JOHN EARMAN* and CLARK GL YMUURT THE GRAVITATIONAL RED SHIFT AS A TEST OF GENERAL RELATIVITY: HISTORY AND ANALYSIS CHARLES St. John, who was in 1921 the most widely respected student of the Fraunhofer lines in the solar spectra, began his contribution to a symposium in Nncure on Einstein’s theories of relativity with the following statement: The agreement of the observed advance of Mercury’s perihelion and of the eclipse results of the British expeditions of 1919 with the deductions from the Einstein law of gravitation gives an increased importance to the observations on the displacements of the absorption lines in the solar spectrum relative to terrestrial sources, as the evidence on this deduction from the Einstein theory is at present contradictory. Particular interest, moreover, attaches to such observations, inasmuch as the mathematical physicists are not in agreement as to the validity of this deduction, and solar observations must eventually furnish the criterion.’ St. John’s statement touches on some of the reasons why the history of the red shift provides such a fascinating case study for those interested in the scientific reception of Einstein’s general theory of relativity. In contrast to the other two ‘classical tests’, the weight of the early observations was not in favor of Einstein’s red shift formula, and the reaction of the scientific community to the threat of disconfirmation reveals much more about the contemporary scientific views of Einstein’s theory. The last sentence of St. John’s statement points to another factor that both complicates and heightens the interest of the situation: in contrast to Einstein’s deductions of the advance of Mercury’s perihelion and of the bending of light, considerable doubt existed as to whether or not the general theory did entail a red shift for the solar spectrum. Even the ablest expositors of the theory seemed unable to give a clear and cogent derivation of the ‘Einstein effect’; indeed, our search of the literature has not turned up a single unproblematic presentation of the correct formula for the red shift prior to the mid 1920s. Many competent physicists naturally found grounds for objecting to the purported derivations, and less competent ones found in them an invitation to raise muddled arguments against the theory. The discussions that followed did not always illuminate the theory, but ‘University of Minnesota, Minnesota Center for Philosophy of Science, 315 Ford Hall, Minneapolis, MN 55455, U.S.A. ? University of Illinois at Chicago Circle, U.S.A. ‘C. E. St. John, ‘The Displacement of thesolar Lines,’ Nurure. 106( t921), 789 - 790; quotarion from p. 789. Stud. Hisf. Phil. Sci., Vol. II, (1980) No. 3, pp. 175-214. Pergamon Press Ltd., Printed in Great Brirain. 175 176 Studies in History and Philosophy of Science they certainly made vivid what the theorists of the day believed about it. In this paper we propose to examine both the attempted derivations of the red shift and reactions of the theorists to the observational tests. The examination engages not only issues in the history and sociology of science but also points of physics and mathematics and questions of scientific methodology. While this approach threatens to make the presentation unwieldy, there is simply no alternative if one wants to understand the real import of the debates about the red shift. The main historical focus of the paper will be on the period from 1907, the date of Einstein's first prediction of the red shift, through the early 1920s, when the opinions of both the theorists and the experimentalists tended to stabilize; but some consideration of both the prior background and later developments is needed to round out the picture. Section 1 examines the attempts by Einstein and others to base the spectral shift directly on the principle of equivalence or on some combination of the quantum principle and the conservation of energy. Section 2 details some of the many fallacious formal derivations of the spectral shift within the completed general theory and traces the confusions and uncertainties they caused. Section 3 present Synge's treatment of the energy of a photon in the gravitational field and his photon frequency definition of the spectral shift. Section 4 reviews the observational evidence up to the 1920s. Section 5 discusses the reaction of the theorists, especially in England, to this evidence. Section 6 assesses the status of the spectral shift prediction as a test of the general theory of relativity and, in particular, the claim that while a negative observational result would count as strong evidence against the theory, a positive result is worth little as evidence f o r the theory. Section 7 discusses attempts to interpret the spectral shift as a Doppler effect or, a!ternatively as a non-Doppler gravitational phenomenon. Section 8 provides a summary and conclusion. Various mathematical details are to be found in the Appendix. 1. Heuristic Derivations of the Red Shift For the early history of Einstein's prediction of the gravitational red shift, as so often in physics, heuristic derivations had a far greater importance than did precise mathematical demonstrations. In the case of the red shift, however, the suggestiveness and fruitfulness of the heuristic derivations were counterbalanced by the confusions they engendered, some of the effects of which linger on even to the present day. Einstein's initial derivations, given several years before the final formulation of the general theory of relativity, rested on the principle of equivalence. This principle was useful both in guiding him towards his goal -- a satisfactory relativistic theory of gravitation -- and in allowing him to make bold predictions before he was even within sight of that The Gravitational Red Shifr 177 goal, But it is exactly the heuristic fruitfulness of the principle of equivalence that tends to obscure what the final theory says about the red shift. Many attempts have been made to remove the looseness and vagueness of the formulations of this principle that preceded the general theory of relativity and to find some precise counterpart within the completed theory;2 but there is still much disagreement on this matter, and in almost all of the attempts, the precision is purchased at the price of the original heuristic power - here, as in many other cases, a certain amount of imprecision and vagueness seems indispensible. Our task, however, is not that of assessing the validity of the principle of equivalence in the light of subsequent developments, but rather that of describing how Einstein used it and how rhis use affected attitudes towards the red shift. The fifth and final part of Einstein’s 1907 essay on the principle of relativity and its consequences gave itself over to speculations about a question which ‘forces itself. , , on the mind of anyone who has followed the previous applications of the principle of relativity’; namely, ‘Is it conceivable that the principle of relativity also holds for systems which are accelerated with respect to each other?‘3 Einstein argued for a positive answer, at least for the case of uniform acceleration: as far as we know, the laws of physics are the same for two systems Z, and X2, where Z;, is accelerated in the direction of its x-axis wirh acceleration y, while Z2 is ‘at rest’ but situated in a homogeneous gravitational field characterized by a gravitational acceleration of y along its negative x-axis. We have therefore no reason to suppose in the present state of our experience thaf the systems 1, and I2 differ in any way, and will therefore assume in what follows the complete physical equivalence of the gravitational field and the corresponding acceleration of the reference system. This assumption extends the principle of relativity to the case of uniformly accelerated translational motion _ _ . The heuristic value of the assumption lies therein that it makes possible the replacement of a homogeneous gravitational field by a uniformly accelerated reference system, the latter case being amenable to theoretical treatment to a certain degree.’ Einstein’s 1907 theoretical treatment of uniform acceleration was fruitful in yielding, in combination with the equivalence principle, the consequence that ‘light coming from the surface of the sun. , . possesses a wavelength that is greater by about a two-millionth part than that of light generated by identical material on the surface of the earth.‘5 The treatment itself, however, was uncharacteristically cumbersome, principally because of the circuitous ‘SeeH. Ohanian. ‘What is the Principle of Equivalence.?‘, Am. J. Phys., 45 (1977). 903 - 909. ‘A. Einstein. ‘Uber das Relativititsprinzip und die aus demselben gezogenen Folgerungen,’ J&I. Rudioukf. Elekrronik, 4 (1907). 41 I - 462;quotation from p. 454; trans. from H. M. Schwartz, ‘Einstein’s Comprehensive 1907Essay on Relativity, Part III,’ Am. J. Phys., 45 (1977), 899-902. ‘Einstein. Ref. 3. p. 454; trans. from H. M. Schwartz. ‘Einstein. Ref. 3, p. 459; trans. from H. M. Schwartz. 178 Studies in History and Philosophy oJScience explanations of the concepts involved. Despite the explanations, some of the meanings remained obscure. Planck complained that Einstein’s concept of uniform acceleration needed clarification since the rate of change of the (three) velocity is not a relativistic invariant. In an addendum published the following year, Einstein replied than the acceleration of a body is to be measured in at ‘acceleration-free’ (i.e. inertial) frame relative to which the body is instantaneously at rest.* While this reply captured one of the key features of the correct relativistic treatment of uniform acceleration, a final clarification was achieved only through the work of Minkowski’ and Born.B Nor did Einstein’s reply do anything to clarify the meanings of the concepts of ‘time’ and ‘local time’ used to describe the accelerating frame and to derive the conclusion that the rate of atomic clocks is affected by gravity in such a way as to give a red shift. It is not surprising then that when four years later Einstein again took up the question of the influence of gravity on the propagation of light, his essay opened with the remark that ‘I return to this theme because my previous presentation does not satisfy me. . .I9 This time Einstein’s strategy was to sidestep the problems of a detailed analysis of accelerated motion and to exploit instead a coupling of the Doppler principle with the equivalence principle. Let the receivers, and the source S, be at rest in the inertial frame K in which there is a homogeneous gravitational field of acceleration y in the negative z-direction (see Fig. 1). In order to deduce the relation between the t t I--S, tr d Y S, / J x Fig. I. frequency u2 of a light signal when emitted at S, and the frequency u, as measured at S,, we can imagine a physically equivalent system K’ which is ‘A. Einstein, ‘Berichtigungen zu der Arbeit “&er das Relativitatsprinzip und die aus demselben gezogenen Folgerungen” ’ , Jb. Radiwkt. Elektronik, 5 (1908), 98 - 99. ‘H. Minkowski. ‘Raumund &it’, Whys. Z., 10 (I-), 104-111. “M. Born. ‘Die Thebrie des starren Elektrons in der Kinematik des Reiativit&tsprinzips,’ Ann/n. Phys., 30 (l&9). 1 -56. *A. Einstein, ‘tiber den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes,’ Ann/n. Phys., 35 (191 I), 898 - 908; quotation from p. 905; trans. from W. Pcrrett and G. B. Jeffrey, The Principle of Relutiviry (New York: Dover Publications, 1922). The Gravitational Red Shift 179 gravitation-free and which moves with uniform acceleration y in the direction of the positive z-axis to which S, and S, are rigidly attached. At the moment of emission of the signal from &. let K’ be instantaneously al rest with respect to an inertial frame KO. Then at the time of arriva1 of the light signal, S, has a velocity (yd)/c relative to K,, so that by the Doppler principle 111 = u2[f + (yd)/cq. (1.1) Using the equivalence principle again to transfer the result back to K, we can rewrite (1. I) as II, = u*(1 + 0/c’) (1.2) where 9 is the gravitational potential. Einstein was quick to note that this Doppler argument seemed to involve an ‘absurdity’. For ‘if there is a constant transmission of light from S, to S,, how can any other number of periods per second arrive at S, than are emitted at S,?’ His resolution employed an idea already broached in his 1907 paper: . . . the two clocks in S, and S, do not both give ‘time’ correctly. If we measure time in S, with a clock U, then we must measure time in S, with a clock that goes 1 + O/c’ times more slowly than a clock U when compared at one and the same place.lD This way of avoiding the absurdity seemed to imply that one of the basic tenets of the special theory of relativity - the constancy of the velocity of light had to be abandoned when gravity is taken into account. Einstein embraced this consequence and made it the basis of a further prediction, the bending of light passing near a massive body. Though the predicted value for the deflection of light by the Sun was only a half of the final general relativity value, the existence of the deflection effect was crucially important in providing Einstein with a seIection principle for judging the acceptability of theories of gravitation. The notion of a variable speed of light was also the basis on which Einstein attempted to construct a theory of the static gravitational field in 1912; but before turning to this development, another aspect of his 1911 paper needs examination. The second section of Einstein’s 1911 paper focuses on energy considerations; again, the focus is provided by the lens of the equivalence principle. Using the results of special relativity, Einstein concluded that for the construction in Fig. 1, the relation between the energies is, to first order, E, = Ez (1 + v/c) = -Ez[l + (yd)/cZ]. (1.3) ‘OEinstein, Ref. 9, p. 905; trans. from W. Perrett and G. B. Jeffrey, 180 Studies in History and Philosophy ofScience The equivalence principle was then used to rewrite (1.3) as E, = E,(l + m/F). (1.4) If the quantum relation E = hv had been used at this point, the red shift formula (1.2) would have been derived. Or alternatively, the combination of (1.2) and (1.4) couWhave been used to argue that if a frequency is to be associated with a quantum of energy, then the energy and frequency must be proportional. But Einstein did neither of these things, and it might seem a little puzzling that he did not, It may be, of course, that since the derivation of the red shift formula was beside the main point of the section, which was to show that from the equivalence principle one could obtain the equivalence of gravitational and inertial mass, Einstein did not want to introduce an irrelevancy. More likely, Einstein did not want to contaminate his work on gravitation with the almost universal skepticism with which his light quantum hypothesis had been greeted; in other publications of this period he tended to keep relativity theory separate from quantum considerations.” In any case, it is clear in retrospect that there are problems in applying Einstein’s equivalence principle to the light quantum. Einstein wanted to attribute the excess in energy arriving at S, to the potential energy (EJc1)@ of the radiation at S2, But since the photon, as we now caI1 Einstein’s light quantum, has no rest mass, it is questionable whether the usual expression m@ for the potential energy can legitimately be applied. This embarrassment can be avoided by reversion to the original attitude that the quantum principle is to be limited to the emission and absorption processes, and, thus, by focusing on the final states of the emitter & and the absorber S, after the photon has been absorbed at S,.12 The change in energy of S, is Am,? + Am,@, where Am2 = - (Izv~)/c?, while the change in energy at S, is Am,? + Am,+, where Am, = (hu,)/C2. Setting the sum of these changes to zero, by conservation of energy, we have u, = uJ(l + @,/P)/(l t @l/C’)]. (1.5) For 0, /c’ 4 1, this relation is approximately u, = UJl + (@a - @,)/c’]. (1.6) This approach succeeds by treating the photon instrumentally and, thus, by sidestepping the problem of how to represent the energy of the photon itself in “See A. Pais. ‘Einstein on Particles, Fields and the Quantum Theory,’ forthcoming in the Proceedings of the Einsrein Centennial Symposium, lnstitute for Advanced Study, March 1979. “The following argument is found in S. Weinberg, Grnvirurion and Cosmology (New York: John Wiley, 1972), pp. 84-85. The Gravitational Red Shift 181 the gravitational field. That problem was not solved until almost a quarter of a century later by J. L. Synge: Synge’s contribution will be discussed in Section 3 below, where we will see that only under restrictive conditions can the red shift be interpreted in terms of the change of the energy of the photon as it climbs through the gravitational field. In 1912 Einstein tried once again to analyze uniformly accelerating motion and to characterize the state of a static gravitational field by means of the concept of a variable speed of light.13 The attempt ended in a perplexing failure, for he was forced to conclude that his theory could consistently accommodate the equivalence principle only in infinitesimal regions. The difficulty derived from Einstein’s proposed field equations, the details of which are not relevant here, What is important is that the failure of his 1912 theory seems to have convinced Einstein that he could no longer regard mathematics in its subtler forms ‘as a pure 1uxury,“4 and later that year he began in earnest to study the absolute differential calculus of Ricci and LeviCivita. While Einstein had finally chosen the correct mathematical tooi for building a relativistic theory of gravitation, he was over the next three years quite lost in the tensors; for he believed that a natural causality requirement precluded generally co-variant field equations. It was only at the end of 1915 that he succeeded in formulating the correct field equations.‘s After the completion of the general theory, Einsteih continued to see, and encouraged others to see, the theory as embodying and as being a direct outcome of the principle of equivalence. The effects of such an attitude on the interpretation of the red shift will become evident in the following sections. All of the heuristic derivations of the red shift can be faulted on various technical grounds. But to raise such objections is to miss the purpose of heuristic arguments, which is not to provide logically seamless proofs but rather to give a feel for the underlying physical mechanisms. It is precisely here that most of the heuristic red shift derivations fail - they are not good heuristics. For they are set in Newtonian or special relativistic space-time; but the red shift strongly suggests that gravitation cannot be adequately treated in a flat space-time.” Einstein’s resort to the notions of a variable speed of light “A. Einstein. ‘Lichtgeschwindigkeir und Statik dcs Gravitations&Ides,’ Ann/n. P&s., 38 (1912), 355-369, and ‘Zur Theorie des stat&hen Gravitationsfeldes,’ ibid.. 443-458. Einstein’s theory is analyzed in our paper ‘The Failure of the Principle of Equivalence in Einstein’s 1912 Variable Speed of Light Theory,’ forthcoming in the Proceedings of rhe Memphis Stare University Einstein Centenary Conference, March 1979. “From a letter to Sommerfeld, dated October 29, 1912. See A. Hermann (cd.), AIberr Einstein/Arnold Sommerfeld Briefwechsel (BaseI: Schabe, 1%8), p. 26. “For details, see our paper ‘Einstein and Hilbert: Two Months in the History of General Relativity,’ Arch. EM. exact Sci.. 19 (1978). 291-307, and ‘Lost in the Tensors: Einstein’s Struggles with Covariance Principles,’ Stud. I-I&. Phil. Sci.. 9 (1978), 251 - 278. “See A. Schild, ‘Gravitational Theories of the Whitehead Type,’ Evidencefor Gruvitutional Theories, C. M#ler (cd.) (New York: Academic Press, 1%2). 182 Studies in History and Philosophy of Science and variable clock rates in a gravitational field can be seen as an acknowledgement, albeit unconscious, of this point; but as we will now see, these notions served to obscure the role of the curvature of space-time as the light ray moves from source to receiver. 2. Attempts at 8 Formal Derivation of the Red Shift In 1916 Einstein presented the first derivation of the red shift from his newly completed general theory. ” Though couched in terms of the formalism of the new theory, the derivation actually relied on the same ideas as the 1907 and I91 i heuristic derivations, especially the idea that the rate of clocks is affected by the gravitational field. The gist of Einstein’s construction is as follows. In a static gravitational field, the co-ordinate system can be chosen so that the line element has the form ds2 = -g,,dx,dx, + g,,(#)2 a. /I = 1,2, 3, (2.1) where the metric potentials g, may depend upon the spatial co-ordinates Y but are independent of the time co-ordinate x’. (For the sake of simplicity we have set c = 1.) For a ‘unit clock’ which is ‘at rest’ in the field, we have ds = 1 and dx’ = dx2 = dx” = 0. It follows, therefore, from (2.1) that For the case of a sphericaliy symmetric gravitation field generated by a mass M, Einstein assumed that g., = (I- CM/r), where C is a constant and r is the radial co-ordinate.‘B Einstein’s conclusion is that in such a field, (2.2) entails that The clock goes more slowly if set up in the neighborhood of ponderable masses. From this it follows that the spectral lines of light reaching us from the surface of large stars must appear displaced towards the red end of the spectrum’* To the modern eye, Einstein’s derivation is no derivation at all, for the formula (2.2) expresses only a co-ordinate effect, and in contrast to the case of the bending of light, Einstein provided no deduction from the theory to explain what happens to a light ray or a photon as it passes through the gravitational field on its way from the Sun to the Earth. “A. Einstein, ‘Die Grundlage der allgemcinen Relativititstheoric,’ An& Phys., 49 (1916), 769 - 822. “The ‘Schwarzschild solution’ of Einstein’s field equations did not appear until later that year; see K. Schwarzschitd, ‘uber das Gravitationsfcld eines Massenpunktes nach der Einsteinschen Theorie.’ Sk. pre-. Akud. Wis., 7 (l916), 189-I%. ‘#Einstein, Ref. 17, p. 820; trans. from W. Perrett and G. B. Jeffrey. The Gravitational Red Shift 183 Unfortunately, Einstein’s ‘derivation’ was dressed up by the expositors of the general theory, and it quickly became codified in the literature as the official derivation. In the English-language literature the sequence was initiated by Willem de Sitter, who provided the first detailed review of Einstein’s general theory to appear in English.20 De Sitter transmitted his article to Arthur Stanley Eddington, then Secretary of the Royal Astronomical Society, along with a reprint of Einstein’s 1916 ‘Grundlage’ paper, apparently the only copy of this essay to reach England during the war years.” Eddington’s subsequent rapid immersion in relativistic gravitation is evidenced by his numerous publications and lectures and his participation in the English eclipse expedition of 1919.2z What is more important for our purposes, however, is Eddington’s assumption of the role of chief expositor and defender of Einstein’s theory in England; the quasi-official recognition of this role can readily be confirmed by turning the pages of Nature and Observarory, where letters raising objections to the theory were often followed by a response from Eddington. Thus, it is hardly surprising that Eddington’s version of the red shift derivation, a version which closely parallels de Sitter’s, was reproduced in most the English-language text books of the period.‘$ Eddington’s treatment, as it appears in his Report, starts with the premise that an atom ‘is a natural clock which ought to give an invariant measure of an interval ds; that is to say, the interval ds corresponding to one vibration of the atom is always the same’.” So if (#), and (M), are the periods of two similar atoms ‘at rest’ in the field at points I and 2 respectively, the application of (2.1) and the relation (ds), = (ds), (the Premise) gives If 1 refers to the Sun and 2 to the Earth, then since m, < m2 we have (dx*), > (dx4)? so that ‘the solar atom thus vibrates more slowly, and its spectral lines will be displaced towards the red’.25 ‘“W.de Sitrer, ‘On Einstein’sTheory of Gravitation and Its Astronomical Consequences,’ Mon. Nor. R. Asrr. Sot.. 76 (1916), 699-728. and 77 (1917). 155 - 184. “See ‘Arthur Stanley Eddington,’ Obi!. Nor. F&L R:-SW. Land., 5 (1945), 113 ~ 125. Prof. S. Chandrasekhar has informed us rhal in conversation Eddington confirmed these details. We are most grateful to Prof. Chandrasekhar for his illuminating comments on this period. *lln a letter to Einstein. dated December I. 1919, Eddington wrote: ‘1 have been kept busy lecturing and writing on your theory. My Report on Relativity is sold out and is being reprinted . . . 1had a huge audience at the Cambridge Philosophical Society a few days ago, and hundreds were turned away unable to get near the room.’ (Einsrein Papers, Princeton University. ISI microfilm Reel No. 9.) ‘Qe, for example, E. Cunningham, Refurivity, the Electron Theory, and Gravitation (London: Longmans, Green, 1921). pp. I21 - 122; and L. Silberstein, The Theory of Relativity and Gravitation (New York: Van Nostrand, 1922). pp. lO2- 105. “A. S. Eddington. Report on the Relativity Theory of Gravitation (London: Fleetwood Press), 1st edn. 1918, 2nd edn. 1920; quoration from p. 56 of the 2nd edn. “Eddington, Ref. 24, 2nd edn., p. 57. 184 Studies in History and Philosophy of Science This treatment is exactly backwards from what is wanted; namely, the coordinate time intervals equal and the proper time intervals different. Henceforth we will refer to it as the ‘backwards derivation’. The confusion lies in Eddington’s misapplication of the Premise. Grant, for the moment, the Premise. What needs to be compared in the first instance are not the proper time intervals (A), and (ds), corresponding respectively to the vibrations of similar atoms to points 1 and 2; rather, the most immediate task is to compare the proper time interval (ds), for a vibration at I with the proper time interval [dsll between the reception of two light signals sent from 1 as markers of the beginning and end of the vibration (see Fig. 2). Fig. 2. In a stationary co-ordinate system the co-ordinate time interval for a vibration is transmitted without change from 1 to 2 so that (~3% for the emission at 1 is the same as [dx$ for the reception at 2. In effect, this was demonstrated in 1920 by Max von Laue for the more restrictive case of a static gravitational field;26 he proved that in such a case, Maxwell’s equations admit solutions in which the time co-ordinate enters in the form exp(ivx’), where the co-ordinate frequency v is a constant, independent of x’ and fl.l’ For a source and receiver ‘at rest’, (2.1) then gives = V-zJhi-iJn. (2.4) or in terms of the proper frequency, which is inversely proportional to the proper time interval, ‘5~ the Appendix for definitions of stationary and static. “M. von taue, ‘Theorctisches Uber neuere optische Beobachtungen zur ReIativit&stheorie,’ Phys. 2.. 21 (19201, 659-662, and 22 (19211, 332. The Gravitational Red Shryt 185 Now is the place for the Premise to be applied, at the end and not at the beginning of the argument: since the proper frequency Y, of the atom at 1 is the same as the proper frequency of a similar atom at 2, the observer at 2 will see a shift towards the red in the spectral lines coming from atoms at I, as measured relative to the frequency of similar atoms at 2. In order to avoid ambiguity, double subscripts on the frequency could be used, the first subscript denoting the source and the second the point of measurement. Thus, (2.5) would read /u VI, 2 I,1 = d-iGwzl2. (2.5’) In this notation, the Premise reads v~,~ = u~,~, and the red shift effect is then If the reader thinks that this notation is overly fussy and pedantic, he need only read on. The first and, as far as we have been able to determine, the only explicit query of the backwards derivation to appear in the literature of the period was posed by James Rice in a letter to Naturc2’ Rice was puzzled, as well he should have been, as to why Einstein’s theory predicted any shift in the spectral lines if, as seems implicit in the backwards derivation, the proper time interval ds and not the co-ordinate interval G!? is transmitted unchanged. Eddington’s reply pinpointed the essential fact; The rule deduced from Einstein’s theory for comparing the passage of two light pulses at the points A and A ’ respectively is not ds = ds ‘, but dt = dt ‘, provided that the coordinates used are such that the velocity of right does not change with taaD But Eddington managed to leave the reader with a confused impression: At a point in the laboratory (r = constant), df, for a light vibration from a solar atom differs from dt, for a terrestrial atom. It follows from the formula (A) [the formula for the line element] that ds, and ds, will differ in the same ratio since we are now concerned only with the relation of dt and ds on earth. The intermediary quantity I is thus eliminated; and the difference in the light received from solar and terrestrial sources is an absolute one, which it is hoped the spectroscope will detez30 The elimination of the ‘intermediate quantity’ t (our 9) is indeed the crucial aspect of the argument leading to (2.5); but the elimination is not achieved, as Eddington seems to indicate, by considering dr, for a solar atom and dt, for a terrestrial atom. Further, we are told that ds, and ds, differ in the same ratio as “J. Rice, ‘The Predicted Shift of the Fraunhofer Lines,’ Nature, Lord. 104 (1920), 598. “A. S. Eddington, Nuture, Land, 104 (19X)), 598; the italics are Eddington’s. 10Eddington, Ref. 29, p. 599. 186 Studies in History and Philosophy of Science df, and df,; but we are not told what the ratio is - the upside ratio of the backwards derivation or the correct ratio of (2.4) - nor for that matter is it made clear just what ds, and ds2 now denote. And nowhere does Eddington explicitly state the correct formula for the red shift. A similar unclear impression is left by Eddington’s remarks at a meeting of the Royal Society of London, held on the day his response to Rice was published in Narure.” Two months later in a response to Guillaume,f2 Eddington swept away some of the misimpressions and reemphasized the essential point: It is perhaps unfortunate that in the best known discussions of this problem the question of what happens along the light-wave from Sun to Earth, through the nonEuclidean space - time, is scantily treated. From an absolute point of view, it is here that the cause of the spectral shift occurs, since ds changes continuously along the path (dt remaining constant). This completion of the argument is one of those things which are obvious if you happen to approach them in the right way; but it causes a great deal of difficutty if you have no! grasped the full significance of the fact that the co-ordinate system has been so chosen that the velocity of light at any point does not involve the co-ordinate f.” One wonders whether Eddington was indulging in some mild self-criticism since the ‘best known discussions of this problem’ were Eddington’s own and since ‘a great deal of difficulty’ has resulted from his not approaching the problem in the right way. If so, he did not take the criticism to heart, for the ‘completion of the argument’ was something which Eddington never fuhy carried out. All of the editions of his widely read and admired Space, Time, and GruvitaliotP contain a treatment of the red shift much closer to the backwards derivation than to the correct one. This is perhaps attributable to the semi-popular nature of that work and to the understandable desire not to burden the general reader with too much detail. But then one would expect that the highly technical Mathematical Theory of Relativity would provide Eddington the proper setting for putting the matter straight once and for all. One is disappointed, however. There, Eddington emphasized that ds ‘becomes gradually modified as the waves take their course through non-Euclidean space - time’ and that this modification is the source of the red shift.35 But still the promise to complete the argument is not fulfilled, and Eddington continued to refer to dl as the ‘time’ of vibration of an atom, and “A. S. Eddingron, in ‘Discussion on Ihe Theory of Relativity,’ Proc. R. Sot.. 97 (19X3),72- 74. ‘*E. Guillaume. ‘Displacement of the Solar Lines and the Einstein Effect,’ Ubservutory, 43 (1920). 227-228. JJA. S. Eddington, Observatory, 43 (1920). 229. In a nonstationary gravitational field the time coordinate I cannot be chosen so that dt is preserved; consequently, the derivation of the red-shift formula is more complicated in such a case (see section 6 below). “A. S. Eddington, Space. Time und Gruvitulion (Cambridge: Cambridge University Press), 1st edn. 1920, reprinted 1921, 1923, 1929, etc. “A. S. Eddington, The Mathematid Theory of Relativity (Cambridge: Cambridge University Press), 1st edn. 1923, 2nd edn. 1924; quotation from p. 92 of the 2nd edn. The Gravitational Red Shift 187 from the fact that the line element assumed the form d9 = ydt* for an atom ‘at rest’, he inferred that ‘the times of vibration of similar atoms will be inversely proportional to $y.‘3E As an ironic and sad footnote, it should be added that Rice, whose letter to Nature called attention to the emperor’s new clothes, published his own textbook three years later; in it he repeated the backwards derivation.3’ The miasma which hung over the theoretical treatment of the red shift even affected Hermann Weyl’s otherwise elegant Raum - Zeit - Materie. In all five editions,38 the red shift was approached within the confines of a static gravitational field, using the relation ds = fdt (2.7) between the proper time ds and the ‘cosmic’ (co-ordinate) time t at a fixed point of space.3Q It is assumed that If two sodium atoms at rest are objectively fully alike, then the events that give rise to light-waves of the D-line in each must have the same frequency, as measured in proper time.” The conclusion drawn in the first three editions is that if f has the values f,and f2 respectively at the locations 1 and 2 and if T, and 7? are respectively the frequencies, as measured in ‘cosmic time’, of the atoms at 1 and 2, then there will exist the relationship fi7, = f*T*; 71/r* = fdf,. Weyl’s commentary on this formula is as follows: . ..the light waves emitted by an atom will have, of course, the same frequency, measured in cosmic time, at ali points of space, Consequently, if we compare the sodium D-line produced in a spectroscope by the light sent from a star of great mass with the same line sent by an earth-source into the same spectroscope, there should be a slight displacement of the former line towards the red as compared with the latter.” In the fourth edition, two crucial changes are made. First, the symbol Y is substituted for the previously used T; and second, the formula (2.8) is inverted to read’? “Eddingtan. Ref. 35. p. 92. “J. Rice. Relarivily: A Systemark Trealment of Einstein’s Theory (London: Longman. Green, 1923). pp, 287 - 289. 3*H.Weyl, Rum-Zeit-Murerie (Berlin: Julius Springer), 1st edn. 1918,2nd edn. 1918, 3rd edn. 1919, 4th edn. 1921, 5th edn. 1923. ‘*In our notation, I = X’ andf = 6. “Weyl. Ref. 38, 3rd edn., p. 211; the italics are Weyl’s. “Weyl, Ref. 38, 3rd edn., p. 212; the italics are Weyl’s. “Weyl, Ref. 38, 4th edn., p. 223. 188 Studies in History und Philosophy uJScience h/f, = vJf2. (2.9) What accounts for the apparent sleight-of-hand involved in these changes? The textual evidence is not sufficient to provide a firm answer, but it invites the following speculation. Instinctively, Weyl must have seen that (2.8) is the right form for the red shift, but at the same time he must have sensed that (2.8) was not justified by the considerations he adduced - perhaps this accounts for the use of the symbol T for frequency in spite of the fact that T is more standardiy used to denote period rather than frequency. And in addition, he would have wanted, if only unconsciously, his formula for the red shift to agree with the then current backwards formula. In any case, Weyl’s corrected argument, as given in the 4th and 5th edns, seems to be this. Since the proper time periods of the two atoms are the same, we have from (2.7) fidr, = fzdr,. (2.10) And since frequency is inversely proportional to period, (2.9) follows. At first glance, this seems to be just the old backwards derivation, but in Weyl’s version, there is a difference; namely, he is explicit that II in (2.9) is ‘cosmic’ (co-ordinate) frequency and that in a static field the cosmic frequency of the light wave emitted by the atom at 1 does not change in transmission to 2. Hence, in (2.9) vi can be taken to be the cosmic frequency, as measured at 2, of the light wave emitted at 1. But since we are now referring everything to a single point - location 2-the ratios of the co-ordinate frequencies are the same as the ratios of the proper frequencies (since theffactor cancels out), and so (2.9) should hold for proper frequencies as well. The resulting formula is beautifully ambiguous: it seems to agree with everyone else’s upside down ratio and to contradict the correct ratio (2.5); but on the other hand, v, and u2 can be interpreted respectively as the v,,* and ulr2 of the pedantic notation, turning his (2.9) into the correct relation (2.5”). One is tempted to postulate a kind of demonic possession or mass hysteria in order to explain how some of the acutest minds of 2Oth-century physics, in possession of all the facts needed to arrive at a correct conclusion, could uniformly produce an incorrect or ambiguous result. In the case of von Laue, the temptation is almost irresistible. In Vol. 2 of his Relurivitiitstheo&, von Laue follows the route leading to (2.4). But obviously wanting his final result to agree with everyone else’s (backwards) formula, he concludes that since ‘the number of oscillations serves directly as the measure of proper time’,“3 we obtain V,h? = ii-E&/~,. (2.1 I) ‘*M. van Laue, Die Relutivit&fheorie (Braunschweig: F. Vieweg. 1921), Vol. 2, p. 188. The Gravirational Red Shift 189 The upshot, of course, is that the spectral rays of the Sun come to us with a frequency less than that of similar terrestrial sources. This, says von Laue, is what is called the displacement towards the red. The ambiguity which allows the backwards formula to pass muster is resolved by the more pedantic notation. Von Laue’s conclusion vsun< uEarrh,which seems, prima facie, to express a displacement towards the red, is really uSunS. un 0 and x2 > P define the vector field X whose components are x’ = (t, 0, 0, x). It is easy to verify that X is a timelike, nonrotating, Killing vector.“’ Thus, k’s hX, h = (X.X-)-“*, is a static frame. But since the normalization factor h is not constant, there will be a spectral shift for source and receiver at rest in the frame. Again, however, the effect is not a gravitational one since no real gravitational field is present. This example, and the preceding one also, can be ruled out by the imposition of global requirements on frames because the frames in question cannot be defined on all of Minkowski space-time; but such a move has an ad hoc flavor since the red shift is normally discussed in terms of local conditions. The additional requirement that the frame be non-accelerating is a local condition which will rule out the troublesome example of the preceding paragraph, but only at disastrous expense. For in any space - time, flat or not, a frame V which is both geodesic and stationary is one in which there is no spectral shift for source and receiver co-moving with the frame (see Appendix). The spectra1 shift predicted by general relativity theory is a complicated function of the states of motion of the source and receiver and the curvature of “?We are indebted to Robert Geroch for bringing this example to our attention. lq3The space - time metric g, induces on the spaceLike hypersurface a positive definite metric h,. = g, + V, V, (see Appendix); spatial distances are measured in h,. “‘This example was supplied by David Malamenr. The Gravitational Red Shift 211 the space- time between them, and the task of separating these variables is a difficult if not impossible one. The use of stationary and static frames might have been thought to quash the effects of motion, but as the above examples show, this is clearly not so since the spectral shift in these cases can have no other cause. On the other hand, the motion is not of the sort that happily lends itself to a Doppler interpretation. Unless the frame is co-variantly constant (i.e. vJ = 0), the world lines of the reference points can be considered as being non-parallel to one another , and thus, the reference points can be regarded as being in relative motion. Taking advantage of this fact, Synge”5 has defined a concept of relative velocity between the source and receiver of a light signal: the four-velocity vector of the source at the instant of emission is parallel transported along the light ray to the receiver; the resulting vector is projected onto the hyperplane orthogonal to the four velocity of the receiver at the point of reception; and the projection is then normalized to give the relative velocity.“’ Synge was able to show that the theoretical value of the spectral shift can be expressed directly in terms of this quantity. However, Synge’s relative velocity may bear only a distant resemblance to the velocity concepts used in classical and special relativistic expressions for the Doppler shift; for instance, it may be non-zero for source and receiver at rest in a static frame where. as we saw above, there is a natural sense in which the spatial distances do not change. Moreover, Synge’s inference from the fact that the Riemann tensor does not appear in his Doppler-like formula for the spectral shift to the conclusion that the effect is not a gravitational one can be misleading. For although the Riemann tensor may not make an explicit appearance, the curvature can make itself felt in Synge’s formula through its effects on the parallel transport of vectors. Further investigations may pinpoint various classes of cases where the Doppler or gravitational labels can be happiiy applied. But the above considerations are enough to show that in general it is wise to speak of the spectral shift without attaching these labels - otherwise one runs the risk of being guilty of windy warfare. 8. Conclusion In recent years the red shift has typically been treated, both by physicists and historians, as a distinctly minor issue in the development of gravitational theory. Our view is rather different: the red shift is a litmus, and its coloring ‘Tynge, Ref. 1IO, pp. 119 ff. ““If tbere is more than one null geodesic connecting the points of emission and reception, then Synge’s concept of relative velocity must be relativized to a null path if parallel transport along different null geodesics leads to different results. 212 Studies in History and Philosophy of Science reveals most of the major themes that dominated the development avD reception of general relativity. Einstein’s early derivations of the red shift show his most characteristic style of work - heuristic, allusive, sometimes baffling, but unfailingly fruitful. His derivation of the red shift from within general relativity shows something of the same characteristics, but it reveals rather more directly the difficulty Einstein had in treating the relations between physical quantities and co-ordinate expressions in a coherent way, In this, as we have seen, Einstein was joined by many of the best mathematical physicists at work on gravitation at the time. In this regard, the red shift is only an extreme case of a difficulty that was common enough for other crucial tests as well. One can find all manner of confusion about the perihelion advance and the bending of light, but for some reason, hard to put one’s finger on, the confusions about these tests were not nearly so widespread as were those concerning the red shift. These very confusions and misunderstandings about how to apply the theory, misunderstandings shared by many of its advocates, doubtless helped to make the scientific debate over general relativity a little more furious and chaotic. Anti-relativists did not have to create their own misunderstandings of the theory (although many of them did not hesitate); the misunderstandings were already there to be enjoyed and used in the battle. More than a litmus for the historians, the red shift debates illustrate, besides, how much more intricate and delicate issues of confirmation can be and were than the representations of physicists and philosophers sometimes lead us to believe. Historically, we find neither clear-cut agreement nor disagreement between measurements and theory, but instead a dispersion of results which could be interpreted either in favor of the theory or against it, according to one’s determination, We find, besides, a kind of psychological dependence of the red shift on the eclipse results, since it was only after the 1919 eclipse expedition that a number of solar scientists found the will to interpret their results in favor of the ‘Einstein effect’. And we find more power for testing general relativity than is usually attributed to the red shift. Altogether, there may be no other single topic which so vividly illustrates the intellectual ferment, the styles of work, the profundity and the confusion, associated with general theory of relativity. Appendix Notation Throughout the paper we use modern notation - e.g. ordinary and covariant derivatives are denoted respectively by comma and semi-colon and the standard conventions that go with it, e.g. the Einstein summation convention on repeated indices. Round and square brackets denote respectively The Gravitational Red Shifr 213 symmetrization and anti-symmetrization. Definitions Since reference frames are crucial to the derivation and interpretation of the red shift, it will be usefu1 to review some of the basic concepts. Let (M, g) be a relativistic space-time, where M is a four-dimensional differentiable manifold and g is pseudo-Riemannian metric of signature (--- +). A reference frame for (n/l, g) is a unit timelike vector field V on Iw, i.e. V- V = g( V, v) = 1. The frame V is stationary iff it is proportional to a Killing vector field; that is, V = hX where h is a positive function and X is a vector fieId satisfying fXg = 0 (or, in co-ordinate terminology, X, i;i, = 0). V is geodesic iff its acceleration A’ = Vi,, P vanishes. V is non-rotating iff its rotation matrix OJ V) = himhi”VlmZnl vanishes, where h;, s gij + ViVJis the projection tensor. V is static iff it is both stationary and non-rotating. I/ is paruliel or covariuntly constant iff PCj = 0. A space-time is said to be stationary (respectively, static) if it admits a global stationary (static) frame. Lemmas. We recaI1 a standard lemma relating these definitions to co-ordinate characterizations, Lemma 1. Suppose that (M, g) admits a stationary frame V. Then for every point p E A4 in the domain of definition of V there exists an open neighborhood N(p) and a co-ordinate chart xi, i = 1, 2, 3, 4, covering N@) such that V u a /a~’ and gU;, = 0. If Y is static, then the co-ordinate system can be chosen to have the additional property that ga4 = 0, Q = 1, 2, 3. Another important but non-trivial lemma about stationary frames was proved recently by MUlIer zum Hagen,“’ Lemma 2. A frame Vis stationary iff the round trip proper time of a light signal between any two of the trajectories of V is constant in time. As discussed in section 6, however, lemma 2 is not sufficient to justify interpreting the red shift in a stationary frame as a non-Doppier effect, Two further lemmas link stationary frames and the red shift. Lemma 3. Let X be a Killing vector field and P the tangent vector field of a geodesic. Then P-X is constant along the geodesic. Proof. Let A be an affine parameter of the geodesic. Then = (P;*P)xi f P’PkX,,,. (A.1) The first term on the rhs of the second equation vanishes because P is the “‘H. MNler zum Hagen, ‘A New Physical Characterization of Stationary and Static Space-Times,’ Proc. Cumb. Phil. Sot. Math. Phys. 11 (1972), 381 - 389. 214 Studies in History and Philosophy of Science tangent vector of a geodesic. In the second term, only the symmetric part of X,;, contributes to the sum, and this part vanishes because of Killing’s equations. Lemma 4. In a stationary frame V, the photon frequency is independent of the path the photon takes between the source and the receiver; thus one can properly speak of the red shift for the frame V. Proof. Let V = hX, where X is a Killing vector field. Killing’s equations give V,ij, = (logh)& Contracting with VJand using the identities (A.21 gives k$vJ = 1, Vi;$r = 0 (A.31 Ai = (log/z),, I- (jz/h)VJ, tr s f$V. 64.4) Contracting again and using Ai I/i = 0 yields ir=0. (A.9 So the proportionality factor is constant along the trajectories of I/. From lemma 3 and the discussion of section 3, it follows that the photon frequency ratio Y,/v,, for source S and receiver R co-moving with V, is h],/hlR. Hence, the frequency ratio is independent of when the photon leaves S and also of the (nonbroken) null geodesic connecting S to R. Another consequence for the red shift is given in: Lemma 5. If the frame Vis both stationary and geodesic, then there is no red shift for source and receiver co-moving with V. Proof. Combining (A.4) and (A.51 gives for a stationary frame Ai = (log/z),,. (A.61 Thus, if V is geodesic as well, h = constant, and the photon frequency ratio is unity. This lemma does not necessarily mean that there is no red shift for a source S and receiver R which are both at rest in a stationary frame and which are both in geodesic motion, But this consequence does hold if a mild form of Copernicanism is added - namely, S and R are not privileged in that all the other observers at rest in the frame are also in geodesic motion.“a “#The research for this paper was supported by National Science Foundation Grants Sot. 78-01076 and 78-03887.