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9 Gravitation Theory: Opportunities Theoretical research depends most importantly on its human re- sources. Theorists are much more able than experimentalists to redirect their research programs when important new opportunities appear. Consequently, the needs and the health of theory are best discussed in terms of the vitality and diversity of the research programs of individuals. Similarly, any list of the most important problems in theory must be descriptive rather than prescriptive. The research problems discussed here are selected from the menu of topics that theorists currently consider important. CLASSICAL GRAVITATION, SINGULARITIES, ASYMPTOTIC STRUCTURE Although we now seem to have a decent understanding of the basic physics of the General Theory of Relativity in the nonquantum regime, outstanding problems of great significance remain. The most important of these is the Cosmic Censorship Conjecture (see section in Chapter 8 on Gravitational Collapse and Black Holes). The proof of this conjecture would confirm the already widely accepted and applied theory of classical black-hole dynamics, while its overturn would throw black-hole dynamics into serious doubt. A number of related issues about asymptotic properties of space- time remain to be settled, although there has been enormous progress 72
GRAVITATION THEORY: OPPORTUNITIES 73 in the last decade in this area. The measurement and even the definition of angular momentum at null infinity needs further clarification. It is impossible to give a local and covariant definition of energy density for the gravitational field, owing basically to the Principle of Equivalence, which says that space-time is everywhere locally flat. Nevertheless, significant progress has come in quasi-local definitions, in which one attempts to measure the total mass energy within a closed surface, and further development of these ideas will be useful. One conjectured extension of the Positive Energy Theorem still remains unproved, namely, that the total mass of an isolated system containing black holes must not only be positive but must exceed the sum of the irreducible (Area Theorem) masses of the black holes. QUANTUM GRAVITY The unification of gravitation physics with quantum physics or the construction of a completely new theory incorporating both is one of the greatest challenges in theoretical physics. The challenge confronts us not so much because of the possibility of immediate experimental test (simple order-of-magnitude estimates indicate that laboratory tests of a quantum theory of gravity are not likely within the decade covered by this report); rather, the challenge of quantum gravity confronts us, first, because we observe a system for which we can be sure quantum gravity is important. This is the universe itself. Quantum gravitational ejects are significant in the extreme conditions of the big bang, and there can be no understanding of the complete history of our universe without an understanding of quantum gravity. Second, the present vision of a unity of all particle interactions will not be complete until gravity is incorporated in that unity. Indeed, it may be that gravity enters in an essential way into any fundamental understanding of matter. Third, there are some explicitly observational problems that will require a deeper theory, as we shall see below. There is no lack of issues in quantum gravity; throughout the field there are unresolved problems and issues of principle. Working out the quantum mechanics of Einstein's classical theory would seem a reasonable starting point in the study of the quantum theory of gravitation. Not only are we unable to calculate effectively with the resulting theory (it is not renormalizable), but fundamental issues such as identifying the variable that plays the role of time and the construc- tion of the Hilbert space of states are still not satisfactorily resolved. It may be that the Lagrangian for general relativity, so unique and successful in the classical regime, does not correctly describe the
74 GRA VITA TION .; ~' .~.,~ FIGURE 9.1 Space-time foam. On length scales of the order of (~GIc3)~72 ~10-33 cm space-time undergoes enormous fluctuations in curvature with associated energy density c5hiG2 = 5 x 1093 g/cm3. Of the same order of magnitude is the negative energy density due to gravitational attraction of the wormholes. Space-time foam illustrates the geometric approach to quantum gravity. quantum mechanics of space-time on distances of 10-33 cm (see Figure 9.11; rather, it may be an effective model good only on longer scales. Perhaps the correct Lagrangian is one in which gravity is unified with matter theories, or perhaps there is no gravitational Lagrangian at all. Lagrangian theories of gravity tend to share common problems. Perhaps the most important is the problem of the cosmological constant, or energy density, of the vacuum state. Calculation of quantum corrections to typical field theories suggests a cosmological constant of the order of unity on the Planck scale; observation tells us it is 10~2° times smaller. Understanding these 120 orders of magnitude is one of the most significant challenges confronting any quantum gravitational theory. It may be that local Lagrangian field theory is not the correct approach to quantum gravity. Perhaps, as some believe, the basic quantum quantities are not the variables describing a space-time continuum but a more discrete structure. Finally, it may be that the laws of quantum mechanics themselves require modification in the
GRAVITATION THEOR Y.: OPPORTUNITIES 75 extreme physical regime where quantum gravitational effects are important. There are many avenues of approach that promise to shed light on a quantum theory of gravity and its applications. A partial list of them includes the canonical approach, covariant perturbation theory, Euclidean quantum gravity, quantum field theory in curved space-time, geometrical quantization, twister theory, discrete gravity, curvature- squared theories, nonlinear quantum mechanics, spin networks, in- duced gravity, asymptotic quantization, quantum cosmology, super- gravity theories, Kaluza-Klein theories, and superstring theories. One could perhaps even attempt to assess their prospects viewed from some present perspective. To do so, however, would not provide a guide for the future of the area. There are many diverse approaches because there are many ideas and deep unsolved problems. There is no obvious single approach, and there should be none at this stage. The best hope for substantial progress is to encourage a variety of ap- proaches and to encourage cross-fertilization between them and with other relevant areas quantum field theory, particle physics, and mathematics, on the one hand, and cosmology and astrophysics on the other. One can expect developments in the area to proceed by fits and starts. New ideas will be proposed, tested, and either abandoned or added as pieces of an as yet incomplete structure. New techniques will produce new objectives, and new objectives will produce new tech- niques. Taking greater risks will be necessary to support diversity and encourage innovation, but the payoff will be a deeper understanding of perhaps the most fundamental problem of physics. ASTROPHYSICAL PROPERTIES OF NEUTRON STARS AND BLACK HOLES Work should continue on modeling of astrophysical properties for neutron stars and black holes. Here the relativity physics is fairly well understood, but the interaction between general relativity and other phenomena such as hydrodynamics, electrodynamics, and radiative transfer remains to be understood in detail. The construction of models for active galactic nuclei and quasars, both of which involve accretion onto black holes, and their confrontation with observation, is an active and quite challenging problem in relativistic astrophysics. A crucial lack is the absence of currently available observational means to distinguish between black-hole models (see Figure 9.2) and other sorts of models.
76 GRA VITA TION ...:,: i.. .~N, ,I~!/,~//i,///,Y76~ 5' :~7`''~: ~,'; 2 ~ '~'in. "a t~ '~'~~~ ' Hi; I';' ·~. ; ', ', . ,/ I ~_,~;.·, .. ,,, . ~.,Ma4H~c ~ - -, // ~ ~ A/ A/ A" /iDt: my/ / / ~ ~ ~~/,,` ,_ ED Mod ~~W FIGURE 9.2 One possible model for generating the jets seen coming from some radio galaxies and quasars. An accretion disk orbiting a supermassive (109 MSun) black hole deposits chaotic magnetic field onto the hole, which "cleans" the magnetic field lines that thread it. The ordered field interacts with the hole's rotation-induced gravitomagnetic field to produce ~102°-V potentials that accelerate relativistic particles out the poles, forming jets. This model exemplifies the complexity and variety of physics possible for black holes in an astrophysical setting and the importance of more detailed observations.
GRA VI TA TION THEOR Y.: OPPORTUNI TIES 77 COMPUTATION The Einstein equations form a difficult system of nonlinear partial differential equations. Lacking a general solution by analytic means, we must rely on numerical solutions for many applications of the Einstein equations, notably for gravitational collapse, black-hole col- lisions, and inhomogeneous cosmology. Great progress has been achieved in the last decade on numerical relativity using large-scale computers, but the equations are difficult enough that the significant computational problems remain untouched. The most difficult prob- lems, those involving full general relativity in three space dimensions and one time dimension, will be in reach with supercomputers of the capability projected for the next decade, although substantial develop- ment of numerical algorithms will also be required. An example is the problem of the black-hole binary, in which one follows the orbital decay and final coalescence of two black holes in a binary system with energy loss by gravitational waves. As possibly the strongest gravity- wave source in the universe, this mechanism holds great promise for testing relativity in the regime of highly dynamical strong fields, if the wave forms can be detected and measured. A second important use of computers in relativity is for symbolic manipulations. The analytic computations in relativity are often ex- traordinarily intricate, and computer assistance is often useful or even essential. Symbolic manipulation packages for algebra and calculus have gradually become more and more significant owing to the increased availability of hardware and to great advances in software algorithms for symbolic manipulations. The development of supercom- puters, and provision of access to them by researchers, will play an increasingly important role for research on certain important problems in gravitation theory. NEW KINDS OF EXPERIMENTAL TESTS Solar-system tests of relativity are now approaching a precision of one part in 103 of the first post-Newtonian terms in effects such as time delay and light bending. To reach the level of second-order post- Newtonian effects will require a further factor of 103 improvement; as we have seen (see section on Measurement of Second-Order Solar- System Effects in Chapter 3), experiments at this level are under study. Further theoretical work on second-order post-Newtonian effects, in general relativity and especially in alternative theories, will be needed. New theoretical proposals may also be needed to interpret current tests
78 GRAVITATION of the R-2 law for Newtonian gravity in the laboratory and on Earth, over ranges of millimeters to kilometers. For instance, axion forces that arise in certain field theories of elementary-particle physics give some additional motivation for such experiments and suggest possible anomalous effects, such as spin-dependent forces and forces that violate time-reversal invariance. Relativity predicts the evolution of the universe, and, therefore, observations in cosmology may someday be used to test the theory. At present, the theory is used to interpret the data rather than the data used to test the theory. However, as cosmological data become more extensive and precise, the situation could be reversed. Analysis of the consistency of cosmological models with observations, therefore, continues to be an important theoretical question. Current speculations in quantum gravity suggest exotic effects, such as violation of CPT invariance, evolution of pure quantum-mechanical states to mixed states, and baryon decay mediated by gravitational effects. Supergravity yields a number of effects of its own. At present all these seem far too weak to measure, but the possibility exists that some such effect will turn up that is within experimental reach. So far, no actually or potentially observable phenomena in high-energy phys- ics have been tied to gravity, but modern Grand Unified Theories are importantly influenced by virtual processes that transpire at the grand unification mass scale, which may be only 2 to 4 orders of magnitude below the Planck mass scale of quantum gravity. One may optimisti- cally hope for direct connections between the observable phenomena of high-energy physics and quantum gravity sometime in the next decade or two. COMMUNICATION WITH OTHER SUBFIELDS: GRAVITATION EXPERIMENT, ASTRONOMY AND ASTROPHYSICS, FIELD THEORY AND ELEMENTARY-PARTICLE PHYSICS, PURE MATHEMATICS General relativity theory has experienced a period of great growth over the past 20 years. An important stimulus for this growth has been the interchange of ideas and problems with other subfields. The discovery of pulsars and quasars by astronomers has focused much attention on theoretical studies of neutron stars and black holes. In turn, the discovery and observation of gravitational-wave sources may provide a new window for astronomical observations of compact objects. Tests of relativity have stimulated much work on alternative theories as well as on the observable predictions of the Theory of
GRAVITATION THEOR Y.: OPPORTUNITIES 79 General Relativity, and tests have now ruled out important classes of alternative theories. The example of general relativity has provided an important stimulus over the past 60 years to field theory; and in particular the supergravity theories and Kaluza-Klein theories, considered important hopes for unification, grew out of general relativity. The problems of frontier particle physics have to a significant extent become those of gravitation physics. In turn, developments in field theory have given rise to new directions in gravity by providing new techniques and new theories in which gravity plays a part. One can expect this close relationship between general relativity and particle physics to grow even more rapidly in the coming decade. Communication with pure mathematicians led to the proof of the Positive Energy Theorem, one of the most important results in gravity theory in the past decade. Modern ideas from algebraic geometry have significantly influenced and contributed to the progress of the twister program and to the study of complex spaces at asymptotic null infinity. One can also expect this close relationship with mathematics to grow as mathematical tools become even more important in the exploration of theoretical ideas. Continued strong relations of gravitation theory with other subfields such as those just mentioned will be essential for its continued vitality, and indeed for the vitality of theoretical physics as a whole.
