Feynman Lectures On Gravitation / Edition 1

by Richard Feynman, Fernando Morinigo

Overview

The Feynman Lectures on Gravitation are based on notes prepared during a course on gravitational physics that Richard Feynman taught at Caltech during the 1962-63 academic year. For several years prior to these lectures, Feynman thought long and hard about the fundamental problems in gravitational physics, yet he published very little. These lectures represent a useful record of his viewpoints and some of his insights into gravity and its application to cosmology, superstars, wormholes, and gravitational waves at that particular time. The lectures also contain a number of fascinating digressions and asides on the foundations of physics and other issues.Characteristically, Feynman took an untraditional non-geometric approach to gravitation and general relativity based on the underlying quantum aspects of gravity. Hence, these lectures contain a unique pedagogical account of the development of Einstein's general theory of relativity as the inevitable result of the demand for a self-consistent theory of a massless spin-2 field (the graviton) coupled to the energy-momentum tensor of matter. This approach also demonstrates the intimate and fundamental connection between gauge invariance and the principle of equivalence.

Product Details

ISBN-13:	9780813340388
Publisher:	Taylor & Francis
Publication date:	06/20/2002
Series:	Frontiers in Physics
Pages:	280
Product dimensions:	6.00(w) x 9.00(h) x (d)

About the Author

d matter and nuclear physics, and to theoretical astrophysics. Editor of Perseus' Frontiers in Physics series and former editor of American Physical Society's Reviews of Modern Physics, Dr. Pines is a member of the National Academy of Sciences, the American Philosophical Society, a foreign member of the USSR Academy of Sciences, a fellow of the American Academy of Arts and Sciences, and of the American Association for the Advancement of Science. Dr. Pines has received a number of awards, including the Eugene Feenberg Memorial Medal for Contributions to Many-Body Theory; the P.A.M. Dirac Silver Medal for the Advancement of Theoretical Physics; and the Friemann Prize in Condensed Matter Physics.

Read an Excerpt

Lecture 13

13.2 ON THE POSSIBILITY OF A NONUNIFORM AND NONSPHERICAL UNIVERSE

.Since all our conclusions have been so heavily based on the postulate of a uniform cosmology, we might review the nature of the evidence. If we examine a region of the universe within 1.3 x 10⁸ light-years from us, we find simply one cluster, the Virgo cluster-in other words, the matter distributed in a strongly unsymmetric fashion. The lack of symmetry in this region is so large that it can't be explained away. For example, it cannot be due to obscuring by galactic dust, which could affect only that region of the sky lying within a few degrees of the galactic equator. The uniformity must show up in examining regions which are large compared to 10⁸ ly, since the number of galaxies in the usual cluster is much too large to be comfortably described as a fluctuation of the density. What is encouraging is that in very faraway regions the sky seems to be populated by very regular and compact clusters of clusters, involving perhaps thousands of nebulae, all swarming about as bees, relative to the center of gravity of the supercluster. The dispersion of the red shifts shows that the velocities relative to the center of mass must be of the order of 1000 km/sec. This dispersion serves as a very useful measure of the mass of the clusters. It is evident that the clusters are formed by gravitational attractions, and that they are entities which have a long life-they are stable bound systems. From this information we deduce the mass, because we know that velocities of 1000 km/sec are insufficient for escape. There are some worries in this correction; for example if we make a similar computation for the mass of the Virgo cluster, we get a mass 30 times smaller than by other means. Yet the faraway superclusters are larger so that the computation may be more reliable.

The very existence of clusters shows that nebulae attract each other with sufficient strength that these systems are held together for times comparable to the age of the universe. It is very interesting to note that almost all galaxies are in clusters; only a very small fraction appear to exist not in a cluster. The conclusion is that nearly all of the visible matter of the universe does not have sufficient kinetic energy to escape from other matter nearby. In view of this fact, it seems to me that it is very unlikely that the average density of the universe is much smaller than the critical density. If the density is much smaller, the formation of the clusters must be ascribed to local fluctuations which made matter denser in some regions. But on a statistical basis it would be very difficult that there should be enough local fluctuations of the right kind so that nearly all matter occurs in clusters. It might be argued that at an earlier time when the density was greater such fluctuations might be easier-but I have not seen a quantitative argument based on this idea. The inescapable conclusion is that most matter gets pulled into clusters because the gravitational energy is of the same order as the kinetic energy of the expansion-this to me suggests that the average density must be very nearly the critical density everywhere.

The preceding guesses about the average density do not lend any support to the hypothesis of uniformity. It could still be that the question as to whether the matter in given regions holds together is purely local, and the situation varies from region to region. The cosmological principle can be studied only by making a detailed comparison of the density of matter and the nonlinearity of the red shift, which represents an acceleration of the galaxies. These are quantities which are in principle independently measurable, yet a definite relation between them is predicted by theories adopting the cosmological principle. If the variation of density with radial distance were to be measured with great accuracy, we might find that maybe matter is too dense in the inner region, in such a way that the cosmological principle cannot hold.

Even if the cosmological principle were wrong, it would be possible for the universe to have a spherical symmetry-a prejudice in favor of the cosmological principle reflects our surprise at finding that we are at what looks like the center of the universe. Let us assume that the visible region is nearly symmetric but that outside there is more matter, distributed in a lopsided fashion. What would be the difference? We would expect a first order correction to the motion of the distant nebulae to be due to the tidal forces as illustrated in Figure 13.1. (If the visible region were accelerating as a whole we would not be aware of the acceleration!) The result would be a correction to the red shift having a quadrupole character; the red shift should be redder in two polar regions and bluer in a equatorial region.

The preceding discussions show us how little the relativistic theory of gravitation tells us about cosmology. The central problems of cosmology can be solved only when we actually know what the universe "really" looks like, when we have accurate plots of red shifts and densities as a function of distance and position in the galactic sky.

13.3 DISAPPEARING GALAXIES AND ENERGY CONSERVATION

Let us briefly mention some of the other interesting puzzles of cosmology and cosmological models. Is it possible for certain nebulae to vanish from view forever? With our present theory, the answer is no; regardless of whether the density is greater or smaller than critical, no galaxy which is now visible will ever disappear entirely, although it may get redder and redder. It makes no sense to worry about the possibility of galaxies receding from us faster than light, whatever that means since they would never be observable by hypothesis.

In Hoyle's theory, the nebulae fax away do vanish; his expression for the recession velocity is such that the volume of a cube defined by eight given galaxies doubles in every time interval of a certain size. The velocities are thus postulated to increase so that some do accelerate to speeds faster than c. However, Hoyle's theory does not pretend to explain the motion of the galaxies in terms of forces. It simply states a kinematic rule for calculating specified positions as a function of time, and the specifications are such that galaxies do disappear from view eventually. There is little point in searching for a force in nature which might result in Hoyle's expansion-It would have to be of an unfamiliar type, since no finite forces could ever result in such a motion in the framework of the present mechanics.

I have inherited a prejudice from my teacher, Dr. Wheeler, to consider it against the rules to explain a result by making a convenient change in the theory, when the present theory has not been fully investigated. I say this because there has been speculation that a force of expansion, which would tend to accelerate galaxies as Hoyle wants, would result if the positive charges were not exactly equal to the negative charges. Since we have an excellent theory of electrodynamics in which e⁺ is precisely the opposite of e^-, such a speculation seems useless. We are not going to overthrow a beautiful theory of electrodynamics to provide a mechanism for a kinematic model which might easily be found wrong as soon as we make accurate observations. It is still quite possible for Hoyle to be right but we shall find it out first of all from observations on the universe as it is.

Let me say also something that people who worry about mathematical proofs and inconsistencies seem not to know. There is no way of showing mathematically that a physical conclusion is wrong or inconsistent. All that can be shown is that the mathematical assumptions are wrong. If we find that certain mathematical assumptions lead to a logically inconsistent description of Nature, we change the assumptions, not Nature.

I say all this because I am not sure in which ways the theory of the Universe according to Hoyle may not quite coincide with many other assumptions we physicists ordinarily make. For example, there may be some trouble in the possibility of signalling which is so much a part of the thinking in relativistic theories. If a given galaxy has disappeared from view, is it really out of the universe, for us? Is it possible that we may ask a friend in the outer fringe how this galaxy is doing, while he is still in our view and the other galaxy is in his view? Or is there a general conspiracy of the detailed kinematics to preserve the relativistic rule, that a velocity near c in a coordinate system moving relative to us with a speed near c always yields a sum less than c?

Let us speak for a moment about the conservation of energy. In Hoyle's theory matter is created at rest at the "center" of the universe and we have said that there is no over-all energy creation because the negative gravitational energy just balances the rest mass energy. This kind of conservation of energy is that if we take a box of finite size anywhere, no matter appears inside except by an energy flow from the outside. In other words, only a law of local conservation is meaningful. If energy could disappear in one place and simultaneously reappear in some other place, without the flow of anything in between, we could deduce no physical consequences from the over-all "conservation." Let us therefore interpret Hoyle's creation of matter as follows. We imagine a finite universe having large masses distributed in a spherical shell as illustrated in Figure 13.2. We imagine pairs of particles of zero energy dropping in from the outside of the shell toward the inside. These we might think of as photons or gravitons or neutrinos, or maybe some new particles, some shmootrinos which don't worry about baryon conservation. When they meet another shmootrino dropping in from the other side with opposite momentum, these can have enough energy to create a hydrogen atom. In this way we can have both Hoyle's matter creation and a local energy conservation, since the matter is created from energy flowing in by the shmootrinos. If the flux of shmootrinos were very high, and the cross section for matter production be finite just above threshold, we can understand why matter should be created at rest relative to the average of the nebulae. The idea is that if the flux is high enough, as soon as a shmootrino has enough energy it will find another coming in the opposite direction and proceed to create matter; if the maximum energy which shmootrinos can acquire in falling is barely the same as the threshold, the matter will be created "at rest." In some such manner we can have simultaneously local energy conservation and Hoyle's theory. Of course, there are a great many remaining problems, such as we still have to consider why baryon number may not be conserved....

Table of Contents

Foreword Quantum Gravity Lecture 1 * A Field Approach to Gravitation * The Characteristics of Gravitational Phenomena * Quantum Effects in Gravitation * On the Philosophical Problems in Quantizing macroscopic Objects * Gravitation as a Consequence of Other Fields Lecture 2 * Postulates of Statistical Mechanics * Difficulties of Speculative Mechanics * The Exchange of One Neutrino * The Exchange of Two Neutrinos Lecture 3 * The Spine of the Graviton * Amplitudes and Polarizations in Electrodynamics, Our Typical Field Theory * Amplitudes for Exchange of a Graviton * Physical Interpretation of the Terms in the Amplitudes * The Lagrangian for the Gravitational Field * The Equations for the Gravitational Field * Definition of Symbols Lecture 4 * The Connection Between the Tensor Rank and the Sign of a Field * The Stress-Energy Tensor for Scalar Matter * Amplitudes for Scattering (Scalar Theory) * Detailed Properties for Plane Waves, Compton Effect * Nonlinear Diagrams for Gravitons * The Classical Equations of Motion of a Gravitating Particle * Orbital Motion of Particle About a Star Lecture 5 * Planetary Orbits and the Precession of Mercury * Time Dilation in a Gravitational Field * Cosmological Effects of the Time Dilation. Machs Principle * Machs Principle in Quantum Mechanics * The Self Energy of the Gravitational Field Lecture 6 * The Bilinear Terms of the Stress-Energy Tensor * Formulation of a Theory Correct to All Orders * The Construction of Invariants with Respect to Infinitesimal Transformations * The Lagrangian of the Theory Correct to All Orders * The Einstein Equation for the Stress-Energy Tensor Lecture 7 * The Principle of Equivalence * Some Consequences of the Principle of Equivalence * Maximum Clock Rates in Gravity Fields * The Proper Time in General Coordinates * The Geometrical Interpretation of the Metric Tensor * Curvatures in Two and Four Dimensions * The Number of Quantities Invariant under General Transformations Lecture 8 * Transformations of Tensor Components in Nonorthogonal Coordinates * The Equations to Determine Invariants of g(( * On the Assumption that Space is Truly Flat * On the Relations Between Different Approaches to Gravity Theory * The Curvatures as Referred to Tangent Spaces * The Curvatures Referred to Arbitrary Coordinates * Properties of the Grand Curvature Tensor Lecture 9 * Modifications of Electrodynamics Required by the Principle of Equivalence * Covariant Derivatives of Tensors * Parallel Displacement of a Vector * The Connection between Curvatures and Matter Lecture 10 * The Field Equations of Gravity * The Action for Classical Particles in a Gravitational Field * The Action for matter Fields in a Gravitational Field Lecture 11 * The Curvature in the Vicinity of a Spherical Star * On the Connection Between matter and the Curvatures * The Scwarzschild Metric, the Field Outside a Spherical Star * The Schwarzschild Singularity * Speculations on the Wormhole Concept * Problems for Theoretical Investigations of the Wormholes Lecture 12 * Problems of Cosmology * Assumptions Leading to Cosmological Models * The Interpretation of the Cosmological Metric * The Measurements of Cosmological Distances * On the Characteristics of a Bounded or Open Universe Lecture 13 * On the Role of the Density of the Universe in Cosmology * On the Possibility of a Nonuniform and Nonspherical Universe * Disappearing Galaxies and Energy Conservation * Machs Principle and Boundary Conditions * Mysteries in the Heavens Lecture 14 * The Problem of Superstars in General Relativity * The Significance of Solutions and their Parameters * Some Numerical Results * Projects and Conjectures for Future Investigations of Superstars Lecture 15 * The Physical Topology of the Schwarzschild Solutions * Particle Orbits in a Schwarzschild Field * On the Future of Geometrodynamics Lecture 16 * The Coupling Between Matter Fields and Gravity * Completion of the Theory: A Simple Example of Gravitational Radiation * Radiation of Gravitons with Particle Decays * Radiation of Gravitons with Particle Scattering * The Sources of Classical Gravitational Waves Bibliography Index

Editorial Reviews

In 1962-63 Feynman offered a lecture course on gravitational physics for advanced graduate and postdoctoral fellows at the California Institute of Technology. Two students took notes on the 27 lectures, and Feynman edited and corrected the first 11; these were typed up and distributed to students and sold in the college bookstore. A further five lectures were added in a 1971 edition, though the editing was not as rigorous. Here they are again, indexed and referenced. Feynman never got around to editing the rest of the lectures, so they are not included. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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