GRAVITATION. AND COSMOLOGY: PRINCIPLES AND APPLICATIONS. OF THE GENERAL THEORY. OF RELATIVITY. STEVEN WEINBERG. Massachusetts. Gravitation and cosmology principles and applications of the general theory of relativity - Weinberg musicmarkup.info - Ebook download as PDF File .pdf), Text File .txt) or . Weinberg S. Gravitation and cosmology principles and applications of the general theory of musicmarkup.info - Ebook download as PDF File .pdf) or read book.

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Weinberg, Steven. Gravitation and cosmology. Includes bibliographies. 1. General relativity (Physics). 2. Gravitation. 3. Cosmology. QC6.W47 I'l Gravitation And Cosmology: Principles. And Applications Of The General Theory Of Relativity. Steven Weinberg. Page 2. Page 3. Page 4. Page 5. Page 6. Library of Congress Cataloging in Publication Data: Weinberg, Steven. Gravitation and cosmology Includes bibliographies. 1. General relativity (Physics) . 2.

Sign in Create an account. Syntax Advanced Search. About us. Editorial team. General Editors: Steven Weinberg.

W ithout exception, their efforts only succeeded in replacing the fifth postulate with some other equivalent postulate, which might or might not seem more self-evident, but which in any case could not be proved from Euclids other postulates either. Thus, the Athenian neo-Platonist Proclos offered the substitute postulate: I f a straight line intersects one o f two parallels, it will intersect the other also.

That is, if we define parallel lines as straight lines that do not intersect however far extended, then there can be at most one line that passes through any given point and is parallel to a given line. John Wallis, Savillian Professor at Oxford, showed that Euclids fifth postulate could be replaced with the equivalent statement Given any figure there exists a figure, similar to it, of any size.

And Legendre proved the equivalence o f the fifth 1 History o f Non-Euclidean Geometry postulate with the statement There is a triangle in which the sum o f the three angles is equal to two right angles.

In the Jesuit Geralamo Saccheri published a detailed study o f what geometry would be like if the fifth postulate were false. He particularly examined the consequences o f what he called the hypothesis o f the acute angle, that is, that a straight line being given, there can be drawn a perpendicular to it and a line cutting it at an acute angle, which do not intersect each other.

Similar tentative explorations o f non-Euclidean geometry were begun by Lambert and Legendre. It seems to have been Carl Friedrich Gauss who first had the courage to accept non-Euclidean geometry as a logical possibility. His gradual enlightenment is recorded in a series o f letters4 to W. In a letter dated he begged Taurinus to keep silent about the heretical opinions he had revealed. Gauss even went to the extent o f surveying a triangle40 in the Harz mountains formed by Inselberg, Brocken, and Hoher Hagen to see if the sum o f its interior angles was !

It was. Then, in , Gauss received a letter from his friend Wolfgang Bolyai, describing the non-Euclidean geometry developed by his son, Janos Bolyai , an Austrian army officer. He subsequently also learned that a professor in the Kazan, Nikolai Ivanovich Lobachevski , had obtained similar results in Gauss, Bolyai, and Lobachevski had independently discovered what in modern terms is called the two-dimensional space of constant negative curvature.

Such spaces are still very interesting; we shall see in the chapter on cosmography that the space in which we actually live may be a three-dimensional space of constant curvature. But to its discoverers the important thing about their new geometry was that it describes an infinite two-dimensional space in which all o f Euclids assumptions are satisfied-except the fifth postulate!

In this it is unique, which perhaps explains why it was discovered more or less independently in Germany, Austria, and Russia. The surface o f a sphere also satisfies Euclidean geometry without the fifth postulate, but being finite it does not have room for parallel lines. We shall see in Chapter 13, on symmetric spaces, that the two-dimensional space o f constant negative curvature cannot be realized as a surface in ordinary three-dimensional Euclidean space, which is doubtless why it took two millennia to find it.

And of course it also violates the alternative common-sense versions o f Euclids fifth postulate given by Proclos, Wallis, and Legendre through a given point there can be drawn infinitely many lines parallel to any given line; no figures o f different size are similar; and the sum o f the angles of any triangle is less than Newton tested this possibility by experiments with pendulums of equal length but different composition.

L evi-Civita. This result was later verified more accurately by Friedrich Wilhelm Bessel in Roland von E otvos7 succeeded by a different method. I f this were the case. His tools were an inclined plane to slow the fall. These observations were later improved by Christaan Huygens No twist was detected. At equilibrium the beam would sag in such a way that lAimgA9 - Figure 1.

Eotvos hung two weights A and B from the ends o f a cm beam suspended on a fine wire at its center. He knew that the moon falls toward the earth a distance 0.

Newton did not publish this calculation for twenty years. The advantage is that the angle between the direction of the sun and the balance arm changed with a hr period.

The first suggestion o f an inverse-square law may have been made around by Ismael Bullialdus Einstein was very impressed with the observed equality o f gravitational and inertial mass8. Newton in proved that planets moving under the influence o f an inverse-square-law force would.

That is. It also sets very stringent limits on any possible nongravitational forces that might exist. Johannus Scotus Erigena c. Adelard also translated Euclid from Arabic into Latin. Christopher Wren This theory was taken up by Adelard o f Bath twelfth century. This idea was not entirely original with Newton. Some irregularities in the orbit of Uranus remained unexplained until. In H. Today we know that this model is totally wrong.

These stupendous accomplishments were published on July 5. A year before his prediction o f Neptune. Seeliger constructed an elaborate model o f the zodiacal light. One problem remained. This discrepancy was confirmed in by Simon Newcomb What really impressed Newton about all this was that there are a great many more transformations that do not leave the equations o f motion invariant.

If 0 and O' use the unprimed and primed coordinate system. In his words It is a simple matter to check that these equations take the same form when written in terms o f a new set of space-time coordinates: What then determines which reference frames are inertial frames?

Newton answered that there must exist an absolute space. The equations o f motion can hold in their usual form in only a limited class o f coordinate systems.

Relative space is some movable dimension or measure of absolute space. Samuel Clarke But afterwards. No one is competent to say how the experiment would turn out if the sides o f the vessel increased in thickness and mass until they were several leagues thick. This experiment I have made m yself. The most famous is the rotating bucket A t first. First stand still.. The stars will seem to rotate around the zenith. The surface o f the earth is not exactly an inertial frame.

There is a simple experiment that anyone can perform on a starry night. The theory o f electrodynamics presented in by James Clark Maxwell clearly did not satisfy the principle o f Galilean relativity. It would surely be a remarkable coincidence if the inertial frame. Then pirouette. This argument can be made more precise.

The issue is not closed. For one thing. And if Mach is right. Observe that the stars are more or less unmoving. I have not yet mentioned special relativity because.

Maxwell himself thought that electromagnetic waves were carried by a medium. Either we admit that there is a Newtonian absolute space-time. Poincare in particular seems to have glimpsed the revolutionary implications that this would have for mechanics. The most important experiment was that o f Albert Abraham Michelson and E. W ithout entering this controversy. Hendrik Antoon Lorentz26 As discussed in Chapters 4 and 5.

A collaboration with the mathematician Marcel Grossman led Einstein by to the view 38 that the gravitational field must be identified with the 10 components o f the metric tensor o f Riemannian space-time geometry.

The laws o f physics in the form given them by Maxwell and Einstein could still only be true in a limited class o f inertial reference frames. The new physics. A number o f attempts were made in by Einstein. It is not clear that Einstein was directly influenced by the MichelsonMorley experiment itself.

After Einstein. The Lorentz group o f transformations is not in any way larger than the Galileo group. Einstein proposed that the Galilean transformation 1.

Before Maxwell. As we shall see in Chapter 3. The gravitational deflection o f light by the sun had not yet been measured. A crucial step toward this goal was taken in It remained to construct a relativistic theory o f gravitation. These developments are discussed in detail in the next chapter. I have been content to base this chapter on secondary sources, aside from the works of Newton, Mach, Maxwell, Newcomb, and Einstein quoted in the text.

The authorities on whom I have drawn most heavily are listed below. Weyl, Space, Time, Matter, 4th ed. Carmeli, S. Fickler, and L. W itten Plenum Press, New York, , p. II, Chapter V. Einstein, Annalen der Phys. The leading English edition is Euclid's Elements, translated with an introduction and commentary by T. Heath rev. Quoted by R. Bonola, in Non-Euclidean Geometry, trans. Carslaw Dover Press, , pp. Klein, Math. Weyl, in Space-Time-Matter, trans.

Brose Dover Press, , p. Beltrami, Saggio di interpretazione della geometria non-euclidea, quoted by J. North in The Measure of the Universe Oxford, , p. Cajori University o f California Press, , p. Ungarn, 8, 65 ; R. Fekete, Ann. Also see J. See, for example, A. Einstein, The Meaning of Relativity 2nd ed.

Lee and C. Yang, Phys. Dicke, in Relativity, Groups, and Topology, ed. DeWitt and B. Krotkov, and R. Dicke, Ann. Dobbs, J. Harvey, D. Paya, and H. Horstmann, Phys. W itteborn and W. Fairbank, Phys. Letters, 19, The most accessible edition is that of Florian Carjori, ref. Newcomb, Astronomical Papers of the American Ephemeris, 1, Excerpts are quoted by A.

Mach, The Science of Mechanics, trans. McCormack 2nd ed. Schiff, Rev. Clemence, Rev. Niven Dover Publications, , p. For an account o f these experiments, see C. Michelson and E. Morley, Am. Its Origins and Impact on Modern Thought, ed. Jaseia, A. Javan, J. Murray, and C. Townes, Phys.

Fitzgerald, quoted by O. Lodge, Nature, 46, Also see O. Lodge, Phil. Lorentz, Zittungsverslagen der Akad. Brill, Leiden, ; Proc. Amsterdam English version , 6, The third reference, and a translated excerpt from the second, are available in The Principle of Relativity, ref. Louis International Exposition in , trans. Halstead, The Monist, 15, 1 , reprinted in Relativity Theory: Its Origins and Impact on Modern Thought, ref.

Palermo, 21, For a balanced view o f this question, see G. Holton, Am. Einstein, Ann. Physik, 17, ; 18, Translations are given in The Principle o f Relativity, ref.

Holton, ref. See ref. Feigl and G. Einstein, Jahrb. Planck, Sitzungsber. Leipzig, 26 Leipzig, 35, For an English translation, see The Principle of Relativity, ref. A Einstein, Ann. Leipzig, 38, , Abraham, Lincei Atti, 20, ; Phys. Nordstrom, Phys.

Leipzig, 40, ; 42, ; 43, ; Phys. Einstein, Phys. Einstein and M.

Grossmann, Z. Einstein, Vierteljahr Nat. Einstein, Sitzungsber. Also see D. Hilbert, Nachschr. This famous experiment may in fact be a myth. See A. Miller, Isis, to be published Wells, The Time Machine. This chapter, while self-contained, is only a brief summary, and aims primarily at establishing our notation and collecting some formulas that will be useful later.

The reader who needs a more extensive introduction to special relativity is advised to turn to one o f the books listed at the end o f this chapter, and then return. The reader who feels completely at home with the subject may find it desirable to move on immediately to Chapter 3.

The Principle o f Special Relativity states that the laws o f nature are invariant under a particular group o f space-time coordinate transformations, called Lorentz transformations.

I shall not continue this discussion in historical terms, but shall simply define the Lorentz transformations, and then show how Lorentz invariance guides our search for the laws o f nature. In our notation a, 3, y, and so on. We shall use natural units in which the speed o f light is unity, so all x a have the dimension o f length. It is this property that accounts for the observation by Michelson and Morley that the speed o f light isthe same in all inertial systems.

W e can also show that the Lorentz transformations 2. The general solution o f 2. This proof is an elementary example o f the sort o f thing we do in Chapter 13, on symmetric spaces. Incidentally, if we had only assumed that the transformations. But the statement that a free particle moves at constant velocity would not be an invariant statement unless the velocity were that o f light, and since there are massive particles in the world, we must reject the conformal group as a possible invariance o f nature.

The set o f all Lorentz transformations of the form 2. Equation 2. W e are dealing almost exclusively with proper Lorentz transformations, and unless otherwise noted, any Lorentz transformation is assumed to satisfy Eq.

The difference arises only in those transformations, called boosts, that change the velocity o f the coordinate. Suppose that one observer 0 sees a particle at rest, and a second observer O' sees it moving with velocity v. From 2.

One convenient choice that satisfies Eq. It can easily be seen that any proper homogeneous Lorentz transformation may be expressed as the product o f a boost A v times a rotation R. Although the Lorentz transformations were invented to account for the invariance o f the speed o f light, the change from Galilean relativity to special relativity had immediate kinematic consequences for material objects moving at speeds less than that o f light.

The simplest and most important is the time dilation o f moving clocks. He will calculate the proper time interval 2. Such particles o f course do not tick; instead 2. The time dilation 2. However, during this time the distance from the observer to the light source will have increased by an amount vr dt', where vr is the component o f v along the direction from observer to light source.

The transition from violet to red shift occurs for a source moving at an angle between straight toward the observer and at right angles to the line o f sight. Let us suppose that a particle moves in a field o f force at a velocity so high that Newtonian mechanics does not suffice to calculate its motion. Let us also suppose, as in the case o f electrodynamics, that we know how to calculate the force F on our particle in any Lorentz frame in which, at a given moment, it is at rest.

Fortunately, there is an easier way. Let us define the relativistic force f a acting on a particle with coordinates x a r by r. B Under a general Lorentz transformation 2. Hence, according to A , the force four-vector in the. Now that we know how to calculate we can use thedifferential equations 2. However, the initial values o f dxajdx must be chosen so that dx really is the proper time, that is, so that. Note that 2.

That this is true can be seen either directly from 2. The relativistic form 2. Recall that in our units 1 sec equals 3 x cm. The unique feature o f our p and E is that. I do not follow this custom here. W hy do we call p and E the relativistic momentum and energy? W e can use these names for anything we like. Sometimes the factor myis called the relativistic mass m. At zero velocity theenergy E has the finite value m.

A p'na will vanish. I f the total mass is conserved in a reaction as in elastic scattering. The velocity can be eliminated from Eqs. The conservation o f p and E in the original inertial frame tells us that A Pn vanishes.

I shall not show here that p and E are the only functions o f velocity whose conservation is Lorentz-invariant. For suppose that momentum is conserved in two different coordinate systems related by a Lorentz transformation. This notation will be extended in Chapter 4. There are several ways o f forming tensors out o f other tensors: Although any vector can be written in a contravariant or a covariant form.

Many physical quantities are not scalars or vectors. Note also that 2. A linear combination o f tensors with the same upper and lower indices is a tensor with these indices. The product o f the components o f two tensors yields a tensor whose upper and lower indices consist o f all the upper and lower indices of the two original tensors.

Setting an upper and lower index equal and summing it over its values 0. Thus the constant o f proportionality is unity. The definition o f Lorentz transformations tells us immediately that rjaj3 is a covariant tensor. Recall that rjali and rjal3 are numerically the same matrix.

We can form a mixed tensor by lowering one index on or raising one index on rja i. Aside from the scalars. We can define a tensor with an arbitrary pattern of upper and lower indices by setting all its components equal to zero. Since and r]ap are tensors. See Section 2. The left-hand side is then simply the determinant o f A. To find the constant of proportionality. Lowering all the indices gives back the same numerical quantity except for a minus sign: The fundamental theorem is that if two tensors.

The current and charge densities are usually defined by J x. The formalism outlined in this section is nothing but a description o f the representations o f the homogeneous Lorentz group. W e shall explore these representations in greater generality in Section 2. To see this. Whenever any current J a x satisfies the invariant conservation law 2. Hence we can apply the four-dimensional Gauss theorem.

Q is a scalar. Note incidentally that 2. There is a useful alternate form to the homogeneous equations 2. First consider a system o f particles labeled n. We now give a similar definition for the density and current o f the energy-momentum four-vector p a. W e can change A y by a term djp without affecting F yd. W e note from 2. Returning to 2. In this case 2. But each collision conserves momentum.

The energy-momentum tensor 2. Then 2. M is also conserved: We further note that and since T j0 is the density o f the jth component o f momentum. Consider first an isolated system. A systematic method for constructing these terms is presented in Chapter This will be the case if the mean free path between collisions is small compared with the scale o f lengths used by the observer. In order to isolate the internal part o f J ap.

Even when the velocity U is not zero. A perfect fluid is defined as having at each point a velocity v. Because o f the antisymmetry o f safiyd. But in the lab frame it is is a tensor. Now go into a reference frame at rest in the laboratory. First suppose that we are in a frame o f reference distinguished by a tilde in which the fluid is at rest at some particular position and time. We shall translate the above definition o f a perfect fluid into a statement about the energy-momentum tensor.

At this space-time point. Apart from energy and momentum. Let us consider one such conserved quantity. In order to gain some insight into the possible equations o f state.

Our scalar equation 2. As shown in Section 2. T ap will have the isotropic form 2. Xjv 2. As an example. For all such fluids. The proportionality expressed in Eq. In the unperturbed state. In this case. The sound wave produces small changes n l.

The speed o f sound is much less than the speed o f light i. Pem 2. To first order in small quantities. W e suppose that the presence o f weak space-time gradients in an imperfect fluid has the effect o f modifying the energy-momentum tensor and particle current vector by terms AT alt and AN a. Instead o f 2. In the approach o f Eckart. The general practice is to define p and n as the total energy density and particle number density in a comoving fra m e: In such fluids.

In the approach o f Landau and Lifshitz. The correct treatment o f dissipative effects for relativistic fluids raises certain delicate questions o f principle.

In practice. Hence 2. W where T and ah are the temperature and entropy per particle. Our task is now to construct the most general possible dissipative tensor AT alt allowed by Eq. With this definition o f Ua. The two approaches are perfectly equivalent. As in the last sections. To this end. It is convenient at this point to go over to a comoving frame. Setting Ul. Note that this is only possible because we have included the second term in Eq.

Note also that A T ali is not allowed to involve gradients o f p. Let us define a shear tensor. In general. It now only remains to translate our results from the forms 2.

To see when this applies. As we have seen. Note that. T aa cannot be expressed in the form 2. Under the general Lorentz transformation rule. In fact. W e shall see in Section We can compile a catalogue o f all possible Lorentz transformation rules by constructing the most general representation o f the homogeneous Lorentz group.

The infinitesimal Lorentz group consists o f Lorentz transformations infinitesimally close to the identity. The transformation property o f any object under ordinary spatial rotations is determined by its behavior with respect to infinitesimal Lorentz transformations 2.

The most familiar example is the Dirac electron field. B equal to 1. The problem o f finding the general representations o f the infinitesimal homogeneous Lorentz group is thus reduced to the problem of finding all matrices that satisfy the commutation relations 2. The rules for constructing such matrices can be found in any book on nonrelativistic quantum mechanics Finite Lorentz transformations can be built up by multiplying together an infinite number o f infinitesimal Lorentz transformations.

The appearance o f these minus signs means that a spinor field itself cannot be a physical observable. It follows then from 2. From these components. Applying 2. In the same way. Such space-time intervals are very small even for elementary particle masses. But in general the particle seen by the second observer will then necessarily be different from that seen by the first.

The uncertainty principle tells us that when we specify that a particle is at position xx at time tx. Does the second observer then see B absorbed at x2 before it is emitted at x x? The paradox disappears if we note that the speed v characterizing any Lorentz transformation A v must be less than unity.

In consequence there is a certain chance o f a particle getting from x x to x2 even if x x — x2 is spacelike. There is only one known way out o f this paradox. The second observer must see a particle emitted at x2 and absorbed at xx. To put it another way.

W e are thus faced again with our paradox. The Special Theory Interscience Publishers. Fluid Mechanics. Chapters Sykes and W.

B ibliography 63 pi-meson at x x and then sees the pi-meson and some other neutron turn into a proton at x 2.

Note that this conclusion does not obtain in nonrelativistic quantum mechanics or in relativistic classical mechanics. Theory of Relativity. Special Relativity 2nd ed. Principles of Relativity Physics Academic Press. Landau and E. Chapters I-V II. Since mass is a Lorentz invariant. There is such a particle.

Reid Pergamon Press. Part I. Chapters X V. This reasoning leads us to the conclusion that for every type o f charged particle there is an oppositely charged particle of equal mass. Oliver and Boyd. Chapter X V. Field Pergamon Press. Friedman and V. April Quantum Mechanics 3rd ed.

Israel and J. Misner and D. Chapman and T.. Chapter For x. Note B and Chapter Bludman and M.. Section Sykes and W.. Griffin Academic Press. Cambridge University Press. Nuovo Cimento Letters. For a rigorous discussion o f the necessity o f antiparticles in relativistic quantum mechanics.

For rj. Streater and A. W itten Plenum Press. Group Theory. W e shall first see what this principle says. Einstein reflected that. This can be easily proved for a system o f particles N. See Section 1. The attentive reader may have noticed a certain resemblance between the Principle o f Equivalence and the axiom which Gauss took as the basis o f nonEuclidean geometry.

Had g depended on x or i. There is a little vagueness here about what we mean by "the same form as in unaccelerated Cartesian coordinate systems.

Although inertial forces do not exactly cancel gravitational forces for freely falling systems in an inhomogeneous or time-dependent gravitational field. For example. W e are not yet ready to state the Principle o f Equivalence in its final form.

See the end of Section 4. The equivalence principle says that this cancellation o f gravitational by inertial force and hence their equivalence will obtain for all freely falling systems. It would be difficult to conceive o f a theory that satisfies this requirement and does not go all the way to the strong principle that no gravitational effects of any sort can be felt in a locally inertial frame.

Because o f this deep analogy. Wapstra and N ijgh1 have shown that the limits set by Eotvos on any possible inequality in the ratio o f gravitational to inertial mass for glass. The mass o f different substances arises in different proportions from the masses o f the neutrons and protons plus electrons o f which they are composed.

We might. The experiments o f Eotvos. In particular. The weak principle is the same. We saw in Chapter 1 that Gauss assumed that at any point on a curved surface we may erect a locally Cartesian coordinate system in which distances obey the law o f Pythagoras.

To this accuracy. Certainly the experiments o f Eotvos and Dicke are not accurate enough to. This is a pity. This question might be settled by studying the motion o f a small body in orbit about a large body that is itself in free fall in a gravitational field. According to the Principle o f Equivalence. The earth is in free fall. The freely falling coordinates are functions o f the x and Eq.

The purpose o f 3. W e now show that g is also the gravitational potential. Using 3. The bax are determined by Eq. W e first recall the formula for the metric teneor. In order to use this information. Thus Eqs. The second kind o f term arises because Cxx x carries a label X.

Hence we shall interpret the Principle o f Equivalence as meaning that the locally inertial coordinates that we construct at a given point X can be chosen so that the first derivatives o f the metric tensor vanish at X. Add to Eq. Now we return to our previous compact notation.

Let us introduce an arbitrary parameter p to describe the path. An additional consequence o f the relation 3. I f the particle is sufficiently slow. Such paths are called geodesics.

Xv dr2 j is symmetric in its lower indices. The equivalence principle tells us that its rate is unaffected by the gravitational field if we observe the clock from a locally inertial coordinate system c x. I f points 1 and 2 are at rest in a stationary gravitational field.

Let us apply Eq. The red shifts are much larger for white dwarf stars like Sirius B and 40 Eridani B. The convection tends to be vertical. Such stars have masses typically o f the order o f one solar mass.

For instance the mass o f Sirius B is determined by calculating the total mass o f Sirius A and B from their separation and period. Although this alleviates problems arising from convective Doppler shifts or temperature or pressure. I f we know the mass o f a white dwarf star. Until recently the best result that could be achieved in this way was that the solar gravitational red shift is o f the order o f 2 parts per million.

Thermal effects are more serious. It is not the rotation o f the earth or the sun that bothers u s.

The really bothersome problems arise from unknown Doppler shifts owing to the convection o f gases in the solar atmosphere.

This would seem to make the experiment impossible. Normally resonant absorption is impossible for such a narrow y-ray line. Note that T appears here rather than Tj2. To the gravitational violet shift AvG there is then. This experiment was made possible by the Mossbauer effect. Their idea was to move the y-ray source up and down with velocity vQ cos cot.

Taking account o f Stark shifts in the spectrum of 40 Eridani B appears to improve the agreement. A t a point directly below perigee there is no firstorder Doppler shift because the time for the light to reach us from the satellite is momentarily constant. This discrepancy was actually an intrinsic frequency shift owing to the difference between the source and target crystals including differences in their temperature and was removed by subtracting the asymmetry in y-ray counts when the source is below the target from the asymmetry when the target is below the source.

In this case the frequency shift o f the emitted light must be determined from 3. The agreement between theory and experiment has since8 been improved to about 1 percent. There have also been proposals8a to measure the gravitational red shift o f light from an artificial satellite.

It follows that the frequency vs o f a given atomic line from the satellite will be related to the frequency ve o f the same line on earth by 3. Suppose that the photon is then absorbed at point 2 by a second heavy apparatus. When a photon is produced at point 1 by some heavy nonrelativistic apparatus. Hence even if we suppose that the EotvosDicke experiments could improve to an unlimited accuracy. I have insisted on including a nonre lativistic emitter and absorber in the above calculation.

This derivation rests on the Principle o f Equivalence in tivo respects: It assumes that the change in gravitational mass o f the apparatus equals the change in its inertial mass and hence its internal energy. W e conclude then that the metric tensor g must like r]yji have three positive eigenvalues.

Newton believed that inertial forces. Einstein considered himself a follower o f Mach. The fact that g is related to rjap by the congruence 3. The inertial frames. The distinction is not one o f metaphysics but o f physics. It follows that there exists a matrix D. The celestial bodies play a role here because the gravitational field equations for gfiv need boundary conditions at infinity. Cocconi and Salpeter pointed o u t10 that there is a large mass near us.

In this case the three transitions among neighboring states should have the same energy and the photon absorption coefficient should show a single sharp peak at this energy.

In this sense. These points are so important that they are worth repeating. Hughes et al. W e are not yet ready to go into the details o f the field equations and cosmology. See Section In the absence o f nearby matter. This was checked experimentally by Hughes. I f we think o f the L i7 nucleus as a single proton with angular momentum. The issue between Mach and Einstein can be drawn by asking whether in fact the presence o f large nearby masses does affect the laws o f motion.

When a large mass like the sun is brought close. The Theory of Relativity Clarendon Press. Chapter VII. Wapstra and G. The General Theory Interscience Publishers. Kemmer 2nd rev. Englewood Cliffs.

Brose Dover Publications. Wiese and D. Blamont and F. L59 W itten Wiley. On the experimental tests o f the Principle o f Equivalence.

Introduction to the Theory of Relativity Prentice-Hall. Pound and J. Space Sei. Pound and G. The Theory of Space. Greenstein and V. New York..

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