Steve Carlip

Time 2020-11-18 15:04:24

Web Name: Steve Carlip

WebSite: http://carlip.physics.ucdavis.edu

ID:120098

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One of the deepest problems of modern physics is that of reconciling ourwell-established theories of fundamental processes at very small scales, as described by quantum field theory, with those at very large scales, as described by general relativity. Efforts to formulate a consistent quantum theory of gravity date back to at least 1930 --here is a nice history-- but despite eighty years of work, we still seem far from an answer. While such a unification is probably unimportant at laboratory scales, it is vital for understanding the physics of strong gravitational fields: in the cosmology of the very early Universe, for example, and in the formation and evaporation of black holes. But the study of quantum gravity is difficult, and the main thing we have learned in these years of research is that the obvious approaches don't work. The difficulties are partly technical -- general relativity is a complicated, nonlinear theory -- but we face deep conceptual problems as well. According to general relativity, gravity is a consequence of the geometry of spacetime. That means that when we talk about quantizing gravity, we really mean "quantizing space and time themselves." We don't know what a completed quantum theory of gravity will look like, but we will surely end up with a picture of the Universe quite unlike anything we now imagine. In the past few years, two promising new approaches to these problems have emerged. The first is string theory, a model in which elementary particles are not treated as pointlike objects, but instead as extended one-dimensional "strings." (Here is a nice nontechnical introduction.) The second is a reformulation of general relativity in terms of new variables -- "self-dual connections" or "Ashtekar variables" -- that behave more like those of conventional quantum field theories. This approach is now often called "quantum geometry." More recently, a new method, "causal dynamicaltriangulations," has also shown promise. Gary Au has written a nice nontechnical paper based on interviews with physicists working on string theory and quantum geometry, andI have written a more technical review of a variety of approaches to the problem. An alternative general strategy for research is to explore simpler models that share the underlying conceptual features of quantum gravity while avoiding the technical difficulties. For example, general relativity in2+1 dimensions -- two spatial dimensions plus time -- has the same basic structure as the full (3+1)-dimensional theory, but it is technically much simpler, and the implications of quantum gravity can be examined in detail. Similarly, quantum black holes may be simpleenough to allow us to learn concrete lessons about the full theory.For the past few years, I have concentrated on four areas of research: Looking at (2+1)-dimensional quantum gravity as a sort of testing ground for approaches to the full quantum theory; Trying to understand the quantum gravitational basis of black hole thermodynamics; Investigating other "windows" into quantum gravity, such as causal dynamical triangulations -- relatively simple settings, models, and approximations that may offer insight into quantum gravity without requiring a complete theory; and Exploring a variety of other issues involving quantum gravity and "low dimensional physics," including string theory.I have also run a seminar on career prospects and options for physics graduates.(2+1)-Dimensional Quantum GravityGeneral relativity in 2+1 dimensions -- that is, two spatialdimensions plus time -- has proven to be a very useful model for exploring the conceptual foundations of quantum gravity. In three spacetime dimensions, general relativity has finitely many physical degrees of freedom, and there are no freely propagating gravitational waves. As a result, quantum gravity reduces to a special instance of ordinary quantum mechanics, and problems such as nonrenormalizability that are associated with quantum field theory disappear. But the model is still a coordinate-invariant theory of spacetime geometry, and most of the conceptual issues of the full theory remain.In all, roughly 15 different approaches to quantizing (2+1)-dimensional gravity have been developed. Most of these are discussed in abook I wrote in 1998 for Cambridge University Press. The model has offered insight into such issues as the nature of time in quantum gravity, the source of black hole entropy, and the question of whether the topology of space can change. Here is a paperI wrote with Jeanette Nelson comparing two interesting approaches.A recent review article I wrote on general relativity in 2+1 dimensions forthe on-line journal LivingReviews in Relativity can be foundhere. Another review ishere, this one discussing what we know about the microscopic "statistical mechanics" that presumablyunderlies (2+1)-dimensional black hole entropy. An older and more general review I wrote on (2+1)-dimensional black holes is here. For some research papers on the statistical mechanics of the (2+1)-dimensional black hole, look here and here.In some ways, ordinary (2+1)-dimensional gravity may be too simple. Recently, a number of physicists have becomeinterested in a slightly more complicated version, topologically massive gravity, which has a new propagating "graviton." I have been involved in this work; twopapers are here andhere.Black Hole ThermodynamicsThanks to the work of Hawking and Bekenstein, we have known for 25years that black holes are thermal objects, with characteristic temperatures, entropies, and radiation spectra. But we still do not really understand why black holes behave this way -- we don't know what microscopic quantum states are responsible for the "statistical mechanics" that leads to these thermodynamic properties. This problem serves as a key test for any attempt to quantize gravity: a model that cannot reproduce the Bekenstein-Hawking entropy for a black hole in terms of microscopic quantum gravitational states is unlikely to be right. An important focus of my recent work here has been an attempt to understand how much of the statistical mechanics of black holes can be determined purely from general symmetries, independent of the details of quantum gravity. I have shown that a symmetry mechanism is at least plausible. (Hereand here are some papers, and here is a review.) This idea would help explain one of the mysteries of this field, sometimes called the problem of universality: the fact that very different approaches to quantum gravity, with different starting points and different underlying degrees of freedom, all seem to give the same answer. This overview won the2007 GravityResearch Foundation essay prize; here is a less technical summary.Another interesting issue is whether "quasinormal modes" -- the dampedoscillations of a disturbed black hole -- can tell us anything about black hole quantum mechanics. I've written two papers on this subject, one on (2+1)-dimensionalblack holes and anotheron the higher-dimensional black holes that are understood in string theory."Windows"When faced with a question that seems too hard to answer, a physicist'sfirst reaction is likely to be, "Let's find a simpler question." In the absenceof a full-fledged quantum theory of gravity -- a complete, self-consistent theory that agrees with observations -- a natural strategy is to look intosimpler "windows" into quantum gravity that might give us useful clueswithout requiring a complete answer. Black hole thermodynamics, forinstance, offers a simple setting in which to probe complex problems;(2+1)-dimensional gravity provides another simple model.Lattice quantum gravity may be another such window. The basic idea ofputting a continuous theory on a lattice, approximating it by a simplerdiscrete theory, has had considerable success in quantum chromodynamics(QCD). The gravitational version is similar, but unlike QCD, where fields live on a fixed lattice, gravity is the lattice: just as the flattriangles in a a geodesic dome approximate a sphere, varying edge lengths or patterns of connectivity in higher dimensions can approximate varying curved spacetimes.In particular, several of my students and I have begun to work on a promising lattice approach known as causal dynamical triangulations, in which a causal structure -- a "direction of time" -- is put in from the start. We have found the first independentconfirmation of the pioneering results of Ambjorn, Jurkiewicz, and Loll, who showed that the method gives asensible semiclassical limit that really looks like a four-dimensionalspacetime. With the code now running stably, we are starting tolook at new questions, such as the renormalization group flow of the cosmological constant and the predicted patterns of quantum fluctuations in the early Universe. One intriguing prediction of the causal dynamical triangulations method is that while spacetime appears four-dimensional at large scales, it undergoes a "dimensional reduction" to two dimensions at very smallscales. If this is a general feature of quantum gravity, and not just apeculiarity of this particular approximation, it could be telling us something very important. A lecture of mine on this topic may befound here.I also work on a variety of other issues involving quantum gravity and "low dimensional physics," and on other areas in which geometry and topology are important to physics. Some of the questions I have studied include: Can quantum fluctuations in spacetime solve the "cosmological constant problem"? See here and here for two papers, and here for a write-up in the American Institute of Physics newsletter, Physics News Update. Can quantum fluctuations cause the topology of space to change in time? Here is a paper written with a graduate student, Russell Cosgrove, that gives some answers in (2+1)-dimensional spacetime. What does the "wave function of the Universe" really mean? In this paper I address some issues involved in interpreting the "no boundary" proposal of Hartle and Hawking. Here I work with a bunch of mathematicians to try to understand whether quantum fluctuations in the topology of the early Universe can affect that wave function. (The answer seems to be "yes" -- they might even help explain how inflation got started.) Can string theory be related to three-dimensional topological field theories? This paper with Ian Kogan discusses some of the problems in making such a connection. A few less technical areas I've worked in are the following: Debunking "creationist" cosmology: in this paper, Ryan Scranton and I demolish an article by Robert Gentry, a well-known "young Earth creationist," that is, a person who believes on religious grounds that the Universe is only about 10,000 years old, and who attempts to shoehorn science into that picture. (For more on the struggle to keep "creation science" out of the science classroom and to defend the teaching of evolution, see the National Center for Science Education Web pages. I am proud to be one of the early Steves in Project Steve.) Weighing kinetic energy: in this paper I discuss the experimental and theoretical foundation for the statement that "a hot brick weighs more than a cold one." Investigating the "speed of gravity": in Newton's theory, gravity propagates instantaneously. This is testable: if one puts a finite propagation speed into Newtonian gravity, the forces between two orbiting bodies no longer point toward their center of mass, and this "aberration" would lead to observable orbital instabilities. In general relativity, on the other hand, gravity (like everything else) cannot propagate faster than light. In this paper I show how this light-speed propagation can be reconciled with the observed lack of aberration, and correct some errors in the literature. Here I weigh in on the question of whether a recent observation of deflection of quasar light by Jupiter gives us observational information about the speed of gravity. Looking at "varying constants": there is an old idea, dating back to Dirac, that fundamental "constants" such as the fine structure constant may actually vary in time. Here and here are two papers in which I look at the question of whether black holes can tell us anything about the allowed variations. (Probably not much, unfortunately...) Answering Frequently Asked Questions: I have written portions of the Usenet Physics FAQs and Astronomy FAQs.My StudentsI generally expect my graduate students to be fairly independent, and to complete at least one major project largely on their own (although withme as a resource, of course). Here are some papers written by my studentswhile they were at Davis:Michael Ashworth worked on coherent states in quantum gravity, and on conformal field theoryfrom (2+1)-dimensional gravity.Sayan Basu worked on covariantcanonical quantization and, with a postdoc, on observational searches forviolations of Lorentzinvariance of a sort that might be produced by quantum gravity.Yujun Chen made major progress in quantizing Liouville theory, particularly in the sector that is probablyrelevant to black hole entropy.Russell Cosgrove studied the problem of time in quantum gravity. (See the discussionhere of such conceptual problems.)Eric Minassian worked on the question of singularities in (2+1)-dimensional quantum gravityPeter Salzman and I wrote a paper on a possible experimental test of whether gravity must be quantized, or whetherit can be treated as a fundamentally classical theory. If we are right -- we're awaitingindependent confirmation -- the need for quantum gravity could be tested in thenext generation of molecular interferometry experiments.Jim Van Meter didn't finish his paper on approximation methods for the Einstein field equations until he graduated, but he went on to work on one of the pioneeringefforts to numerically simulateblack hole mergers.Careers for PhysicistsOver the past decade, the career options available to graduates with Ph.D.s in physics have shifted dramatically. While the unemployment rate among new graduates is still low, the "traditional" academic path,graduate school to a postdoc to a faculty position, has become muchmore difficult. At the same time, industrial support for long termresearch and development in physics has declined, and government labs are, at best, hiring very slowly. Most current faculty members, onthe other hand, followed a traditional path, and few are very familiarwith the choices now available.My "Careers in Physics" seminar for graduate students brings in a variety of speakers with physics degrees who have jobs outside academia, to describe their work and also to give some nuts-and-bolts advice about how to get a job. Past speakers have ranged from stock analysts to high school teachers to microwave engineers to radiation safety experts. We also invite occasional "skill" speakers, to address such issues as how to give a good talk and whatto expect at a job interview.For statistics and information about jobs in physics, look at the AmericanInstitute of Physics Career Services and Statistical Researchpages, and at Joanne Cohn's very nice Physics and Astronomy Job Hunting Resources site. Job-hunting help is available atUC Davis from the Internship and Career Center.Some General Relativity LinksIf you're interested in learning more about general relativity and quantum gravity, here are some good starting places: General Physics Links SLAC Spires can be used to search for published papers in high energy physics. The preprint arXiv contain preprints of most new papers in general relativity (gr-qc), high energy theory (hep-th), and a number of other fields of physics. Ned Wright's Cosmology tutorial gives an excellent introduction to modern cosmology. The American Physical Society Home Page is a good source for news and information. So is the Institute of Physics site and their Physics Web. The online American Physical Society journal/newsletter Physics is a good source of summaries and commentaries on important new research results. For a more general audience, there's also Physics Central. Contributions of 20th Century Women to Physics is a great site for overcoming stereotypes of physics as "men's work." Nobel Prize winner Gerard 't Hooft has an interesting site with advice to students about "how to become a good theoretical physicist." Relativity John Baez's gravity tutorial gives a very good introduction to the basic ideas of general relativity, at an understandable but sophisticated level. The Topical Group in Gravitation is the American Physical Society's gravity group, and has a link to Matters of Gravity, a newsletter that covers recent developments in general relativity and gravity, along with links to many other general relativity sites. Living Reviews is a refereed electronic journal of review papers in general relativity and gravity. Syracuse University relativity links offer long lists of links to relativity groups, conferences, tutorials, software, names and addresses, and popular and technical relativity sites. MPEG movie trips to Black Holes and Neutron Stars Page is what it sounds like, and is a lot of fun. Andrew Hamilton's black hole pages provide more simulations. Classical and Quantum Gravity is the major specialized journal in the field. Another important journal is General Relativity and Gravitation. The web-based journal Living Reviews in Relativity offers review articles that are periodically updated by their authors (thus "living"). Hyperspace contains information about conferences, job openings, and general news about research in gravity. A Book List The site Book recommendations from the relativity FAQs is a good place to get suggestions for further reading at all levels.Fellow, Institute of Physics (UK)Editorial Board member, Proceedings of the Royal Society of London ADivisional Associate Editor, Physical Review LettersKramers Professor, Utrecht University, 2007Member, Nominating Committee, International Society on General Relativity and Gravitation, 2004-Member, Executive Committee, American Physical Society Topical Group in Gravitation, 1998-2001Editorial Board member, Classical and Quantum Gravity, 1995-2004National Science Foundation Young Investigator Award (NYI), 1993Department of Energy Outstanding Junior Investigator Award, 1991Elected to Phi Beta Kappa, 1974Referee for about 35 different physics journalsGrant reviewer for national science agencies of nine countriesInvited speaker at 4-5 conferences a yearMemberships American Physical Society International Society on General Relativity and Gravitation Institute of PhysicsHere are some slightly longer explanations of a few of the ideas I've mentioned elsewhere. Be warned -- the explanations here are, for the most part, drastic oversimplifications, and shouldn't be taken too literally.This section is still under construction (and will be for a long time...).Black hole thermodynamicsDrop a box of gas into a black hole, and you end up with no gas and abigger black hole. But a box of gas has entropy, and the second law ofthermodynamics tells us that entropy cannot decrease. Based on thought experiments of this sort, and on the known result that the area of a blackhole event horizon can never decrease, Bekenstein proposed in the early1970s that a black hole should have an entropy proportional to its area.This argument was at first dismissed because it was believed that black holes were truly "black," that is, that they emitted no radiation. Thiswould mean their temperature was absolute zero -- anything with a finite temperature radiates -- and an object with zero temperature cannot have a changing entropy. All this changed, though, when Hawking discovered that black holes actually do radiate, with a black body spectrum.(See "Hawking radiation.") The lawsof thermodynamics were quickly extended to include black holes.In ordinary thermodynamic systems, though, thermal properties appearas a consequence of statistical mechanics, that is, as a collective result of the behavior of large numbers of "microscopic" degrees of freedom.The temperature of a box of gas, for instance, reflects the kinetic energyof the molecules of gas, and the entropy measures the number ofaccessible physical states. There have been many attempts to formulate a "statistical mechanical" explanation of black hole thermodynamics,but it's safe to say that a complete picture has not yet been found.For a recent review, see my Lecturesat the 2007 Aegean Summer School on Black Holes.In the path integral approach to quantum gravity, the quantum amplitudefor a transition between an initial spatial geometry and a final spatialgeometry is obtained as a "sum over histories," where each "history" isa spacetime that interpolates between the chosen fixed initial and finalspaces. If we knew how to perform this not-very-well-defined infinitesum, we could, in principle, compute anything we wanted to know, althoughthe problems of interpretation might not disappear.We don't know how to carry out this sum, though, and the usual approximationsused in quantum field theory apparently fail -- that's what it means to saythat general relativity is nonrenormalizable. Analternative is to approximate the path integral by putting the theory ona lattice, essentially breaking an infinite sum into a discrete, finitecollection of "paths."For ordinary field theory on a lattice -- quantum chromodynamics, for example -- the lattice is held fixed, with fields placed at vertices or along edges. For general relativity, on the other hand, the lattice itself provides thedynamics. Just as flat triangles can be put together to form the curvedsurface of a geodesic dome, so flat simplices can be put together to form a curved spacetime.The idea of this kind of discrete approach to quantum gravity is not at allnew. Until recently, though, the resulting path integrals didn't seem togive anything close to a good approximation of a classical spacetime.Instead of a smooth, nearly flat spacetime, the results looked like either "crumpled" or "branched polymer" geometries, not at all like the world welive in. Recently, though, a new approach to the problem has been proposed. Called "causal dynamical triangulations" (or "Lorentzian dynamical triangulations"), this programtreats time in a new way, choosing new "gluing" rules that guarantee awell-behaved direction of time and, in the process, rule out certain kindsof quantum fluctuation of topology. While it is too soon to tell whether this approach will work, there are some encouragingsigns -- at least we seem to be able to obtain a good "nearly classical"four-dimensional spacetime from the simulations.Conceptual problems in quantum gravityWhy is quantum gravity hard? There are a lot of particularanswers, but most, if not all, of them, have the same root.According to general relativity, gravity is a characteristicof the structure of spacetime, so quantum gravity means quantizingspacetime itself. In a very basic sense, we have no idea what thismeans. For instance: As a probabilistic theory, quantum mechanics gives time a specialrole: we would like to say, for example, that an electron has atotal probability of one of being somewhere in the Universe at agiven time. But if spacetime is quantized, we don't know what"at a given time" means. We can try to define time as "the reading of a clock." But anyquantum mechanical clock made of ordinary matter has a finite (thoughsmall) probability of occasionally running backwards. We would like to require that causes precede effects; this is, infact, a basic axiom in the construction of quantum field theories.But if spacetime is subject to quantum fluctuations, the notion of"preceding" gets smeared out, and we no longer know how to make senseof causality. We can't say "A comes before B" any more; the best weseem to be able to do is to say "A probably comes before B." Typical observables in quantum theories are things like "thefield at a given point x." But if spacetime is quantized, we nolonger know what it means to talk about "a given point." It isprobably true that all, or almost all, observables in quantumgravity are nonlocal, and we know very little about how to dealwith such objects. It's likely that the structure of spacetime itself at very smalldistances is quite different from what we're used to. For instance,there are variations on the uncertainty principle that seem to implythat there is a minimum observable length. (Observing small distancesrequires high momenta, which then curve spacetime and distort thedistances.) But we don't know what to replace our ordinary picturewith. Typically, quantum field theories can't be solved exactly, and mustbe approached with systematic approximation methods, or "perturbation theory." For quantum gravity, ordinary perturbative methods startwith flat spacetime and treat curvature as a small distortion. But thereis no reason to believe that flat spacetime -- or any simple, smoothmanifold at all -- is a good approximate solution to quantum gravity.I should stress that people who work on quantum gravity don't, for themost part, spend much time thinking about such problems in the abstract.Rather, we try various approaches that work elsewhere, run into somedifficulty that's rooted in these conceptual problems, and try to solve it in a concrete instance; or we look for ways to reformulate generalrelativity or quantum mechanics to make these issues less important;or we look for simpler models in which some of these problems occur butmay be solvable. There are lots of vague ideas about what quantum gravity might look like; the hard part is in getting any of them to actually work in detail.(For a nice review paper by Chris Isham on some of the conceptual issues in quantum gravity, gohere.)The cosmological constantIf you want to derive the Einstein field equations from scratch, you cando so without making very many assumptions. You must assume thatthe geometry of spacetime is dynamical; there are no extra fixed, nondynamical "background structures" that influence the geometry;special relativity becomes a good approximation when gravitationalfields are weak;the field equations can be derived from a Lagrangian, or an actionprinciple; andthe field equations involve no more than second derivatives; that is,they determine "accelerations" rather than requiring accelerations asinitial data.These assumptions lead almost uniquely to a set of field equations withtwo undetermined constants. One of these is Newton's constant, whichdetermines the strength of the gravitational interaction. The other is thecosmological constant, Lambda.In modern terms, the cosmological constant looks like a very peculiar kind of gravitating matter, one that pervades the Universe with a constant density and an extremely high, negative pressure. Einstein originally introduced this term in order to allow static cosmological solutions to the field equations, and he later called it his biggest blunder. But it's not so easy to put this genie back in the bottle. The cosmological constantcan be interpreted as the energy density of the vacuum, and can arise for at least two reasons: In quantum field theory, the vacuum is not really "empty," but isfilled with fields that briefly appear because of quantum fluctuations. If you start with a theory with no cosmological constant, these vacuumfluctuations will create one. Conversely, to end up with no cosmological constant, you have to cancel off the vacuum fluctuations very precisely. This is tricky, because the fluctuations occur at all energy scales; you either need a mechanism that can simultaneously deal with all scales of distance and energy at once, or you have to postulate extraordinarily accurate "fine tuning" that allows effects at one scale to exactly cancel effects at very different scales. The vacuum energy changes when matter fields undergo phase transitions and spontaneous symmetry-breaking, essentially because spontaneous symmetry-breaking determines the nature of the physical ground state. Many "inflationary universe" models rely on this mechanism --they require a cosmological constant in the very early Universe to produce exponential expansion, followed by a phase transition that eliminates Lambda. The question is again why the present value should be exactly (or very nearly) zero. We can measure Lambda by looking at its effect on cosmology. A value significantly different from zero in "particle physics" units would lead, depending on sign, to exponential expansion and a cold, empty universe, or to a universe that would have recollapsed long before its present age. A nonzero value of Lambda is not ruled out observationally, and infact there is some evidence for a small positive cosmological constant. But the value of the vacuum energy density must be, at most, comparable in magnitude to the density of ordinary matter in the Universe, and this is is tinycompared to the values one expects from particle physics -- it's some 120orders of magnitude smaller than one would expect from simpledimensional analysis. While a number of more or less exotic suggestions have been floated,no one really knows why Lambda should be so small. This "cosmological constant problem" is one of the biggest mysteries in modern physics.See Ned Wright's cosmology tutorial and Eli Michael's cosmological constant page for more about Lambda.Physics describes evolution: it tells how, given some initial state, a systemwill look at later times. To describe the earliest moments of the Universe inquantum cosmology, we need to have some idea about the initial conditions, or "boundary conditions" at the beginning of time. One proposal, due to Hartle and Hawking, is that "the initial boundary condition of the Universe is that it had no boundary." Popular books describe this proposal with varying degrees of accuracy. (See PBS's"Stephen Hawking's Universe" for a fairly good example.) By forgettingthat the "no boundary proposal" is a quantum mechanicaldescription, though, these popularizations can sometimes be misleading.In particular, it's worth remembering that a quantum mechanical objectdoes not have a unique, well-defined "history." For a particle, for instance, such a history would be a trajectory -- position as a function of time -- and would determine both the particle's position and its momentum at all times. But by the Heisenberg uncertainty relations, this cannot be done: we can never simultaneously exactly specify a particle's position and momentum.The Hartle-Hawking "no boundary" proposal is based on the path integral,or "sum over histories," approach to quantum mechanics, in which aprobability amplitude is computed by taking a weighted sum over all possiblehistories that lead from an initial condition (in this case, "nothing") to afinal state. In a certain approximation, this sum is dominated by a"history" in which the Universe initially has a positive-definite metric -- thus the frequent references to "imaginary time." But neither this nor any other single history represents "the way the Universe really evolved."Hawking radiationThere are a number of ways of describing the mechanism responsible forHawking radiation. Here's one:The vacuum in quantum field theory is not really empty; it's filled with"virtual pairs" of particles and antiparticles that pop in and out of existence, with lifetimes determined by the Heisenberg uncertainty principle. When such pairs forms near the event horizon of a black hole, though, they are pulled apart by the tidal forces of gravity. Sometimes one member of a pair crosses the horizon, and can no longer recombine with its partner. The partner can then escape to infinity, and since it carries off positive energy, the energy (and thus the mass) of the black hole must decrease.There is something a bit mysterious about this explanation: it requiresthat the particle that falls into the black hole have negative energy. Here's one way to understand what's going on. (This argument is based roughly on section 11.4 of Schutz's book, A first course in general relativity.)To start, since we're talking about quantum field theory, let's understand what "energy" means in this context. The basicanswer is that energy is determined by Planck's relation, E=hf,where f is frequency. Of course, a classical configuration ofa field typically does not have a single frequency, but it can be Fourier decomposed into modes with fixed frequencies. Inquantum field theory, modes with positive frequencies correspondto particles, and those with negative frequencies correspond toantiparticles.Now, here's the key observation: frequency depends on time, andin particular on the choice of a time coordinate. We know thisfrom special relativity, of course -- two observers in relativemotion will see different frequencies for the same source. Inspecial relativity, though, while Lorentz transformations canchange the magnitude of frequency, they can't change the sign, so observers moving relative to each other with constant velocitieswill at least agree on the difference between particles andantiparticles.For accelerated motion this is no longer true, even in a flatspacetime. A state that looks like a vacuum to an unacceleratedobserver will be seen by an accelerated observer as a thermal bath of particle-antiparticle pairs. This predicted effect, the Unruheffect, is unfortunately too small to see with presently achievable accelerations, though some physicists, most notably Schwinger, have speculated that it might have something to do with thermoluminescence.(Most physicists are unconvinced.)The next ingredient in the mix is the observation that, as it issometimes put, "space and time change roles inside a black holehorizon." That is, the timelike direction inside the horizon isthe radial direction; motion "forward in time" is motion "radially inward" toward the singularity, and has nothing to do with whathappens relative to the Schwarzschild time coordinate t.The final ingredient is a description of vacuum fluctuations. Oneuseful way to look at these is to say that when a virtual particle-antiparticle pair is created in the vacuum, the total energy remainszero, but one of the particles has positive energy while the otherhas negative energy. (For clarity: either the particle or theantiparticle can have negative energy; there's no preference forone over the other.) Now, negative-energy particles are classicallyforbidden, but as long as the virtual pair annihilates in a timeless than h/E, the uncertainty principle allows such fluctuations.Now, finally, here's a way to understand Hawking radiation. Picturea virtual pair created outside a black hole event horizon. One of the particles will have a positive energy E, the other a negative energy -E, with energy defined in terms of a time coordinate outside the horizon. As long as both particles stay outside the horizon, they have to recombine in a time less than h/E. Suppose, though, that in this time the negative-energy particle crosses the horizon. The criterion for it to continue to exist as a real particle is nowthat it must have positive energy relative to the timelike coordinate inside the horizon, i.e., that it must be moving radially inward. This can occur regardless of its energy relative to an external time coordinate. So the black hole can absorb the negative-energy particle from a vacuumfluctuation without violating the uncertainty principle, leaving its positive-energy partner free to escape to infinity. The effect on the energy of the black hole, as seen from the outside (that is, relative to an external timelike coordinate) is that it decreases by an amountequal to the energy carried off to infinity by the positive-energy particle. Total energy is conserved, because it always was, throughoutthe process -- the net energy of the particle-antiparticle pair was zero.Note that this doesn't work in the other direction -- you can't have the positive-energy particle cross the horizon and leaves the negative-energy particle stranded outside, since a negative-energy particlecan't continue to exist outside the horizon for a time longer than h/E.So the black hole can lose energy to vacuum fluctuations, but it can'tgain energy.(See the relativity FAQs, here,for a related but slightly different description of black hole thermodynamics and Hawking radiation.)Loop quantum gravitySee "quantum geometry"Lower dimensional gravityMany of the fundamental conceptual issues inquantum gravity involve general features of the theory, which don't dependon "details" like the number of dimensions of spacetime. In fewer thanfour dimensions, though, the mathematics of general relativity becomesvastly simpler. Lower dimensional models therefore become useful testinggrounds for quantum gravity.One way to understand the simplification is the following. In n dimensions, the phase space of general relativity -- the space of generalized positions and momenta, or equivalently the space of initial data -- is characterized by a spatial metric on a constant-time hypersurface, which has n(n-1)/2 independent components, and its time derivative (or conjugate momentum), which adds another n(n-1)/2 degrees of freedom per spacetime point. It is a standard result of general relativity, however, that n of the Einstein field equations are constraints on initial conditions rather than dynamical equations. These constraints eliminate n degrees of freedom per point. Another n degrees of freedom per point can be removed by using the freedom to choose n coordinates. We are thus left with n(n-1)-2n = n(n-3) physical degrees of freedom per spacetime point. In four spacetime dimensions, this gives the four phase space degrees of freedom of ordinary general relativity, two gravitational wave polarizations and their time derivatives. If n=3, on the other hand, there are no field degrees of freedom: up to a finite number of possible global degrees of freedom, the geometry is completely determined by the constraints. Equivalently, it can be shown from basic Riemannian geometry that the fullRiemann curvature tensor in 2+1 dimensions is algebraically determined bythe Einstein tensor. The Einstein tensor, in turn, is fixed uniquely, through the Einstein field equations, by the distribution of matter. As a result, there are no propagating gravitational degrees of freedom -- the geometry of spacetime at a point is (almost) entirely determined by the amount and type of matter at that point. In particular, if there is no matter, the Einstein field equations in 2+1 dimensions imply that spacetime is flat.At first sight, this is too strong a restriction. It's not much of a test to be able to quantize a theory with no degrees of freedom, and general relativity is supposed to be about curved spacetimes, not flat ones. But this sort of counting argument can miss a finite number of "global" degrees of freedom. The simplest example of such degrees of freedom is the following:Consider a flat, square piece of paper, with the following "gluing" rule: a point at any edge is to be considered the same as the corresponding point at the opposite edge. Such a space is topologically a torus, and is sometimes called the "video game model" of the torus. (When you reach one edge, you automatically pop back in at the opposite edge, as in many video games.) Geometrically, this space is flat -- all of the rules of Euclidean geometry hold in any small finite region -- because, after all, any small region looks just like a region of the piece of paper you started with. It's a fun exercise to convince yourself that this is true even for regions that contain an "edge."Now change this model a little bit by starting with a parallelogram rather than a square. This gives you a different manifold for each different choice of parallelogram, up to some subtle symmetries (the "mapping class group"). Each of these manifolds is flat, but they are geometrically distinguishable. In fact, this construction gives you a three-parameter family of flat spaces with torus topology: there's one parameter for the length of each side of the parallelogram, plus one for the angle between two adjacent sides.One of the simplest nontrivial solutions to (2+1)-dimensional general relativity is precisely such a torus universe. The constraints fix the overall scale in terms of two parameters (say, one side length and one angle), but these two parameters have an interesting and nontrivial evolution. More complicated topologies give more parameters. So do point particles, which can be represented as conical "defects" in space. With a negative cosmological constant, (2+1)-dimensionalgeneral relativity even admits black hole solutions, which behave almost like ordinary (3+1)-dimensional black holes.Note that this does not contradict the earlier counting argument. There are still only finitely many total degrees of freedom, rather than one or more degrees of freedom per point. But this finite numberof degrees of freedom still has a very interesting dynamics, and the theoryis rich enough to test many standard approaches to quantum gravity.Two-dimensional spacetimes also provide an interesting testing ground for quantum gravity. As you might guess from the earlier counting argument,ordinary general relativity does not make sense in two spacetime dimensions-- the count gives "-2 degrees of freedom per point." But there are simplemodifications that lead to interesting models of what is called "dilaton gravity."For a beautiful nontechnical introduction to topologies like the "video gametorus," see Jeff Weeks' book, The Shape of Space.Quantum cosmologyAt laboratory scales, quantum gravity is unlikely to be important -- gravityis such a weak force that quantum corrections to general relativity will probably be too small to measure. But there are a few places where quantumgravity is likely to be unavoidable. One of these is the extremely earlyUniverse."Quantum cosmology" is the effort to use quantum gravity to predict someof the properties of the very early Universe -- its topology, for instance, and its initial distribution of matter and energy. This task is rather difficult, sincewe don't yet have a quantum theory of gravity. But there may be reasonableapproximations that can be used to obtain partial information. Among thepopular approaches are various saddle point approximations to the path integral(including approximations of the no boundary proposal)and "minisuperspace models," models in which all but a finite number ofdegrees of freedom of the gravitational field are "frozen out" and held fixed. The quantum geometry program has recentlymade some interesting progress in such minisuperspace cosmology -- see,for example this review by Bojowald.Loop quantum gravity is the leading purely gravitational approach toquantizing general relativity ("purely gravitational" as opposed tostring theory, in which quantum gravity comes as one piece of a larger structure). Its aim is to construct a theory of "quantum geometry"in which geometrical objects such as length and areas -- the metricalinformation fundamental to general relativity -- appear as operators acting on quantum states. I will give a very incomplete summary; formore, see this paperand the articles, talks, and videos under "Outreach" at the Penn State gravity center Web site. For a technical review, see Carlo Rovelli's article in Living Reviews.The states of loop quantum gravity are described by "spin networks,"graphs whose edges are labeled by spins and whose vertices are labeledby "intertwiners" (think Clebsch-Gordan coefficients) that tell how tocombine spins. Geometric operators like area, constructed from thespacetime metric, act on these states, changing the network. Thedynamics of general relativity comes in through the Hamiltonian constraint,a not-fully-understood operator condition that determines the admissible spin networks. An alternative view describes three-dimensional spinnetworks as tracing out spinfoams in (3+1)-dimensional spacetime, providing a setting for a pathintegral approach. The (2+1)-dimensional version of this spin foampicture, the Turaev-Viro model, is a well-understood quantization of(2+1)-dimensional gravity.Loop quantum gravity has had important successes in black hole physicsand in quantum cosmology. Furthermore, unlike manypast attempts to quantize general relativity, the loop approach is knownto be mathematically well-defined, and it is "background-free," avoiding some of the conceptual problems of other methods.But although it is a quantum theory based on general relativity, it is not entirely clear that it is really a "quantum theory of gravity" -- it is not yet understood how to recover a good classical limit that looks like classical general relativity.Renormalization To describe a quantum field theory, one typically starts with a Lagrangian or an action, which contains terms that describe the fundamental fields and their interactions. The Lagrangian contains "bare" coupling constants -- the charge of an electron, for instance -- that determine the strengths of the interactions.The couplings in the Lagrangian, however, are not the same as the couplings weactually measure. The physical interactions receive quantum corrections. The charge of an electron, for example, is screened by "vacuum polarization." This effect occurs because the vacuum is full of virtual pairs of particles and antiparticles(see the entry for Hawking radiation), which act,roughly, as a sort of dielectric medium. An electron in the vacuum attracts the positively charged members of virtual particle pairs and repels the negatively charged members, so the effective charge of the electron, as seen from a distance, is reduced. The amount of screening depends on distance (or energy) -- the closer you can get to an electron, the fewer virtual pairs lie between you and the electron, and the less screening occurs. Other kinds of interactions can lead to "antiscreening,"which occurs in quantum chromodynamics. This means that the effective value of a coupling constant will normally depend on the energy at which you probe the interaction. At shortdistances/high energies, the charge of an electron, for example, becomeshigher -- it is less screened by virtual particle-antiparticle pairs. Thecolor charge of a quark, in contrast, becomes lower, approaching zeroat infinite energy in a phenomenon called "asymptotic freedom." Thisvariation of coupling constants with energy or scale is called the"renormalization group flow."The important thing to keep in mind is that the observed coupling constants are not the same as the bare ones that occur in the Lagrangian.In fact, even if a coupling constant is zero in the Lagrangian, itseffective value can be nonzero once quantum corrections are accounted for.In general relativity, for example, a cosmologicalconstant will be induced by quantum fluctuations even if the "bare"cosmological constant is set to zero. A renormalizable theory is one in which the number of undetermined couplingconstants is finite, even after quantum corrections are taken into account. Anonrenormalizable theory is one in which the number of undetermined effectivecoupling constants is infinite. General relativity, when treated as an ordinary quantum field theory, is nonrenormalizable.Now, even a nonrenormalizable theory can have some predictive power -- forgeneral relativity, see for example, work by Donoghue on "effective field theory." But given its infinite number of free parameters, a nonrenormalizable theory probably cannot serve as a complete description of physics.(One possible loophole, suggested by Steven Weinberg, might make certain nonrenormalizable theories more acceptable. As I noted earlier, thecoupling constants in any theory "flow" with energy. It could be that even if a theory has infinitely many coupling constants, they flow to a finite-dimensional surface at high energies. In such an "asymptotically safe"theory, the coupling constants, although infinite in number, would be determined by a finite number of parameters of the high-energy theory. It is an open question whether general relativity is asymptotically safe; hereis a sympathetic review.)For a much more detailed, but entertaining and not-too-technical, description,see John Baez'sRenormalization Made Easy.String theoryString theory is a big subject. Here's a brief introduction; for more, tryPatricia Schwarz'sstring theory page or the Cambridge relativity group'sstring page.The basic idea of string theory is to replace point particles with "strings,"one-dimensional objects that can come as loops (closed strings) or segments(open strings). This gives a very rich structure -- string theories typicallyinvolve more than four spacetime dimensions, and strings can both vibrate and"wrap around" extra compact dimensions, leading to an enormous number ofpossible quantum states. The hope is that these states can unify gravity andelementary particle physics into a single framework, and that, if we are lucky,only one self-consistent theory will exist.Gravity comes into string theory in two closely related ways. First, one ofthe states of a closed string is a "graviton," a massless, self-interacting spin two particle. There are general results that any theory describing sucha particle has to look like general relativity at low energies. Second, if onelooks at a string propagating in a curved spacetime background, one finds that aconsistent description is only possible if the background obeys certain restrictions,which again look like the Einstein field equations at low energies. These tworesults seem independent, but they are actually linked -- the consistent backgroundin which strings can propagate can be described as a quantum state (technically, a"coherent state") of string excitations, including gravitons.For many years, it looked as if there were several distinct self-consistentstring theories. But the "duality revolution" of the last decade showedthat they all seem to be related by a set of duality transformations, whichtypically relate the strongly-coupled behavior of one string theory with the weakly-coupled behavior of another. This development has given us a glimpseof a larger landscape, in which the string theories we know are only smallpieces. The hypothetical big picture is called M theory (M for "mystery,""matrix," "membrane," and a number of other possibilities). We don't knowmuch about M theory as a whole, though we do know that it is not only a theoryof strings -- higher-dimensional objects, membranes or "branes" for short,also play essential roles. A further revolution has come with the AdS/CFTcorrespondence, a remarkable relation between string theory, including gravity, in certain spacetimes ("asymptotically anti-de Sitter spaces") and lower dimensional conformal field theories that don't include gravity.String theory has shown enough striking coincidences -- surprising internalconsistencies that have even led to deep new results in mathematics -- thatmost practitioners are convinced that a deep underlying structure exists.But we don't yet know what that structure is. Ed Witten, for example, hassaid that the one of the most fundamental questions in string theoryis to understand "what is the new kind of geometry that generalises what Einstein used." We also don't know, in a practical sense, how unique stringtheory is: even if there is only one consistent theory, there seem to be anextraordinarily large number of "ground states," each of which gives notonly a different spacetime geometry, but a different particle content and adifferent gauge group for elementary particles. Whether one of these statesdescribes the real Universe, and, if so, whether there is any way to pickit out as being special, remains to be seen.String theory is also just now beginning to address some of the generalconceptual problems of quantum gravity: theproblem of finding local observables to describe spacetime, for example,and the question of what quantum gravity does about singularities. Mostof what we can actually do in string theory involves perturbation theoryaround a fixed classical background, a process that postpones thesedeep issues but cannot entirely remove them.Topological field theoryThe Lagrangian for a quantum field theory typically depends on the metric,that is, on the geometry of spacetime. This is not surprising -- it is naturalthat the propagation of a field on a curved manifold should depend on thecurvature. There are certain special theories, however, whose actions areindependent of the metric. Quantities such as partition functions computedfrom such theories cannot "see" the geometry of a manifold, but only itstopology.Typical field theories have an infinite number of degrees of freedom -- while they involve only a finite number of fields, each field has one or more degrees of freedom per point. In certain cases, though, symmetries are strong enough to reduce the physical degrees of freedom to a finite number. One example of such a reduction occurs in (2+1)-dimensional gravity. Usage varies, but a "topological field theory" is usually defined as one with both of these properties, metric independence and a finite number of physical degrees of freedom. The archetypical topological field theory is Chern-Simons theory, a type of gauge theory in three dimensions. In the past few years, especially because of the work of Ed Witten, such theories have become increasingly important, both in physics and in mathematics. Topologically massive gravityAs noted above, quantum gravity in three spacetimedimensions has no propagating degrees of freedom. This makes it a powerfulmodel for exploring some kinds of conceptual issues, but many other important questions can't be addressed. One step more complicated is a model called "topologically massive gravity,"proposed by Deser, Jackiw, and Templeton in 1982. This model modifies thefield equations of general relativity by adding a new term with threederivatives. This is normally a dangerous thing to do -- "higher derivative"theories in physics usually have negative energies and no stable solutions-- but in this special case it is consistent. In a different context,the extra term is "topological", that is, it dependsonly on the topology of spacetime and not the particular geometry; hence the somewhat confusing name.The addition of a higher-derivative term in the field equations changes thecounting of degrees of freedom of a theory. For topologically massive gravity, the effect is to add a new, propagating degree of freedom, a sort of massivegravitational wave, or, in the quantum theory, a massive graviton. Recently,the model has been a subject of renewed attention because of its interestingproperties in anti-de Sitter space, where it has become a testing groundfor the AdS/CFT correspondence of string theory.Topology is, roughly speaking, the study of those properties of a space thatdon't change under continuous deformations. The standard joke is that "atopologist is someone who can't tell a donut from a coffee cup." If you havea donut-shaped object made of soft clay, you can deform it into a coffeecup-shaped object without cutting or tearing. A donut is not, on the otherhand, topologically the same as a baseball -- to make a baseball-shapedobject into a donut-shaped one, you have to cut a hole in the middle.The field equations of general relativity determine the geometry of spacetimein terms of the matter content. They do not, in general, determine thetopology. The two aren't completely independent; choices of geometryand topology must be compatible, and this places some restrictions onpossible spacetimes. As a simple example, the average curvature of atwo-dimensional manifold with the topology of torus (that is, the surface of a donut) must be zero, while the the average curvature of a two-dimensional sphere must be positive. But the restrictions are weak, and many topologies are consistent with the same geometry. In an earlier section of this glossary, for example, I described a geometrically flat torus. But there are also nonflat geometries on a torus (picture a donut again), and there are other two-dimensional spaces -- the plane, for instance -- with zero curvature.One of the interesting problems in modern cosmology is that of determiningthe topology of space in the real Universe. An interesting discussion of theproblem is here.The September 1998 issue ofClassical and Quantum Gravity has the proceedings of a conference on thetopology of the Universe.Another interesting question is whether quantum fluctuations can causethe topology of space to change in time. John Wheeler has proposed apicture of "spacetime foam," in which the topology of the Universe atthe smallest scales is undergoing complicated, random fluctuations. Whether this picture is correct or not remains an open question.For a nice nontechnical introduction to topology, see Jeff Weeks' book, The Shape of Space.For a little more detail, a good elementary introduction is Klaus Janich's book, Topology. UC Davis has an extremely strong topologygroup in the math department.

TAGS:Steve Carlip 

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