These authors ranked among the greatest astrophysicists of the 20th century, and their work is remarkable for its deep physical insights and clarity of presentation. This book explores general relativity, properties of matter under astrophysical conditions, stars, and stellar systems. It constitutes a valuable resource for today's physicists, astronomers, and graduate students. 1971 edition.
About the Author
Yakov B. Zel'dovich (1914–87) played an important role in the development of Soviet nuclear and thermonuclear weapons. He also made key contributions to the fields of adsorption and catalysis, shock waves, nuclear physics, particle physics, astrophysics, physical cosmology, and general relativity.
Igor Dmitriyevich Novikov (1935–) is a Russian theoretical astrophysicist and cosmologist who formulated the Novikov self-consistency principle, an important contribution to the theory of time travel. He has worked at the Russian Space Research Institute, the Lebedev Physical Institute, Moscow State University, and the University of Copenhagen.
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Stars and Relativity
By Ya. B. Zel'dovich, I. D. Novikov, Kip S. Thorne, W. David Arnett, Eli Arlock
Dover Publications, Inc.Copyright © 1971 The University of Chicago
All rights reserved.
EINSTEIN'S GRAVITATIONAL EQUATIONS
1.1 The Equality of Inertial and Gravitational Mass
The contemporary theory of gravitation, formulated by Albert Einstein in 1916, was an extension of the special theory of relativity (STR); thus it is frequently referred to as the general theory of relativity (GTR).
The essence of STR is contained in the Lorentz transformation for space-time coordinates, and in the corresponding transformation laws for such physical quantities as energy and momentum. All the postulates and conclusions of STR, such as the constancy of the velocity of light, the dependence of mass on velocity, mass defect and its relationship to the energy of a system, time dilation during fast motion as exemplified by the decay of particles—all of these are confirmed experimentally. Indeed, STR has by now found its place in practical engineering calculations. Thus, there are no doubts whatsoever as to the correctness of STR.
The general theory of relativity (GTR) is in an entirely different situation: Experiments which specifically confirm GTR are few at present. They include the precession of Mercury's perihelion, the deflection of light in the Sun's gravitational field, the gravitational redshift, and (as of 1968) the relativistic delay in radar signals passing near the limb of the Sun. Actually, the principal argument in favor of GTR is none of these. Rather, it is a basic fact which is known to every high school student, and which inspired Einstein's work: the proportionality of weight and mass, i.e., the equality of the acceleration of different bodies in a gravitational field.
Newton's law of gravitation, F = -Gm1m2/ r2, is very similar to Coulomb's law of electrostatics, F = e1e2/r2. Naturally, the question arises as to why the theory of the electromagnetic field, which is formulated in Euclidean space, and GTR with its concept of spacetime curvature, are so different. Couldn't gravity also be defined as a field in Euclidean space?
It will be demonstrated in the forthcoming discussion that STR and quantum mechanics make the spacetime curvature, which is characteristic of GTR, logically unavoidable. GTR is not only the most elegant and comprehensive theory from the viewpoint of mathematics, it is also the physically necessary theory of gravitation.
As we have already emphasized, the most characteristic property of a gravitational field is the fact that it acts with complete indifference upon all bodies, imparting to them equal accelerations. This is a fact that was first established by Galileo. Thus, the gravitational field, in its effects, is different from any other field known to physics.
One of the most important tasks of experimental physics, in view of current interest in the theory of gravitation, has been to measure the accuracy with which gravity imparts equal accelerations to different bodies. The principle of equal accelerations can be formulated equivalently as the principle of strict proportionality between inertial and (passive) gravitational mass: In the equations of motion for a body in a gravitational field,
mi(d2r/dt2) = mg[for all]U,
the inertial mass appears on the left-hand side and the gravitational mass on the right. If all bodies satisfy mi = αmg where α is a universal constant, then Galileo's law will follow, since the masses will not enter into the equation of motion at all. The constant α obviously depends only on the normalization of U; and with an appropriate choice it can be set to unity.
In 1890 Eötvös devised an exceedingly precise method to check the proportionality of the gravitational and inertial masses. In essence, his experiment consisted of the following: Any object at rest on the surface of the Earth experiences gravitational attractions not only from the Earth but also from the Sun, the Moon, and other celestial bodies. The object is also acted upon by centrifugal forces resulting from the diurnal rotation of the Earth about its axis, the annual rotation of the Earth about the Sun, and the monthly rotation of the center of the Earth about the center of gravity of the Earth-Moon system.
The attraction by planets and by other celestial bodies can be neglected. Similarly, we can neglect the centrifugal acceleration due to the motion of the Sun in the Galaxy, etc. The acceleration of the Earth's gravity is about 980 cm sec-2; the centrifugal acceleration due to diurnal rotation at the latitude of Moscow is about 1.5 cm sec-2. The acceleration of the Sun's gravitational field on the orbit of the Earth is about 0.5 cm sec2; the centrifugal acceleration due to the Earth's orbital motion is obviously also 0.5 cm sec-2. The effect of the Moon is characterized by an acceleration of 4 × 10-3 cm sec-2.
Visualize two bodies, A and B, of equal mass, balanced on the ends of a thin rod (balance arm) which is suspended at its middle by a thin fiber. The gravitational forces of the Earth, Sun, and Moon are proportional to the gravitational mass, whereas the centrifugal forces are proportional to the inertial mass.
If the two bodies have identically the same ratio of inertial to gravitational mass, the total force acting upon them will be identical. In this case the balance arm will remain in equilibrium with respect to the axis of the Earth, and also with respect to the Sun, regardless of the direction in which it is oriented. (Zero torque acts.) However, if the mass ratios are different, and if the balance arm is set perpendicular to the direction of the centrifugal forces, then the forces on the two bodies will not be equal; they will produce a torque, causing the balance arm to twist beneath its supporting fiber. The centrifugal force of diurnal rotation is larger than the centrifugal force of annual rotations. However, a change in orientation of the balance arm with respect to the Sun takes place naturally during the rotation of the Earth, without a corresponding change in orientation of the balance arm relative to objects in the laboratory that surround it, or relative to the terrain of the Earth. Hence, in practice, it is most convenient to observe whether or not the balance arm is twisted by unbalanced gravitational and centrifugal forces due to the Sun and the annual rotation of the Earth.
In view of the absence of such twisting, Eötvös concluded that the ratio of gravitational to inertial mass for different bodies differs by no more than one part in 108. The Eötvös experiment was recently repeated by Dicke (1961a) and by Roll, Krotkov, and Dicke (1964). By placing the balance arm in a high vacuum and determining the torque on it, using a photocell and automatic feedback to prevent balance-arm twisting, Dicke was able to improve the accuracy of the Eötvös experiment. His findings agree with Eötvös's findings: the ratio between the gravitational and inertial masses of copper and lead are in agreement. But, with Dicke, the accuracy of this agreement is better than one part in 1010!
Consider the significance of this finding. The inertial mass depends on energy—this is a consequence of STR. In particular, we know from STR that when two deuterium atoms are combined into one atom of helium, the inertial mass decreases by an amount equal to about 0.006 of the initial mass, in accordance with the mass defects of helium and deuterium. Precise determinations of inertial atomic masses by using a mass spectrograph on one hand, and direct measurements of the energy of nuclear reactions on the other hand, have confirmed that energy contributes to inertial mass.
What determines the (passive) gravitational mass of a body, and consequently the force which it experiences in a gravitational field? Does the gravitational force depend on the number of baryons in the body, i.e., the baryon charge, in a manner similar to the dependence of electrostatic attraction on electrical charge? Or does this force depend on the total energy of the body? For conventional matter (neither mesons nor antimatter), the baryon number and the inertial mass are approximately proportional to each other with deviations of the order of 10-3. Hence with small accuracy, an Eötvös-type experiment could not have resolved the problem. However, Dicke's experimental accuracy of better than 10-10 leads inescapably to the conclusion that the force of gravity, like the inertial mass, is proportional to the energy of the body. Such extreme accuracy strengthens the foundations of GTR.
The Eötvös experiment shows that the attraction is not determined by the baryon charge of the body; universal gravitation thus cannot be construed as due to an analogue of the electrostatic attraction of electrical charges of opposite sign. (It is not accidental that two electric charges must have opposite signs in order to attract each other; see chapter 2.) Thus, the notion that some particles—for example, the so-called antiparticles (positrons, antiprotons, antineutrons)—can experience "antigravitation" is absolutely erroneous. From experiments on accelerators it is well known that to create antiparticles it is necessary to expend energy. This energy is the source of the mass of the antiparticle. Consequently, the antiparticle has a positive gravitational mass, exactly the same as the corresponding particle. The Eötvös and Dicke experiments provide an indirect proof of this.
Lee and Yang (1955) (see also Dicke 1962) posed the question of whether or not, along with universal gravitation, there might exist a repulsive analogue to the Coulomb force which is proportional to the number of nucleons. The Eötvös and Dicke experiments demonstrate that such a force does not exist; or, more precisely, that if it does exist, it is at least 107 times weaker than the gravitational force, and at least 1043times weaker than the Coulomb force between two protons.
It should be emphasized that, if repulsive forces due to baryon charge were to exist, GTR could be nonetheless valid. However, in such a case it would be considerably more difficult to disentangle the different forces by experiment (just as it is difficult to perform the Cavendish experiment with bodies that are electrically charged).
1.2 The Fundamental Concept of the General Theory of Relativity
To Newton it seemed obvious that physical space is Euclidean, i.e., that there exist parallel lines, that the sum of the angles in a triangle is π, that the circumference of a circle is 2πr, etc., and that time flows always at the same rate everywhere.
The idea that the properties of space could be otherwise (for example, that the sum of the angles in a triangle could depend on its area) occurred considerably later. Mathematically, such spaces were first discovered and investigated by Lobachevski.
According to STR, in an inertial frame of reference the square of the four-dimensional distance (in space and time) between two infinitely close events (interval) takes the form
ds2 = (cdt)2 - (dx)2 - (dy)2 - (dz)2, (1.2.1)
where c is the speed of light, t is time, and x, y, z are Cartesian space coordinates. Such a system of coordinates is said to be a Galilean system.
Expression (1.2.1) is analogous in form to the expression for the square of the distance in Euclidean three-dimensional space as written in Cartesian coordinates (only the number of squared differentials and their signs differ). Hence, expression (1.2.1) describes for us a flat Euclidean spacetime, or, more precisely, a pseudo-Euclidean spacetime (emphasizing the special nature of time; note in expression [1.2.1] the plus sign in front of (cdt)2 as contrasted with the minus signs in front of the space differentials). STR is the theory of physical processes in this flat spacetime, which we shall call Minkowski spacetime.
In this spacetime a free particle moves along a straight line (Fig. 1), called the world line of the particle. We will not pursue these matters any further because we assume that the reader is familiar with the elementary principles of STR.
Einstein's theory of gravitation was inspired by and based on the "principle of equivalence," which states that when gravity is present, as when it is absent, free particles move along extremal (geodesic) lines of spacetime—spacetime now being curved, not flat.
According to Einstein, the gravitational field is nothing more than a deviation of the properties of real spacetime from the properties of a flat manifold. It is masses that create the gravitational field—which curves space-time. A body in this curved spacetime moves along a geodesic line, which is independent of its mass and composition. This geodesic motion in curved spacetime is perceived by us as curved motion with variable velocity. Einstein's theory postulates from the very beginning that the curvature of the trajectory and the variation of speed are spacetime properties, properties of the geodesies; and, hence, that the accelerations of all bodies must be equal. Thus the ratio of gravitational mass to inertial mass (on which the gravitational acceleration of the body depends, according to Newton) is equal for all bodies.
However, historically Einstein proceeded from more obvious assumptions based on a simple physical model of the gravitational field, which had equal accelerations for every body. This was the well-known elevator thought-experiment: In an isolated, accelerated volume, in the absence of a true gravitational field (i.e., in an elevator in space) all phenomena proceed in precisely the same manner as in a real gravitational field in which the volume either is at rest or moves uniformly. Relative to the elevator, freely moving bodies undergo accelerated motion, and the acceleration is precisely the same for every body.
In the accelerated elevator (without real gravity) one can see all the effects of gravitation—for instance, a changing frequency for propagating light. In particular, compare the frequency of emitted light with the frequency which the light possesses after a time interval has elapsed. Because of the acceleration relative to an inertial frame, the velocities of the elevator and of objects at rest in it will change with time. Hence, the velocity of the light receiver at the moment the light is received will differ from the velocity of the emitter at the moment of emission. As a result of the Doppler effect this difference in velocities will produce a similar difference in frequency between the emitted and the received light. This difference will depend on the relative directions of the light ray and the acceleration.
Such an interpretation is given by an external inertial observer who knows that the elevator is being accelerated. An internal observer attributes the frequency shift to a "gravitational field" present inside the elevator. If the acceleration is directed from the emitter toward the receiver, then the light will suffer a blueshift; in the opposite case there will be a redshift. It is easy to convince oneself that a light ray should exhibit a curvature relative to the three-dimensional coordinate system which is tied to the elevator. Thus, phenomena in an accelerated reference frame are identical with those in a "gravitational field."
However, only a homogeneous gravitational field with a constant value and direction throughout all space can be described in such a way. Gravitational fields created by separated bodies are quite different. To cancel the Earth's gravitational field, we need elevators with different directions of acceleration at different points. Observers in the different elevators, communicating with each other, will find out about their relative accelerations, and will thereby discover the presence of a gravitational field which cannot be removed by going to any single accelerated moving coordinate system.
A true gravitational field cannot be eliminated by any coordinate transformations; Fock has particularly emphasized this in his well-known book (1961). All the same, the elevator model describes the most important properties of gravity (equality of all accelerations; the gravitational influence on light) in such a natural way that it is unreasonable to give up this model. To conserve this local use of such a model at every point in spacetime, one can introduce a transformation of the coordinate system in every region, but this transformation cannot be reduced to any particular motion in flat spacetime. This impossibility expresses the "curvature" of spacetime (see below).
The following sections will deal briefly with the mathematical methods of analyzing spacetime curvature which are required for understanding the material to follow. Readers interested in a more detailed exposition of space-time curvature and GTR are referred to the books by Landau and Lifshitz (1962), Synge (1960), and Robertson and Noonan (1968), and also to the contribution by Zel'manov (1959b).
Excerpted from Stars and Relativity by Ya. B. Zel'dovich, I. D. Novikov, Kip S. Thorne, W. David Arnett, Eli Arlock. Copyright © 1971 The University of Chicago. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Preface to the English Edition,
Preface to the First Russian Edition,
I. THE THEORY OF GRAVITATION,
1. EINSTEIN'S GRAVITATIONAL EQUATIONS,
2. INESCAPABILITY OF THE GENERAL THEORY OF RELATIVITY (GTR) AND PROBLEMS IN THE THEORY OF GRAVITATION,
3. THE SPHERICALLY SYMMETRIC GRAVITATIONAL FIELD,
4. NONSPHERICAL GRAVITATIONAL FIELDS,
II. THE EQUATION OF STATE OF MATTER,
5. INTRODUCTION TO PART II,
6. COLD MATTER,
7. PROPERTIES OF MATTER AT HIGH TEMPERATURES,
8. THERMODYNAMIC QUANTITIES AT HIGH TEMPERATURES,
III. RELATIVISTIC STAGES OF EVOLUTION OF COSMIC OBJECTS,
9. INTRODUCTION TO PART III,
10. THE EQUILIBRIUM AND STABILITY OF STARS,
11. STELLAR EVOLUTION,
12. STAR CLUSTERS,
13. PHYSICAL PROCESSES IN THE VICINITIES OF RELATIVISTIC OBJECTS AND A COMPARISON WITH OBSERVATIONS,
14. QUASISTELLAR OBJECTS,
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