two holes coalesce to form a single one, and gravitational waves amounting to about a part in 103 of the total rest mass of the system are radiated to infinity. EMISSION OF GRAVITATIONAL RADIATION Inspired by experiments to detect gravitational radiation, investiga- tors have studied many source models. The calculations carried out include perturbation studies of gravitational collapse and black holes, approximate models of collapsing cores and of colliding neutron stars, and full-scale numerical calculations of gravitational collapse, colliding neutron stars, and colliding black holes (see Figure 8.11. As noted in Chapter 2, the results of such calculations are essential for making the important estimates of the strengths and frequencies of gravitational waves near the Earth (Figures 6.2 and 6.3 in Chapter 61. Doubts were raised about the validity of the quadrupole formula for gravitational-wave luminosity and radiation reaction of weak-field, slow-motion systems. Careful investigation of this formula by tech- niques of applied mathematics have strongly reinforced the belief in its validity. The experimental confirmation of the prediction of this formula for the binary pulsar PSR 1913+16 has emphasized its importance.

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68 GRAVITATION A__ ~ _.: - ~~) - ,~., - ~:.~ FIGURE 8.1 Numerical simulation of gravitational radiation from two colliding black holes of equal mass. The right axis is the axis of symmetry for the collision, and the left axis lies in the equator. Wave amplitude is plotted upward. This is the outgoing wave at a time of about t = 37 (mass) after the collision. THE POSITIVE ENERGY THEOREM Gravitational binding energy is negative, because gravity is an attractive force. When a body of given mass becomes so compact that the effects of general relativity become significant for its structure, the binding energy becomes comparable with the total rest energy of the matter making up the body. The possibility thus arises that the total energy of the body could become negative, should the binding energy actually dominate. It was conjectured 20 years ago that the total energy of a body could never become negative in the General Theory of Relativity. Heuristically one expects that any body attempting to violate this condition would lose stability and collapse to form a black hole before its total energy could become negative. A general form of this conjecture was finally proved in 1979 by two mathematicians using sophisticated arguments from differential geometry, and several gen-

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GRA VITA TION THEOR Y.: HIGHLIGHTS 69 eral forms of this positive energy theorem have now been proved. Mathematicians were attracted to the problem after relativists publi- cized the importance and apparent difficulty of the conjecture. A quite different and more direct proof of the positive energy theorem was given in 1981 by a particle physicist, using an argument motivated by supergravity theories (see section below on quantum gravity). Two relativists had earlier shown that the Hamiltonian of supergravity the expectation value of which is the total energy is formally nonnegative because it is a sum of perfect squares of certain fermionic charges. When this formal argument is made concrete, it indeed yields a rigorous proof of positive energy in general relativity. QUANTUM FIELD THEORY IN CURVED SPACE-TIME The discovery of the Hawking process by which black holes radiate particles quantum mechanically led to extensive development of the theory of quantum-matter fields in curved background space-time. A deeper understanding of the Hawking process was achieved together with a compelling and suggestive unification of the laws of black-hole mechanics and the laws of thermodynamics. The theory served as a laboratory in which ideas eventually to be important in a quantum theory of gravity could be tested in a simpler situation. Many concep- tually interesting and unanticipated ideas emerged. The reaction of a moving particle detector to a curved space-time, the possibility of CPT nonconservation in quantum gravity, and the possibility of quantum- mechanical evolution from pure to mixed states are three examples. QUANTUM GRAVITY The last decade has seen a remarkable growth in the theoretical effort devoted to the construction of a quantum theory of gravity. The unification of gravity and quantum physics had always been under- stood to be a fundamental question. The activity of the past decade was much stimulated by new techniques arising from gauge theories that could be applied to answer new questions in quantum gravity and to the ever more active search in particle physics for a unified theory of all interactions, which must at the end include gravity. The standard approach to field theory in the 1950s and 1960s was through the perturbation theory for scattering amplitudes. This is not always a sufficient tool in non-Abelian gauge theories such as quantum chromodynamics, nor will it suffice for gravity. On the one hand, gravitational scattering processes are too weak for observation. On the

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70 GRA VITATION other, the perturbation theory for these processes has divergences too strong to be controlled by renormalization. New techniques or new ideas were necessary, and they emerged through a fruitful exchange with particle theory ("ghosts" for example, originated in studies of quantum gravity). Euclidean functional integrals, successful in other areas of field theory, were applied to formulate a quantum theory of gravity based on the Lagrangian of general relativity. When a Euclidean formulation is applied to field theories of flat space-time, it is just a different technique; however, when it is applied to gravity it yields a different quantum theory. Further, it yields the theory in a way in which it can be approximated semiclassically in regimes far from the domain of validity of perturbation theory. New questions could thus be asked, and novel results emerged. For example, in this theory pure states can evolve into mixed states in striking contrast to the usual situation in quantum mechanics. There was also progress in the more traditional canonical approach to quantum gravity. Functional integral techniques clarified some of this approach's central problems, and promising new formulations of the canonical framework were worked out. The gravitational measure in the path integral, the existence of trace anomalies for the stress tensor, and solutions describing topological nontrivial configurations are just some examples. More recently, non-Abelian anomalies and the quantum breaking of coordinate invariance provide other striking illustrations involving gravity, gauge theories, and recent mathematics. Relativists have tended to interpret quantum gravity in terms of the quantum version of Einstein's theory. General relativity works well in the classical long-range limit. It has also been shown to be the unique theory of gravity in this limit, on the basis of a few observational facts taken together with the properties of special relativistic quantum mechanics. It is not self-evident, however, that it is correct on the scales of 10-33 cm (10~9 GeV) that characterize strong quantum gravitational phenomena. The 1 970s and 1980s therefore saw the investigation of new theories that were generalizations of Einstein's theory and also some radically different approaches. The twin motivations for these new initiatives were the hopes that a new theory might be more tractable at small distances than Einstein's theory seems to be, and the need for a new theory to realize the goal of the unification of all interactions. The developments of the past decade have seen a dramatic increase in the diversity of approaches to a quantum theory of gravity. Clearly, at present, a variety of approaches offers the best hope for a solution to this fundamental problem. One cannot help but be excited and impressed by the beauty and potential of these ideas.

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GRA VITA TION THEOR Y.: HIGHLIGHTS 71 One of the most significant developments of the past decade has been the emerging close relationship between particle physics and gravita- tion physics on this fundamental frontier. The search by particle physics for a unified theory has led to the problems of gravitational physics, and the search for a quantum gravity has led gravitational physics to field theory. Goals, techniques, and to some extent people are now shared between the cutting edges of these two areas. Supergravity, induced gravity, higher derivative Lagrangians, twis- tor theory, geometric quantization, discrete gravity, Kaluza-Klein theories, and string and superstring theories are just some of the headings under which new theories of quantum gravity might be grouped. It would be inappropriate to review them all here. Each has its promise and successes, but none has succeeded. We shall mention just two approaches that are currently under intense study by particle and gravitation physicists. Supergravity Symmetries between fields of different spins and different statistics are the basis of supergravity theories, which promote this symmetry into a local gauge invariance. The gravitational field is symmetrically related to a larger collection of fields that describe all particles and all interactions. Supersymmetric theories have a number of remarkable properties, such as a less rapidly divergent perturbation theory than ordinary gravity. Despite the absence of immediate direct experimental tests (a situation that is rapidly improving, however), they have captured the imagination of many theorists as one of the few viable avenues leading toward a unification of the forces of nature. Kaluza-Klein Theories There appear to be only four dimensions to space-time, but in the framework of Kaluza-Klein theories appearances are deceiving. These generalizations of Einstein's theory envisage a world of many (e.g., ten or eleven) dimensions in which all but four are curled up so as to be unnoticeable on our macroscopic scales. In such theories the matter degrees of freedom are space-time degrees of freedom in the extra dimensions. Kaluza-Klein theories offer the hope of a purely geometric unification of gravity with other matter interactions and perhaps even the explanation of the four-dimensional character of our physical world.