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The study of analytic hyperbolic geometry begins in Chap. 2 with the study of gyrogroups. A gyrogroup is a most natural extension of a group into the regime of the nonassociative algebra that we need in order to extend analytic Euclidean geometry into analytic hyperbolic geometry. Geometry, according to Herodotus, and the Greek derivation of the word, had its origin in Egypt in the mensuration of land, and xing of boundaries necessitated by the repeated inundations of the Nile. It consisted at rst of isolated facts of observation and crude rules for calculation until it came under the in uence of Greek thought. Following the introduction of geometry from Egypt to Greece by Thales of Miletus, 640 { 546 B.C., geometric objects were abstracted, thus paving the way for attempts to give geometry a connected and logical presentation. The most famous of these attempts is that of Euclid, about 300 B.C. [Sommerville (1914), p. 1].
According to the Euclid parallel postulate, given a line L and a point P not on L there is one and only one line L0 which contains P and is parallel to L. Euclid's parallel postulate does not seem as intuitive as his other axioms. Hence, it was felt for many centuries that it ought to be possible to nd a way of proving it from more intuitive axioms. The history of the study of parallels is full of reproaches against the lack of self-evidence of the Euclid parallel postulate. According to Sommerville [Sommerville (1914), p. 3], Sir Henry Savile referred to it as one of the great blemishes in the beautiful body of geometry [Praelectiones, Oxford, 1621, p. 140]. Following Bolyai and Lobachevsky, however, the parallel postulate became the property that distinguishes Euclidean geometry from non-Euclidean ones. The Hungarian Geometer Janos Bolyai (1802 { 1860) and the Russian Mathematician Nikolai Ivanovich Lobachevsky (1793 { 1856) independently 1
Analytic Hyperbolic Geometry
worked out a geometry that seemed consistent and yet negated the Euclidean parallel postulate, published in 1832 and 1829. Carl Friedrich Gauss (1777 { 1855), who was the dominant gure in the mathematical world at the time, was probably the rst to understand clearly the possibility of a logically and sound geometry di erent from Euclid's. According to Harold E. Wolfe [Wolfe (1945), p. 45], it was Gauss who coined the term non- Euclidean geometry. The contributions of Gauss to the birth of hyperbolic geometry are described by Sonia Ursini in [Ursini (2001)]. According to Duncan M. Y. Sommerville [Sommerville (1914), p. 24], the ideas inaugurated by Bolyai and Lobachevsky did not attain any wide recognition for many years, and it was only after Baltzer had called attention to them in 1867 that non-Euclidean geometry began to be seriously accepted and studied. In 1871 Felix Klein suggested calling the non-Euclidean geometry of Bolyai and Lobachevsky hyperbolic geometry [Sommerville (1914), p. 25]. The discovery of hyperbolic geometry and its development is one of the great stories in the history of mathematics; see, for instance, the accounts of [Rosenfeld (1988)] and [Gray (1989)] for details. For several centuries (Euclidean) geometry and (associative) algebra developed slowly as distinct mathematical disciplines. In 1637 the French mathematician and philosopher Rene Descartes published his La Geometrie which introduced a theory that uni es (associative) algebra and (Euclidean) geometry, where points are modeled by n-tuples of numbers, n being the dimension of the geometry. The unifying theory is now called analytic (Euclidean) geometry. In full analogy, the aim of this book is to introduce analytic hyperbolic geometry, and its applications, as a theory that uni es nonassociative algebra and the hyperbolic geometry of Bolyai and Lobachevsky, where points are modeled by n-tuples of numbers.

A Vector Space Approach to Euclidean Geometry and A Gyrovector Space Approach to Hyperbolic Geometry

Commonly, three methods are used to study Euclidean geometry: (1) The Synthetic Method: This method deals directly with geometric objects ( gures). It derives some of their properties from other properties by logical reasoning.
(2) The Analytic Method: This method uses a coordinate system, expressing properties of geometric objects by numbers (coordinates). It derives properties from other properties by numerical expressions and equations, numerical results being interpreted in terms of geometric objects [Boyer (2004)]. (3) The Vector Method: The vector method occupies a middle position between the synthetic and the analytic method. It deals with geometric objects directly and derives properties from other properties by computation with vector expressions and equations [Hausner (1998)]. Euclid treated his Euclidean geometry synthetically. Similarly, also Bolyai and Lobachevsky treated their hyperbolic geometry synthetically. Because progress in geometry needs computational facilities, the invention of analytic geometry by Descartes (1596{1650) made simple approaches to more geometric problems possible. Later, further simplicity for geometric calculations became possible by the introduction of vectors and their addition by the parallelogram law. The parallelogram law for vector addition is so intuitive that its origin is unknown. It may have appeared in a now lost work of Aristotle (384{322). It was also the rst corollary in Isaac Newton's (1642{1727) \Principia Mathematica" (1687), where Newton dealt extensively with what are now considered vectorial entities, like velocity and force, but never with the concept of a vector. The systematic study and use of vectors were a 19th and early 20th century phenomenon. Vectors were born in the rst two decades of the 19th century with the geometric representations of complex numbers. The development of the algebra of vectors and of vector analysis as we know it today was rst revealed in sets of notes made by J. Willard Gibbs (1839{1903) for his students at Yale University. The synthetic and analytic methods for the study of Euclidean geometry are accessible to the study of hyperbolic geometry as well. Hitherto, however, the vector method had been deemed inaccessible to that study. In the years 1908 { 1914, the period which experienced a dramatic owering of creativity in the special theory of relativity, the Croatian physicist and mathematician Vladimir Varicak (1865 { 1942), professor and rector of Zagreb University, showed that this theory has a natural interpretation in hyperbolic geometry [Varicak (1910a)]. However, much to his chagrin, he had to admit in 1924 [Varicak (1924), p. 80] that the adaption of vector algebra for use in hyperbolic geometry was just not feasible, as ScottWalter notes in [Walter (1999b), p. 121]. Vladimir Varicak's hyperbolic geometry program, cited by Pauli [Pauli (1958), p. 74], is described by Walter in 4 Analytic Hyperbolic Geometry [Walter (1999b), pp. 112{115]. Following Varicak's 1924 realization that, unlike Euclidean geometry, the hyperbolic geometry of Bolyai and Lobachevsky does not admit vectors, there are in the literature no attempts to treat hyperbolic geometry vectorially. There are, however, few attempts to treat hyperbolic geometry analytically [Jackson and Greenspan (1955); Patrick (1986)], dating back to Sommerville's 1914 book [Sommerville (1914); Sommerville (1919)]. Accordingly, following Bolyai and Lobachevsky, most books on hyperbolic geometry treat the geometry synthetically, some treat it analytically, but no book treats it vectorially. Fortunately, some 80 years since Varicak's 1924 realization, the adaption of vectors for use in hyperbolic geometry, where they are called gyrovectors, has been accomplished in [Ungar (2000a); Ungar (2001b)], allowing Euclidean and hyperbolic geometry to be united [Ungar (2004c)]. Following the adaption of vector algebra for use in hyperbolic geometry, the hyperbolic geometry of Bolyai and Lobachevsky is now e ectively regulated by gyrovector spaces just as Euclidean geometry is regulated by vector spaces. Accordingly, we develop in this book a gyrovector space approach to hyperbolic geometry that is fully analogous to the common vector space approach to Euclidean geometry [Hausner (1998)]. In particular, we nd in this book that gyrovectors are equivalence classes of directed gyrosegments, Def. 5.4, p. 133, that add according to the gyroparallelogram law, Figs. 8.25 { 8.26, p. 322, just like vectors, which are equivalence classes of directed segments that add according to the common parallelogram law. It should be remarked here that in applications to Einstein's special theory of relativity, Chaps. 10 { 11 and 13, Einsteinian velocity gyrovectors are 3-dimensional gyrovectors fully analogous to Newtonian velocity vectors. Hence, in particular, relativistic gyrovectors are di erent from the common 4-vectors of relativity physics. In fact, the passage from n- gyrovectors to (n + 1)-vectors is illustrated in Remark 4.21, p. 122, and employed in the study of the special relativistic Lorentz transformation group in Chap. 11. The 4-vectors are important in special relativity and in its extension to general relativity. Early attempts to employ 4-vectors in gravitation, 1905 { 1910, are described in [Walter (2005)]. In the same way that vector spaces are commutative groups of vectors that admit scalar multiplication, gyrovector spaces are gyrocommutative gyrogroups of gyrovectors that admit scalar multiplication. Accordingly, the nonassociative algebra of gyrovector spaces is our framework for anaI ntroduction 5 lytic hyperbolic geometry just as the associative algebra of vector spaces is the framework for analytic Euclidean geometry. Moreover, gyrovector spaces include vector spaces as a special, degenerate case corresponding to trivial gyroautomorphisms. Hence, our gyrovector space approach forms the theoretical framework for uniting Euclidean and hyperbolic geometry.


All the analogies that hyperbolic geometry and classical mechanics share in this book, respectively, with Euclidean geometry and relativistic mechanics arise naturally through a single common mechanism represented by the pre- x \gyro". Indeed, in order to elaborate a precise language for dealing with analytic hyperbolic geometry, which emphasizes analogies with classical notions, we extensively use the pre x \gyro", giving rise to gyrolanguage, the language that we use in this book. The resulting gyrolanguage rests on the uni cation of Euclidean and hyperbolic geometry in terms of analogies they share [Ungar (2004c)]. The pre x \gyro" stems from Thomas gyration. The latter, in turn, is a mathematical abstraction of the peculiar relativistic e ect known as Thomas precession into an operator, called a gyrator and denoted \gyr". The gyrator generates special automorphisms called gyroautomorphisms. The e ects of the gyroautomorphisms are called gyrations in the same way that the e ects of rotation automorphisms are called rotations. The natural emergence of gyrolanguage is well described by a 1991 letter that the author received from Helmuth Urbantke of the Institute for Theoretical Physics, University of Vienna, sharing with him instructive experience [Ungar (1991b), ft. 36]:
\While giving a seminar about your work, the word gyromorphism instead of [Thomas] precession came over my lips. Since it ties in with the many morphisms the mathematicians love, it might appeal to you." Helmuth K. Urbantke, 1991 Indeed, we will nd in this book that the study of hyperbolic geometry by the gyrolanguage of gyrovector spaces is useful, and that the pursuit of this study entails no pain for unlimited pro t. Analytic Euclidean geometry in n dimensions models points by n-tuples of numbers that form an n-dimensional vector space with an inner prod6 Analytic Hyperbolic Geometry uct and the Euclidean distance function. Accordingly, vector spaces algebraically regulate analytic Euclidean geometry, allowing the principles of (associative) algebra to manipulate Euclidean geometric objects. Contrastingly, synthetic Euclidean geometry is the kind of geometry for which Euclid is famous and that the reader learned in high school. Analytic hyperbolic geometry in n dimensions is the subject of this book. It models points by n-tuples of numbers that form an n-dimensional gyrovector space with an inner product and a hyperbolic distance function. Accordingly, gyrovector spaces algebraically regulate analytic hyperbolic geometry, allowing the principles of (nonassociative) algebra to manipulate hyperbolic geometric objects. Contrastingly, synthetic hyperbolic geometry is the kind of geometry for which Bolyai and Lobachevsky are famous and that one learns from the literature on classical hyperbolic geometry. With one exception, proofs are obtained in this book analytically. The exceptional case is the proof of the gyrotriangle defect identity which is the identity shown at the bottom of Fig. 8.13, p. 285. Instructively, this identity is veri ed both analytically, Theorem 8.45, p. 301, and synthetically, Theorem 8.48, p. 304. It is the gyrotriangle defect identity at the bottom of Fig. 8.13 that gives rise to the elegant values of the squared hyperbolic length (gyrolength) of the sides of a hyperbolic triangle (gyrotriangle) in terms of its hyperbolic angles (gyroangles), also shown in Fig. 8.13 as well as in Theorem 8.49 on p. 307.
While Euclidean geometry has a single standard model, hyperbolic geometry is studied in the literature by several standard models. In this book, analytic hyperbolic geometry appears in three mutually isomorphic models, each of which has its advantages and blindness for selected aspects. These are:
(I) The Poincare ball (or disc, in two dimensions) model. (II) The Beltrami-Klein ball (or disc, in two dimensions) model. (III) The Proper Velocity (PV, in short) space (or plane, in two dimensions) model. The PV space model of hyperbolic geometry is also known as Ungar space model [Ungar (2001b)]. The terms \Ungar gyrogroups" and \Ungar gyrovector spaces" were coined by Jing-Ling Chen in [Chen and Ungar (2001)] following the emergence of gyrolanguage in [Ungar (1991b)]. Ungar gyrogroups and gyrovector spaces may be used to describe algebraic structures of relativistic proper velocities. Hence, in this book these are called PV gyrogroups and PV gyrovector spaces.
Before the emergence of gyrolanguage the author coined the term \Kloop" in [Ungar (1989b)] to honor related pioneering work of Karzel in the 1960s, and to emphasize relations with loops that have later been studied in [Krammer (1998); Sabinin, Sabinina and Sbitneva (1998); Issa (1999); Issa (2001)]. With the emergence of gyrolanguage, however, since 1991 the author's K-loops became \gyrocommutative gyrogroups" following the need to accommodate \non-gyrocommutative gyrogroups" and to emphasize analogies with groups. The ultimate fate of mathematical terms depends on their users. Thus, for instance, some like the term \K-loop" that the author coined in 1989 (as recorded in [Kiechle (2002), pp. 169 { 170] and, in more detail, in [Sexl and Urbantke (2001), pp. 141 { 142]), and some prefer using the alternative term \Bruck-loop" (as evidenced, for instance, from MR:2000j:20129 in Math. Rev.). A new term, \dyadic symset", which has recently emerged from an interesting work of Lawson and Lim in [Lawson and Lim (2004)], turns out to be identical to a two-divisible, torsion-free, gyrocommutative gyrogroup according to [Lawson and Lim (2004), Theorem 8.8]. It thus seems that, as Michael Kinyon notes in his MR:2003d:20109 review in Math. Rev. of Hubert Kiechle's nice introductory book on the \Theory of K-loops" [Kiechle (2002)], \It is unlikely that there will be any convergence of terminology in the near future."
Since the models of hyperbolic geometry are regulated algebraically by gyrovector spaces just as the standard model of Euclidean geometry is regulated algebraically by vector spaces, the theory of gyrogroups and gyrovector spaces develops in this book an internal ecology. It includes the special gyrolanguage, key examples, de nitions and theorems, central themes, and a few gems, like those illustrated in various gures in this book, to amaze both the uninitiated and the practicing expert on hyperbolic geometry and special relativity.


When considering the beings living on a sphere it is easy for us to differentiate between the sphere and some plane surface: we actually see the sphere being curved. But when it comes to us, and our curved space, we cannot see it since this would entail our standing outside space and looking down on it. Can we then determine whether space is curved by doing measurements inside it? To see that this can be done let’s go back to the beings on the sphere. Suppose they make a triangle by the following procedure: they go form the equator to the north pole along a great circle (or meridian) of the sphere, at the north pole they turn 90o to the right and go down another great circle until they get to the equator, then they make another 90o turn to the right until they get to the starting point (see Fig. 7.18). They find that all three lines make 90o angles with each other, so that the sum of the angles of this triangle is 270o, knowing that angles in all flat triangles always add up to 180o they conclude that the world they live on is not a flat one. Pythagoras’ theorem only holds on flat surfaces We can do the same thing: by measuring very carefully angles and distances we can determine whether a certain region of space is curved or not. In general the curvature is very slight and so the distances we need to cover to observe it are quite impractical (several light years), still there are some special cases where the curvature of space is observed: if space were flat light would travel in straight lines, but we observe that light does no such thing in regions where the gravitational forces are large; I will discuss this further when we get to the tests of the General Theory of Relativity in the following sections. The curvature of space is real and is generated by the mass of the bodies in it. Correspondingly the curvature of space determines the trajectories of 175 Figure 7.18: A path followed by a determined being living on the surface of a sphere; each turn is at right angles to the previous direction, the sum of the angles in this triangle is then 270o indicating that the surface in which the bug lives is not flat. all bodies moving in it. The Einstein equations are the mathematical embodiment of this idea. Their solutions predict, given the initial positions and velocities of all bodies, their future relative positions and velocities. In the limit where the energies are not too large and when the velocities are significantly below c the predictions of Einstein’s equations are indistinguishable from those obtained using Newton’s theory. At large speeds and/or energies significant deviations occur, and Einstein’s theory, not Newton’s, describes the observations.

null geodesics equation

Geodesic is not in general the shortest distance between two point. The general definition is , given any two points , a geodesic is a curve such that the distance between two points is stationary.

Eddington'd account

A FIRST draft of this book was published in 1921 as a mathematical supplement to the French Edition of Space, Time and Gravitation. During the ensuing eighteen months I have pursued my intention of developing it into a more systematic and comprehensive treatise on the mathematical theory of Relativity. The matter has been rewritten, the sequence of the argument rearranged in many places, and numerous additions made throughout ; so that the work is now expanded to three times its former size. It is hoped that, as now enlarged, it may meet the needs of those who wish to enter fully into these problems of reconstruction of theoretical physics. The reader is expected to have a general acquaintance with the less technical discussion of the theory given in Space, Time and Gravitation, although there is not often occasion to make direct reference to it. But it is eminently desirable to have a general grasp of the revolution of thought associated with the theory of Relativity before approaching it along the narrow lines of strict mathematical deduction. In the former work wc explained how the older conceptions of physics had become untenable, and traced the gradual ascent to the ideas which must supplant them. Here our task is to formulate mathematically this new conception of the world and to follow out the consequences to the fullest extent. The present widespread interest in the theory arose from the verification of certain minute deviations from Newtonian laws. To those who are still hesitating and reluctant to leave the old faith, these deviations will remain the chief centre of interest ; but for those who have caught the spirit of the new ideas the observational predictions form only a minor part of the subject. It is claimed for the theory that it leads to an understanding of the world of physics clearer and more penetrating than that previously attained, and it has been my aim to develop the theory in a form which throws most light on the origin and significance of the great laws of physics. It is hoped that difficulties which are merely analytical have been minimised by giving rather fully the intermediate steps in all the proofs with abundant cross-references to the auxiliary formulae used. For those who do not read the book consecutively attention may be called to the following points in the notation. The summation convention (p. 50) is used. German letters always denote the product of the corresjjonding English letter by V — g (p. 111). Vl is the symbol for " Hamiltonian differentiation" introduced on p. 139. An asterisk is prefixed to symbols generalised so as to be independent of or covariant with the gauge
A selected list of original papers on the subject is given in the Bibliography at the end, and many of these are sources (either directly or at second-hand) of the developments here set forth. To fit these into a continuous chain of deduction has involved considerable modifications from their original form, so that it has not generally been found practicable to indicate the sources of the separate sections. A frequent cause of deviation in treatment is the fact that in the view of most contemporary writers the Principle of Stationary Action is the final governing law of the world ; for reasons explained in the text I am unwilling to accord it so exalted a position. After the original papers of Einstein, and those of de Sitter from which I first acquired an interest in the theory, I am most indebted to Weyl's Raum, Zeit, Materie. Weyl's influence will be especially traced in §§ 49, 58, 59, 61, 63, as well as in the sections referring to his own theory. I am under great obligations to the officers and staff'


The subject of this mathematical treatise is not pure mathematics but physics. The vocabulary of the physicist comprises a number of words such as length, angle, velocity, force, work, potential, current, etc., which we shall call briefly "physical quantities." Some of these terms occur in pure mathematics also ; in that subject they may have a generalised meaning which does not concern us here. The pure mathematician deals with ideal quantities defined as having the properties which he deliberately assigns to them. But in an experimental science we have to discover properties not to assign them ; and physical quantities are defined primarily according to the way in which we recognise them when confronted by them in our observation of the world around us. Consider, for example, a length or distance between two points. It is a numerical quantity associated with the two points; and we all know the procedure followed in practice in assigning this numerical quantity to two points in nature. A definition of distance will be obtained by stating the exact procedure ; that clearly must be the primary definition if we are to make sure of using the word in the sense familiar to everybody. The pure mathematician proceeds differently; he defines distance as an attribute of the two points which obeys certain laws—the axioms of the geometry which he happens to have chosen—and he is not concerned with the question how this "distance" would exhibit itself in practical observation. So far as his own investigations are concerned, he takes care to use the word self-consistently ; but it does not necessarily denote the thing which the rest of mankind are accustomed to recognise as the distance of the two points. To find out any physical quantity we perform certain practical operations followed by calculations ; the operations are called experiments or observations according as the conditions are more or less closely under our control. The physical quantity so discovered is primarily the result of the operations and calculations; it is, so to speak, a manufactured article—manufactured by our operations. But the physicist is not generally content to believe that the quantity he arrives at is something whose nature is inseparable from the kind of operations which led to it ; he has an idea that if he could become a god contemplating the external world, he would see his manufactured physical quantity forming a distinct feature of the picture. By finding that he can lay x unit measuring-rods in a line between two points, he has manufactured the quantity x which he calls the distance between the points ; but he believes that that distance x is something already existing in the picture of the world —a gulf which would be apprehended by a superior intelligence as existing in itself without reference to the notion of operations with measuring-rods.
Yet he makes curious and apparently illogical discriminations. The parallax of a star is found by a well-known series of operations and calculations ; the distance across the room is found by operations with a tape-measure. Both parallax and distance are quantities manufactured by our operations ; but for some reason we do not expect parallax to appear as a distinct element in the true picture of nature in the same way that distance does. Or again, instead of cutting short the astronomical calculations when we reach the parallax, we might go on to take the cube of the result, and so obtain another manufactured quantity, a " cubic parallax." For some obscure reason we expect to see distance appearing plainly as a gulf in the true world-picture ; parallax does not appear directly, though it can be exhibited as an angle by a comparatively simple construction ; and cubic parallax is not in the picture at all. The physicist would say that he finds a length, and manufactures a cubic parallax ; but it is only because he has inherited a preconceived theory of the world that he makes the distinction. We shall venture to challenge this distinction. Distance, parallax and cubic parallax have the same kind of potential existence even when the operations of measurement are not actually made— if you will move sideways you will be able to determine the angular shift, if you will lay measuring-rods in a line to the object you will be able to count their number. Any one of the three is an indication to us of some existent condition or relation in the world outside us—a condition not created by our operations. But there seems no reason to conclude that this world-condition resembles distance any more closely than it resembles parallax or cubic parallax. Indeed any notion of " resemblance " between physical quantities and the world-conditions underlying them seems to be inappropriate. If the length AB is double the length CD, the parallax of B from A is half the parallax of D from C ; there is undoubtedly some world-relation which is different for AB and CD, but there is no reason to regard the world-relation of AB as being better represented by double than by half the world-relatiou of CD. The connection of manufactured physical quantities with the existent world-condition can be expressed by saying that the physical quantities are measure-numbers of the world-condition. Measure-numbers may be assigned according to any code, the only requirement being that the same measurenumber always indicates the same world-condition and that different worldconditions receive different measure-numbers. Two or more physical quantities may thus be measure-numbers of the same world-condition, but in different codes, e.g. parallax and distance; mass and energy; stellar magnitude and luminosity. The constant formulae connecting these pairs of physical quantities give the relation between the respective codes. But in admitting that physical quantities can be used as measure-numbers of world-conditions existing independently of our operations, we do not alter their status as manufactured quantities. The same series of operations will naturally manufacture the
same result when world-conditions are the same, and different results when they are different. (Differences of world-conditions which do not influence the results of experiment and observation are ipso facto excluded from the domain of physical knowledge.) The size to which a crystal grows may be a measure-number of the temperature of the mother-liquor ; but it is none the less a manufactured size, and we do not conclude that the true nature of size is caloric. The study of physical quantities, although they are the results of our own operations (actual or potential), gives us some kind of knowledge of the world-conditions, since the same operations will give different results in different world-conditions. It seems that this indirect knowledge is all that we can ever attain, and that it is only through its influences on such operations that we can represent to ourselves a "condition of the world." Any attempt to describe a condition of the world otherwise is either mathematical symbolism or meaningless jargon. To grasp a condition of the world as completely as it is in our power to grasp it, we must have in our minds a symbol which comprehends at the same time its influence on the results of all possible kinds of operations. Or, what comes to the same thing, we must contemplate its measures according to all possible measure-codes—of course, without confusing the different codes. It might well seem impossible to realise so comprehensive an outlook; but we shall find that the mathematical calculus of tensors does represent and deal with world-conditions precisely in this way. A tensor expresses simultaneously the whole group of measurenumbers associated with any world-condition ; and machinery is provided for keeping the various codes distinct. For this reason the somewhat difficult tensor calculus is not to be regarded as an evil necessity in this subject, which ought if possible to be replaced by simpler analytical devices ; our knowledge of conditions in the external world, as it comes to us through observation and experiment, is precisely of the kind which can be expressed by a tensor and not otherwise. And, just as in arithmetic we can deal freely with a billion objects without trying to visualise the enormous collection ; so the tensor calculus enables us to deal with the world-condition in the totality of its aspects without attempting to picture it. leaving regard to this distinction between physical quantities and worldconditions, we shall not define a physical quantity as though it were a feature in the world-picture which had to be sought out. A physical quantity is defined by the series of operations and calculations of which it is the result. The tendency to this kind of definition had progressed far even in pre-relativity physics. Force had become " mass x acceleration," and was no longer an invisible agent in the world-picture, at least so far as its definition was concerned. Mass is defined by experiments on inertial properties, no longer as ''quantity of matter." But for some terms the older kind of definition (or lack of definition) has been obstinately adhered to ; and for these the relativity
theory must find new definitions. In most cases there is no great difficulty in framing them. We do not need to ask the physicist what conception he attaches to " length " ; we watch him measuring length, and frame our definition according to the operations he performs. There may sometimes be cases in which theory outruns experiment and requires us to decide between two definitions, either of which would be consistent with present experimental practice ; but usually we can foresee which of them corresponds to the ideal which the experimentalist has set before himself. For example, until recently the practical man was never confronted with problems of non-Euclidean space, and it might be suggested that he would be uncertain how to construct a straight line when so confronted ; but as a matter of fact he showed no hesitation, and the eclipse observers measured without ambiguity the bending of light from the " straight line." The appropriate practical definition was so obvious that there was never any danger of different people meaning different loci by this term. Our guiding rule will be that a physical quantity must be defined by prescribing operations and calculations which will lead to an unambiguous result, and that due heed must be paid to existing practice ; the last clause should secure that everyone uses the term to denote the same quantity, however much disagreement there may be as to the conception attached to it. When defined in this way, there can be no inquiry as to whether the operations give us the real physical quantity or whether some theoretical correction (not mentioned in the definition) is needed. The physical quantity is the measure-number of a world-condition in some code ; we cannot assert that a code is right or wrong, or that a measure-number is real or unreal ; what we require is that the code should be the accepted code, and the measurenumber the number in current use. For example, what is the real difference of time between two events at distant places ? The operation of determining time has been entrusted to astronomers, who (perhaps for mistaken reasons) have elaborated a regular procedure. If the times of the two events are found in accordance with this procedure, the difference must be the real difference of time ; the phrase has no other meaning. But there is a certain generalisation to be noticed. In cataloguing the operations of the astronomers, so as to obtain a definition of time, we remark that one condition is adhered to in practice evidently from necessity and not from design—the observer and his apparatus are placed on the earth and move with the earth. This condition is so accidental and parochial that we are reluctant to insist on it in our definition of time ; yet it so happens that the motion of the apparatus makes an important difference in the measurement, and without this restriction the operations lead to no definite result and cannot define anything. We adopt what seems to be the commonsense solution of the difficulty. W e decide that time is relative to an observer ; that is to say, Ave admit that an observer on
as specified in our definition, is also measuring time—not our time, but a time relative to himself. The same relativity affects the great majority of elementary physical quantities*; the description of the operations is insufficient to lead to a unique answer unless we arbitrarily prescribe a particular motion of the observer and his apparatus. In this example we have had a typical illustration of " relativity," the recognition of which has had far-reaching results revolutionising the outlook of physics. Any operation of measurement involves a comparison between a measuring-appliance and the thing measured. Both play an equal part in the comparison and are theoretically, and indeed often practically, interchangeable ; for example, the result of an observation with the meridian circle gives the right ascension of the star or the error of the clock indifferently, and we can regard either the clock or the star as the instrument or the object of measurement. Remembering that physical quantities are results of comparisons of this kind, it is clear that they cannot be considered to belong solely to one partner in the comparison. It is true that we standardise the measuring appliance as far as possible (the method of standardisation being explained or implied in the definition of the physical quantity) so that in general the variability of the measurement can only indicate a variability of the object measured. To that extent there is no great practical harm in regarding the measurement as belonging solely to the second partner in the relation. But even so we have often puzzled ourselves needlessly over paradoxes, which disappear when we realise that the physical quantities are not properties of certain external objects but are relations between these objects and something else. Moreover, we have seen that the standardisation of the measuring-appliance is usually left incomplete, as regards the specification of its motion ; and rather than complete it in a way which would be arbitrary and pernicious, we prefer to recognise explicitly that our physical quantities belong not solely to the objects measured but have reference also to the particular frame of motion that we choose. The principle of relativity goes still further. Even if the measuring appliances were standardised completely, the physical quantities would still involve the properties of the constant standard. We have seen that the world-condition or object which is surveyed can only be apprehended in our knowledge as the sum total of all the measurements in which it can be concerned ; any intrinsic property of the object must appear as a uniformity or law in these measures. When one partner in the comparison is fixed and the other partner varied widely, whatever is common to all the measurements may be ascribed exclusively to the first partner and regarded as an intrinsic property of it. Let us apply this to the converse comparison ; that is to say, keep the measuring-appliance constant or standardised, and vary as widely as possible the objects measured—or, in simpler terms, make a particular * The most important exceptions are number (of discrete entities), action, and entropy.
kind of measurement in all parts of the field. Intrinsic properties of the measuring-appliance should appear as uniformities or laws in these measures. We are familiar with several such uniformities; but we have not generally recognised them as properties of the measuring-appliance. We have called them laws of nature 1 The development of physics is progressive, and as the theories of the external world become crystallised, we often tend to replace the elementary physical quantities defined through operations of measurement by theoretical quantities believed to have a more fundamental significance in the external world. Thus the vis viva mv2 , which is immediately determinable by experiment, becomes replaced by a generalised energy, virtually defined by having the property of conservation ; and our problem becomes inverted—we have not to discover the properties of a thing which we have recognised in nature, but to discover how to recognise in nature a thing whose properties we have assigned. This development seems to be inevitable ; but it has grave drawbacks especially when theories have to be reconstructed. Fuller knowledge may show that there is nothing in nature having precisely the properties assigned ; or it may turn out that the thing having these properties has entirely lost its importance when the new theoretical standpoint is adopted*. When we decide to throw the older theories into the melting-pot and make a clean start, it is best to relegate to the background terminology associated with special hypotheses of physics. Physical quantities defined by operations of measurement are independent of theory, and form the proper starting-point for any new theoretical development. Now that we have explained how physical quantities are to be defined, the reader may be surprised that we do not proceed to give the definitions of the leading physical quantities. But to catalogue all the precautions and provisos in the operation of determining even so simple a thing as length, is a task which we shirk. We might take refuge in the statement that the task though laborious is straightforward, and that the practical physicist knows the whole procedure without our writing it down for him. But it is better to be more cautious. I should be puzzled to say off-hand what is the series of operations and calculations involved in measuring a length of 10~15 cm. ; nevertheless I shall refer to such a length when necessary as though it were a quantity of which the definition is obvious. We cannot be forever examining our foundations ; we look particularly to those places where it is reported to us that they are insecure. I may be laying myself open to the charge that I am doing the very thing I criticise in the older physics—-using terms that * We shall see in § 59 that this has happened in the case of energy. The dead-hand of a superseded theory continues to embarrass us, because in this case the recognised terminology still has implicit reference to it. This, however, is only a slight drawback to set off against the many advantages obtained from the classical generalisation of energy as a step towards the more complete theory.
have no definite observational meaning, and mingling with my physical quantities things which are not the results of any conceivable experimental operation. I would reply— By all means explore this criticism if you regard it as a promising field of inquiry. I here assume that you will probably find me a justification for my 10-15 cm. ; but you may find that there is an insurmountable ambiguity in defining it. In the latter event you may be on the track of something which will give a new insight into the fundamental nature of the world. Indeed it has been suspected that the perplexities of quantum phenomena may arise from the tacit assumption that the notions of length and duration, acquired primarily from experiences in which the average effects of large numbers of quanta are involved, are applicable in the study of individual quanta. There may need to be much more excavation before we have brought to light all that is of value in this critical consideration of experimental knowledge. Meanwhile I want to set before you the treasure which has already been unearthed in this field.

On Arthur Eddington’s Theory of Everything
Helge Kragh
Abstract: From 1929 to his death in 1944, A. Eddington worked on developing a highly ambitious theory of fundamental physics that covered everything in the physical world, from the tiny electron to the universe at large. His unfinished theory included abstract mathematics and spiritual philosophy in a mix which was peculiar to Eddington but hardly intelligible to other scientists. The constants of nature, which he claimed to be able to deduce purely theoretically, were of particular significance to his project. Although highly original, Eddington’s attempt to provide physics with a new foundation had to some extent parallels in the ideas of other British physicists, including P. Dirac and E. A. Milne. Eddington’s project was however a grand failure in so far that it was rejected by the large majority of physicists. A major reason was his unorthodox view of quantum mechanics. 1. Introduction Arthur Stanley Eddington is recognized as one of the most important scientists of the first half of the twentieth century (Douglas 1956). He owes this elevated position primarily to his pioneering work in astronomy and astrophysics, and secondarily to his expositions of and contributions to the general theory of relativity. Eddington developed a standard model of the interior structure of stars and was the first to suggest nuclear reactions as the basic source of stellar energy. In 1919 he rose to public fame when he, together with Frank Dyson, confirmed Einstein’s prediction of the bending of starlight around the Sun. Six years later he applied general relativity to white dwarf stars, and in 1930 he developed one of the first relativistic models of the expanding universe known as the Lemaître-Eddington model.  Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark. E-mail: helge.kragh@nbi.ku.dk. This is a slightly revised version of a contribution to be published in Dean Rickles and Ian Durham, eds., Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge, a book to be published by Springer. 2 Not satisfied with his accomplishments within the astronomical sciences, during the last part of his life Eddington concentrated on developing an ambitious theory of fundamental physics that unified quantum mechanics and cosmology. The present essay is concerned solely with this grand theory, which neither at the time nor later won acceptance. Although Eddington’s fundamental theory has never been fully described in its historical contexts, there are several studies of it from a philosophical or a scientific point of view (e.g. Kilmister 1994; Durham 2006; French 2003). There have also been several attempts to revive scientific interest in Eddington’s theory (such as Wesson 2000), but none of them have been even moderately successful. At any rate, this essay is limited to the theory’s historical course and its reception among physicists.
2. Warming up
Although Eddington only embarked on his ambitious and lonely research programme of unifying the atom and the universe in 1929, some of the features of this programme or theory can be found in his earlier work. When Dirac’s quantum equation of the electron opened his eyes to a whole new foundation of physics, he was well prepared. The significance of combinations of the constants of nature, a key feature in the new theory, entered Eddington’s masterpiece of 1923, The Mathematical Theory of Relativity. Referring to the “very large pure number [given] by the ratio of the radius of the electron to its gravitational mass = 3 × 1042,” Eddington (1923, p. 167) suggested a connection to “the number of particles in the world – a number presumably decided by pure accident.” And in a footnote: “The square of 3 × 1042 might well be of the same order as the total number of positive and negative electrons” (see also Eddington 1920a, p. 178). Of course, Eddington’s reference to the positive electron was not to the positron but to the much heavier proton, a name introduced by Ernest Rutherford in 1920 but not generally used in the early 1920s. While Eddington in 1923 thought that the number of particles in the world was accidental, in his later theory this “cosmical number” appeared as a quantity strictly determined by theory.
A somewhat similar suggestion had been ventured by Hermann Weyl a few years earlier when considering the dimensionless ratio of the electromagnetic radius of the electron (re = e 2 /mc2 ) to its gravitational radius (rg = Gm/c 2 ). According 3 to Weyl (1919), the ratio of the radius of the universe to the electron radius was of the same order as re/rg ≅ 1040 , namely 𝑟e 𝑟g = 𝑒 2 𝐺𝑚2 ≈ 𝑅E 𝑟e , where RE denotes the radius of the closed Einstein universe. The huge value of the electrical force relative to the gravitational force as given by e 2 /Gm2 had been pointed out much earlier (e.g. Davis 1904), but it was only with Weyl and Eddington that the number ca. 1040 was connected to cosmological quantities. Another feature of Eddington’s mature philosophy of physics, apart from its emphasis on the constants of nature, was the fundamental significance he ascribed to mind and consciousness. Physics, he argued, can never reveal the true nature of things but only deals with relations between observables that are subjectively selected by the human mind. As Eddington (1920b, p. 155) stated in his earliest philosophical essay, he was “inclined to attribute the whole responsibility for the laws of mechanics and gravitation to the mind, and deny the external world any share in them.” By contrast, “the laws which we have hitherto been unable to fit into a rational scheme are the true natural laws inherent in the external world, and mind has no chance of moulding them in accordance with its own outlook.” The same theme appeared prominently in his Gifford Lectures delivered in early 1927, where Eddington (1928, p. 281) provocatively concluded that “the substratum of everything is of mental character.” While up to this time Eddington had exclusively relied on the admired theory of general relativity, he now also appealed to the new symbolic quantum theory as a further argument in favour of his view that physicists manufacture the phenomena and laws of nature. Yet, although he took notice of Heisenberg’s and Schrödinger’s quantum mechanics, for a while he was uncertain of how to make sense of it within the framework of a unified theory of relativistic physics. In The Mathematical Theory of Relativity (1923, p. 237) he wrote: “We offer no explanation of the occurrence of electrons or of quanta. … The excluded domain forms a large part of physics, but it is one in which all explanation has apparently been baffled hitherto.” Four years later the situation had not changed materially. 4
3. Eddington meets the Dirac equation
When Eddington studied Paul Dirac’s relativistically invariant wave equation of the electron in early 1928 he was fascinated but also perplexed because it was not written in the language of tensor calculus (Eddington 1936, p. 2). Although he recognized the new quantum equation as a great step forward in unifying physics, he also thought that it did not go far enough and consequently decided to generalize it. For an electron (mass m, charge e) moving in a Coulomb field the positive-energy Dirac equation can be written 𝑖ℎ 2𝜋 𝜕 𝜕𝑡 = 𝑒 2 𝑟  + 𝑐√(𝑖ℎ/2𝜋) 2∆ + 𝑚2𝑐 2 , where  = ( 2 /x 2 ,  2 /y 2 ,  2 /z 2 ) is the Laplace operator. Eddington (1929; 1931) rewrote the equation by introducing two constants, namely 𝛼 −1 = ℎ𝑐 2𝜋𝑒 2 and 𝛾 = 2𝑚𝑐 ℎ , In this way he arrived at −𝛼 𝜕 𝜕𝑡 = 𝑖𝑐 𝑟  + 𝑐√𝛼2∆ − 𝛾 2 The constant α-1 is the inverse of what is normally called the fine-structure constant, but Eddington always (and somewhat confusingly) reserved the name and symbol  for the quantity hc/2πe 2 . In his initial paper of 1929 he applied his version of the Dirac equation and his unorthodox understanding of Pauli’s exclusion principle to derive the value 𝛼 −1 = 16 + ½ × 16 × (16 − 1) = 136
By 1929 the fine-structure constant was far from new, but it was only with Eddington’s work that the dimensionless combination of constants of nature was elevated from an empirical quantity appearing in spectroscopy to a truly fundamental constant (Kragh 2003). Moreover, Eddington was the first to focus on its inverse value and to suggest – indeed to insist – that it must be a whole number. He was also the first to argue that α was of deep cosmological significance and that it should be derivable from fundamental theory. 5 Fig. 1. Dirac (third from left) with Eddington and Schrödinger (third and second from right) at a colloquium in 1942 at the Dublin Institute for Advanced Studies. The other persons, from the left, are Sheila Power, Pádraig de Brún, Eamon de Valera, Arthur Conway, and Albert J. McConnell. The Dublin Institute was established in 1940 by de Valera, the Irish political leader who was trained in mathematics and much interested in the sciences. Source: http://www-history.mcs.st-and.ac.uk/Biographies/Tinney.html. When Eddington realized that the theoretical value α-1 = 136 did not agree with experiment, at first he pretended to be undisturbed. “I cannot persuade myself that the fault lies with the theory,” he wrote in his paper of 1929. All the same, as experiments consistently showed that α-1 ≅ 137 he was forced to look for a fault in his original theory. He soon came to the conclusion that α-1 = 137 exactly, arguing that the extra unit was a consequence of the exclusion principle, which in his interpretation implied the indistinguishability of any pair of elementary particles in the universe. For the rest of his life Eddington (1932, p. 41) stuck to the value 137, which he claimed to have “obtained by pure deduction, employing only hypotheses already accepted as fundamental in wave mechanics.” It should be pointed out that although Dirac’s linear wave equation served as the inspiration for Eddington’s grand theory, for him it was merely a temporary stepping stone towards a higher goal. He felt that relativistic quantum mechanics, whether in Dirac’s version or some of the other established versions, was still characterized by semi-empirical methods that prevented a truly rational foundation of laws unifying the micro-cosmos and the macro-cosmos. In his monograph Relativity Theory of Protons and Electrons (1936, pp. 6-7) Eddington emphasized the difference between his theory and the one of Dirac: 6
Although the present theory owes much to Dirac’s theory of the electron, to the general coordination of quantum theory achieved in his book Quantum Mechanics … it is not “Dirac’s theory”; and indeed it differs fundamentally [from it] on most points which concern relativity. It is definitely opposed to what has commonly been called “relativistic quantum theory,” which, I think, is largely based on a false conception of the principles of relativity theory. While Dirac and other specialists in quantum mechanics disagreed with the last part of the equation, they very much agreed with the first part. 4. Constants of nature The fine-structure constant was not the only constant of nature that attracted the attention of Eddington. On the contrary, he was obsessed by the fundamental constants of nature, which he conceived as the building blocks of the universe and compared to the notes making up a musical scale: “We may … look on the universe as a symphony played on seven primitive constants as music played on the seven notes of a scale” (Eddington 1935, p. 227). The recognition of the importance of constants of nature is of relatively recent origin, going back to the 1880s, and Eddington was instrumental in raising them to the significance they have in modern physics (Barrow 2002; Kragh 2011, pp. 93-99). Whereas the fundamental constants, such as the mass of the electron and Planck’s quantum constant are generally conceived to be irreducible and essentially contingent quantities, according to Eddington this was not the case. Not only did he believe that their numerical values could be calculated, he also believed that he had succeeded in actually calculating them from a purely theoretical basis (see Fig. 2). For example, for Newton’s gravitational constant he deduced G = 6.6665 × 10-11 N m2 kg-2 , in excellent agreement with the experimental value known at the time, which he stated as (6.670 ± 0.005) × 10-11 N m2 kg-2 (Eddington 1946, p. 105). The constants of nature highlighted by Eddington were the mass m and charge e of the electron, the mass of the proton M, Planck’s constant h, the speed of light c, the gravitational constant G, and the cosmological constant Λ. To these he added the cosmical number N* and, on some occasions, the number of dimensions of space-time (3 + 1). Two of the constants were original to him and deserve mention for this reason. The cosmological constant introduced by Einstein in his field equations of 1917 was not normally considered a constant of nature and was in 7 Fig. 2. Eddington’s calculations of constants of nature compared to those observed in the 1940s. Reproduced from Kragh (2011), p. 99. any case ill-regarded by many cosmologists and astronomers in the 1930s. With the recognition of the expanding universe most specialists followed Einstein in declaring the cosmological constant a mistake, meaning that Λ = 0. Eddington emphatically disagreed. He was convinced that the Λ-constant was indispensable and of fundamental importance, for other reasons because he conceived it as a measure of the repulsive force causing the expansion of the universe. Appealing to Einstein’s original relation between the constant and the radius of the static universe, Λ = 1 𝑅E 2 , he considered Λ to be the cosmic yardstick fixing a radius for spherical space. “I would as soon think of reverting to Newtonian theory as of dropping the cosmical constant,” Eddington (1933, p. 24) wrote. To drop the constant, he continued, would be “knocking the bottom out of space.” Whereas the theoretical value of the cosmological constant is today one of physics’ deep and still unsolved problems, to 8 Eddington it was not. In 1931 he provided an answer in terms of other constants of nature: Λ = ( 2𝐺𝑀 𝜋 ) 2 ( 𝑚𝑐 𝑒 2 ) 4
= 9.8 × 10−55 cm−2 While this was believed to be of roughly the right order, unfortunately (or fortunately for Eddington) there were no astronomical determinations of Λ with which the theoretical value could be compared. Eddington used the value of Λ to calculate from first principles the Hubble recession constant, for which he obtained H0 = 528 km s-1 Mpc-1 . Since the figure agreed nicely with the generally accepted observational value, he took it as evidence of the soundness of his theoretical approach (Kragh 2011, p. 97). In his 1929 paper Eddington derived that the fine-structure constant was related in a simple way to constants of a cosmological nature, such as given by the expression 𝛼 = 2𝜋𝑚𝑐𝑅E ℎ√𝑁∗ He considered the cosmical number N* – the number of electrons and protons in the closed universe – to be a most important constant of nature. According to conventional physics and cosmology there was nothing special about the number, which might well have been different, but Eddington not only insisted that it was a constant, he also claimed that it could not have been different from what it is. Moreover, he claimed that he was able to deduce N* rigorously from theory, just as he was able to deduce the other constants of nature. The result was 𝑁 ∗ = 2 × 136 × 2 256 ≅ 3.15 × 1079 Notice that Eddington gave the number precisely. He counted a positron as minus one electron, and a neutron as one proton and one electron. Assuming that the total number of electrons equals the number of protons, he further derived a relation between two of the very large dimensionless constants: 𝑒 2 𝐺𝑚𝑀 = 2 𝜋 √𝑁∗ 9 This was the relation that he had vaguely suggested as early as 1923. For the mass ratio M/m between the proton and the electron, Eddington argued that it could be found from the ratio of the two roots in the equation 10𝑥 2 − 136𝜔𝑥 + 𝜔 2 = 0 The quantity ω is what Eddington called a “standard mass,” the mass of an unspecified neutral particle. In this way he derived the theoretical value M/m = 1847.6 or nearly the same as the experimental value (Durham 2006, pp. 211-218). In the 1930s a few scientists speculated for the first time that some of the constants of nature might not be proper constants but instead quantities that vary slowly in time (Kragh 2011, pp. 167-192). Eddington considered such ideas to be pure nonsense. In 1938 Dirac proposed a cosmological theory based on the radical assumption that G decreased according to 1 𝐺 𝑑𝐺 𝑑𝑡 = −3𝐻0
However, Eddington (1939a) quickly dismissed Dirac’s theory as “unnecessarily complicated and fantastic.” He was not kinder to contemporary speculations that the speed of light might be a varying quantity: “The speculation of various writers that the velocity of light has changed slowly in the long periods of cosmological time … is nonsensical because a change in the velocity of light is self-contradictory” (Eddington 1946, p. 8). Nearly sixty years later the so-called VSL (varying speed of light) theory developed by João Magueijo revived the discussion of whether Eddington’s objection was reasonable or not (Ellis and Uzan 2006; Kragh 2006). 5. Fundamental theory The research project that Eddington pursued with such fervour and persistence during the last 15 years of his life resulted in a long series of scientific papers and a couple of important monographs. Some of his books were highly technical while others were of a philosophical nature and mostly oriented toward a general readership. To Eddington, the latter were no less important than the first. He followed his research programme in splendid isolation, apparently uninterested in the work done by other physicists in the tradition he had initiated. The isolated and closed nature of Eddington’s research is confirmed by bibliometric studies based on 10 the list of publications given by his biographer, the Canadian astronomer Allie Vibert Douglas (1956). Among the references in Eddington’s 14 research papers on his unified theory in the period 1929-1944, no less than 70 per cent are to his own works. By comparison, the average self-reference ratio in physics papers in the period was about 10 per cent (Kragh and Reeves 1991). The first major fruit of Eddington’s efforts appeared in 1936 in the form of Relativity Theory of Protons and Electrons (RTPE) a highly mathematical and personal exposition of his ongoing attempt to create a new basis for cosmology and physics. During the following years he prepared a systematic account of his theory and its
mathematical foundation, but Fundamental Theory only appeared after his death. The title was not Eddington’s, but chosen but the mathematician Edmund Whittaker who edited Eddington’s manuscript and supervised it to publication. Whittaker had closely followed Eddington’s work which fascinated him more from a philosophical than a physical point of view. Like most scientists he remained unconvinced about the physical soundness of the grand project. In an extensive review of RTPE, Whittaker (1937) likened Eddington to a modern Descartes, suggesting that Eddington’s theory did not describe nature any better than the vortex theory of the French rationalist philosopher. Nonetheless, he described Eddington as “a man of genius.” Whittaker was not alone in comparing Eddington to Descartes. According to the philosopher Charlie Broad (1940, p. 312): “For Descartes the laws of motion were deducible from the perfection of God, whilst for Eddington they are deducible from the peculiarities of the human mind.” Moreover, “For both philosophers the experiments are rather a concession to our muddle-headedness and lack of insight.” Eddington’s ambitious project of reconstructing fundamental physics amounted to a theory of everything. The lofty goal was to deduce all laws and, ultimately, all phenomena of nature from epistemological considerations alone, thereby establishing physics on an a priori basis where empirical facts were in principle irrelevant. In RTPE (1936, pp. 3-5) he expressed his ambition as follows: It should be possible to judge whether the mathematical treatment and solutions are correct, without turning up the answer in the book of nature. My task is to show that our theoretical resources are sufficient and our methods powerful enough to calculate the constants exactly – so that the observational test will be the same kind of perfunctory verification that we apply sometimes to theorems in geometry. … I 11 think it will be found that the theory is purely deductive, being based on epistemological principles and not on physical hypotheses. At the end of the book (p. 327) he returned to the theme, now describing his aim in analogy with Laplace’s omniscient intelligence or demon appearing in the Exposition du Système du Monde from 1796. However, there was the difference that Eddington’s demon was essentially human in so far that it had a complete knowledge of our mental faculties. He wrote: An intelligence, unacquainted with our universe, but acquainted with the system of thought by which the human mind interprets to itself the content of its sensory experience, should be able to attain all the knowledge of physics that we have attained by experiment. He would not deduce the particular events or objects of our experience, but he would deduce the generalisations we have based on them. For example, he would infer the existence and properties of radium, but not the dimensions of the earth. Likewise, the intelligence would deduce the exact value of the cosmical number (as Eddington had done) but not, presumably, the value of Avogadro’s number. Eddington’s proud declaration of an aprioristic, non-empirical physics was a double-edged sword. On the one hand, it promised a final theory of fundamental physics in which the laws and constants could not conceivably be violated by experiment. On the other hand, the lack of ordinary empirical testability was also the Achilles-heel of the theory and a main reason why most physicists refused taking it seriously. Eddington was himself somewhat ambivalent with regard to testable predictions and did not always follow his rationalist rhetoric. He could not and did not afford the luxury of ignoring experiments altogether, but tended to accept them only when they agreed with his calculations. If this were not the case he consistently and often arrogantly explained away the disagreement by putting the blame on the measurements rather than the theory. Generally he was unwilling to let a conflict between a beautiful theory and empirical data ruin the theory. “We should not,” Eddington (1935, p. 211) wrote, “put overmuch confidence in the observational results that are put forward until they have been confirmed by theory.” 12 Fig. 3. Eddington in 1932. Credit: Encyclopædia Britannica. 6. Cosmo-physics In order to understand Eddington’s “flight of rationalist fancy” (Singh 1970, pp. 168-191) it is important to consider it in its proper historical context. If seen within the British tradition of so-called cosmo-physics his ambitious research project was not quite as extreme as one would otherwise judge it. A flight of rationalist fancy it was, but in the 1930s there were other fancies of the same or nearly the same scale. To put it briefly, there existed in Britain in the 1930s a fairly strong intellectual and scientific tradition that in general can be characterised as anti-empirical and prorationalist, although in some cases the rationalism was blended with heavy doses of idealism. According to scientists associated with this attempt to rethink the foundation of physical science, physics was inextricably linked to cosmology. In their vision of a future fundamental physics, pure thought counted more heavily than experiment and observation. The leading cosmo-physicists of the interwar period, or as Herbert Dingle (1937) misleadingly called them, the “new Aristotelians,” were Eddington and E. Arthur Milne, but also Dirac, James Jeans and several other scientists held views of a roughly similar kind (Kragh 1982). 13 Although the world system of Milne, a brilliant Oxford astrophysicist and cosmologist, was quite different from the one of Eddington, on the methodological level Milne’s system shared the rationalism and deductivism that characterized Eddington’s system. Among other things, the two natural philosophers had in common that their ideas about the universe – or about fundamental physics – gave high priority to mathematical reasoning and correspondingly low priority to empirical facts. Milne, much like Eddington, claimed that the laws of physics could ultimately be obtained from pure reasoning and processes of inference. His aim was to get rid of all contingencies by turning the laws of nature into statements no more arbitrary than mathematical theorems. As Milne (1948, pp. 10-12) put it: “Just as the mathematician never needs to ask whether a constructed geometry is true, so there is no need to ask whether our kinematical and dynamical theorems are true. It is sufficient that they are free from contradictions.” Despite the undeniable methodological affinity between the views of Milne and Eddington, the Cambridge professor insisted that his ideas were wholly different from those of his colleague in Oxford. Eddington (1939a) either ignored Milne’s theory or he criticized it as contrived and even “perverted from the start.” Dirac’s cosmological theory based on the G(t) assumption was directly inspired by the ideas of Milne and Eddington. His more general view about fundamental physics included the claim of a pre-established harmony between mathematics and physics, or what he saw as an inherent mathematical quality in nature. By the late 1930s Dirac reached the conclusion that ultimately physics and pure mathematics would merge into one single branch of sublime knowledge. In his James Scott Lecture delivered in Edinburgh in early 1939, he suggested that in the physics of the future there would be no contingent quantities at all. Even the number and initial conditions of elementary particles, and also the fundamental constants of nature, must be subjects to calculation. Dirac (1939, p. 129) proposed yet another version of Laplace’s intelligence: It would mean the existence of a scheme in which the whole of the description of the universe has its mathematical counterpart, and we must assume that a person with a complete knowledge of mathematics could deduce, not only astronomical data, but also all the historical events that take place in the world, even the most trivial ones. … The scheme could not be subject to the principle of simplicity since it would have to 14 be extremely complicated, but it may well be subject to the principle of mathematical beauty. Note that Dirac’s version included even “the most trivial” events in the world. This was not Eddington’s view, for he believed that contingent facts – those “which distinguish the actual universe from all other possible universes obeying the same laws” – were “born continually as the universe follows its unpredictable course” (Eddington 1939b, p. 64). Another major difference between the two natural philosophers was Dirac’s belief that the laws of physics, contrary to the rules of mathematics, are chosen by nature herself. This evidently contradicted Eddington’s basic claim that physical knowledge is wholly founded on epistemological considerations.
7. Nature as a product of the mind
Although Eddington’s project had elements in common with the ideas of Milne and other cosmo-physicists of the period, it was unique in the way he interpreted it philosophically. As mentioned in Section 2, Eddington was convinced that the laws of nature were subjective rather than objective. The laws, he maintained, were not summary expressions of regularities in an external world, but essentially the constructions of the physicists. This also applied to the fundamental constants of nature. Eddington (1939b, p. 57) characterized his main exposition of philosophy of physics, The Philosophy of Physical Science, as “a philosophy of subjective natural law.” Referring to the cosmical number N*, elsewhere in the book (p. 60) he explained that “the influence of the sensory equipment with which we observe, and the intellectual equipment with which we formulate the results of observation as knowledge, is so far-reaching that by itself it decides the number of particles into which the matter of the universe appears to be divided.” In agreement with his religious belief as a Quaker, Eddington deeply believed in an open or spiritual world that was separate from the one we have empirical access to (Stanley 2007). He often pointed out that physics is restricted to a small part of what we experience in a wider sense, as it applies only to what can be expressed quantitatively or metrically. “Within the whole domain of experience [only] a selected portion is capable of that exact representation which is requisite for development by the scientific method,” he wrote in The Nature of the Physical World 15 (p. 275). Far from wanting physics to expand its power to the spiritual or nonmetrical
world, as others wanted it, Eddington found it preposterous to believe that this world could be ruled by laws like those known from physics or astronomy (Douglas 1956, p. 131). Given that his theory was limited to the metrical world, it was not really a theory of everything. A key element in Eddington’s epistemology was what he referred to as “selective subjectivism.” With this term he meant that it is the mind which determines the nature and extent of what we think of as the external world. We force the phenomena into forms that reflect the observer’s intellectual equipment and the instrument he uses, much like the bandit Procrustes from Greek mythology. The physicist, Eddington (1936, p. 328) wrote, “might be likened to a scientific Procrustes, whose anthropological studies of the stature of travellers reveal the dimensions of the bed in which he has compelled them to sleep.” As a result of the selective subjectivism, “what we comprehend about the universe is precisely that which we put into the universe to make it comprehensible.” Eddington’s anthropomorphic and constructivist view of laws of nature was related to the conventionalist view of scientists such as Karl Pearson and Henri Poincaré, only did it go much farther. Eddington (1935, p. 1) recognized the similarity to the view of the great French mathematician, from whose book The Value of Science he approvingly quoted: “Does the harmony which human intelligence thinks it discovers in Nature exist apart from such intelligence? Assuredly no. A reality completely independent of the spirit that conceives it, sees or feels it, is an impossibility.” The basic idea of the human mind as an active part in the acquisition of knowledge in the physical sciences, or even as the generator of the fabric of the cosmos, was not a result of Eddington’s fundamental theory developed in the 1930s. More than a decade earlier, in the semi-popular Space, Time and Gravitation (p. 200), he wrote: We have found that where science has progressed the farthest, the mind has bur regained from nature that which the mind has put into nature. We have found a strange foot-print on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded reconstructing the creature that made the foot-print. And Lo! It is our own. 16
8. Quantum objections
Eddington’s numerological and philosophical approach to fundamental physics attracted much attention among British scientists, philosophers and social critics in particular. The general attitude was critical and sometimes dismissive, as illustrated by the philosopher Susan Stebbing (1937), who in a detailed review took Eddington to task for what she considered his naïve philosophical views. He was, she said, a great scientist but an incompetent philosopher. According to the Marxist author Christopher Caudwell (1939, p. 121), Eddington was a scholastic who wanted to “extract truth by mathematical manipulation at the expense of experiment.” Leading theoretical physicists preferred to ignore the British astronomerphilosopher’s excursion into unified physics rather than arguing with him. Many may have shared the view of Wolfgang Pauli, who in a letter of 1929 described Eddington’s ideas as “complete nonsense” and “romantic poetry, not physics” (Kragh 2011, p. 109). Pauli referred specifically to Eddington’s identification of the
fine-structure constant α with the number 1/136. A main reason for the generally unsympathetic response to Eddington’s theory in the physics community was his unorthodox use and understanding of quantum mechanics. I shall limit myself to some facets of this issue. Eddington’s critique of the standards employed in quantum mechanics generally fell on deaf ears among experts in the field. One of the few exceptions was a paper of 1942 in which Dirac, together with Rudolf Peierls and Maurice Pryce, politely but seriously criticized Eddington’s “confused” use of relativistic quantum mechanics. As the three physicists pointed out: “Eddington’s system of mechanics is in many important respects completely different from quantum mechanics … [and] he occasionally makes use of concepts which have no place there” (Dirac, Peierls and Pryce 1942, p. 193). The sharp difference between Eddington’s quantum-cosmological theory and established quantum mechanics had earlier been highlighted at a conference on “New Theories in Physics” held in Warsaw and Cracow in June 1938. On this occasion Eddington (1939c) gave a lecture in front of some of the peers of orthodox quantum mechanics, including Niels Bohr, Léon Rosenfeld, Louis de Broglie, Oskar Klein, Hendrik Kramers, John von Neuman, George Gamow and Eugene Wigner (Fig. 4). 17
Fig. 4. Attendants at the Cracow session in 1938, with Eddington in the lower right corner. Reproduced from Niels Bohr Collected Works, vol. 7 (Amsterdam: Elsevier, 1996), p. 261. None of the distinguished quantum physicists could recognize in Eddington’s presentation what they knew as quantum mechanics. Kramers commented: “When listening to Prof. Eddington’s interesting paper, I had the impression that it concerned another quantum theory, in which we do not find the formulae ordinarily used, but where we find many things in contradiction with the ordinary theory.” In the proceedings of the Polish conference one gets a clear impression of how Eddington on the one hand, and Bohr and his allies on the other, failed to communicate. It was one paradigm challenging another, apparently incommensurable paradigm. The attempt to create a dialogue between Bohr and Eddington led to nothing. According to the proceedings, Bohr “thought that the whole manner of approaching the problem which Professor Eddington had taken was very different from the quantum point of view.” And Eddington, on his side, stated that “he could not understand the attitude of Prof. Bohr.” He somewhat lamely, and somewhat pretentiously, responded that he just tried to do for quantum mechanics what Einstein had done for classical, non-quantum mechanics. Eddington realized that he was scientifically isolated, yet he felt that the lack of appreciation of his ideas was undeserved and would change in the future. Near 18 the end of his life he confided in a letter to Dingle (1945, p. 247) that he was perplexed that physicists almost universally found his theory to be obscure. He defended himself: “I cannot seriously believe that I ever attain the obscurity that Dirac does. But in the case of Einstein and Dirac people have thought it worth while to penetrate the obscurity. I believe they will understand me all right when they realize that they have got to do so.” Although the large majority of physicists dismissed Eddington’s theory there was one notable exception, namely Erwin Schrödinger. In papers from the late 1930s the father of wave mechanics enthusiastically supported Eddington’s quantum-cosmological theory (Kragh 1982; Rüger 1988). Yet, his enthusiasm cooled as it dawned upon him that the theory could not be expressed in a language accessible to the physicists. In an essay originally written in 1940 but only published much later, Schrödinger (1953, p. 73) admitted that an important part of Eddington’s theory “is beyond my understanding.” Still today this is the general verdict of Eddington’s grand attempt to establish fundamental physics on an entirely new basis.


Indeterminateness of the space-time frame.

It has been explained in the early chapters of Space, Time and Gravitation that observers with different motions use different reckonings of space and time, and that no one of these reckonings is more fundamental than another. Our problem is to construct a method of description of the world in which this indeterminateness of the space-time frame of reference is formally recognised. Prior to Einstein's researches no doubt was entertained that there existed a "true even-flowing time" which was unique and universal. The movingobserver, who adopts a time-reckoning different from the unique true time, must have been deluded into accepting a fictitious time with a fictitious space-reckoning modified to correspond. The compensating behaviour of electromagnetic forces and of matter is so perfect that, so far as present knowledge extends, there is no test which will distinguish the true time from the fictitious. But since there are many fictitious times and, according to this view, only one true time, some kind of distinction is implied although its nature is not indicated. Those who still insist on the existence of a unique " true time " generally rely on the possibility that the resources of experiment are not yet exhausted and that some day a discriminating test may be found. But the off-chance that a future generation may discover a significance in our utterances is scarcely an excuse for making meaningless noises. Thus in the phrase true time, " true " is an epithet whose meaning has yet to be discovered. It is a blank label. We do not know what is to be written on the label, nor to which of the apparently indistinguishable time-reckonings it ought to be attached. There is no way of progress here. We return to firmer ground, and note that in the mass of experimental knowledge which has accumulated, the words time and space refer to one of the " fictitious " times and spaces—primarily that adopted by an observer travelling with the earth, or with the sun—and our theory will deal directly with these spacetime frames of reference, which are admittedly fictitious or, in the more usual phrase, relative to an observer with particular motion. The observers are studying the same external events, notwithstanding their different space-time frames. The space-time frame is therefore something overlaid by the observer on the external world ; the partitions representing his space and time reckonings are imaginary surfaces drawn in the world like the lines of latitude and longitude drawn on the earth. They do
not follow the natural lines of structure of the world, any more than the meridians follow the lines of geological structure of the earth. Such a meshsystem is of great utility and convenience in describing phenomena, and we shall continue to employ it ; but we must endeavour not to lose sight of its fictitious and arbitrary nature. It is evident from experience that a four-fold mesh-system must be used ; and accordingly an event is located by four coordinates, generally taken as x, y, z, t. To understand the significance of this location, we first consider the simple case of two dimensions. If we describe the points of a plane figure by their rectangular coordinates x, y, the description of the figure is complete and would enable anyone to construct it ; but it is also more than complete, because it specifies an arbitrary element, the orientation, which is irrelevant to the intrinsic properties of the figure and ought to be cast aside from a description of those properties. Alternatively we can describe the figure by stating the distances between the various pairs of points in it ; this description is also complete, and it has the merit that it does not prescribe the orientation or contain anything else irrelevant to the intrinsic properties of the figure. The drawback is that it is usually too cumbersome to use in practice for any but the simplest figures. Similarly our four coordinates x, y, z, t may be expected to contain an arbitrary element, analogous to an orientation, which has nothing to do with the properties of the configuration of events. A different set of values of x, y, z, t may be chosen in which this arbitrary element of the description is altered, bub the configuration of events remains unchanged. It is this arbitrariness in coordinate specification which appears as the indeterminateness of the space-time frame. The other method of description, by giving the distances between every pair of events (or rather certain relations between pairs of events which are analogous to distance), contains all that is relevant to the configuration of events and nothing that is irrelevant. By adopting this latter method Ave can strip away the arbitrary part of the description, leaving only that which has an exact counterpart in the configuration of the external world. To put the contrast in another form, in our common outlook the idea of position or location seems to be fundamental. From it we derive distance or extension as a subsidiary notion, which covers part but not all of the conceptions which we associate with location. Position is looked upon as the physical fact—a coincidence with what is vaguely conceived of as an identifiable point of space—whereas distance is looked upon as an abstraction or a computational result calculable when the positions are known. The view which we are going to adopt reverses this. Extension (distance, interval) is now fundamental; and the location of an object is a computational result summarising the physical fact that it is at certain intervals from the other objects in the world. Any idea contained in the concept location which is not expressible by reference to distances from other objects, must be dismissed from our minds. Our ultimate analysis of space leads us not to a "here" and a " there," but to an extension such as that which relates " here " and " there." To put the conclusion rather crudely—space is not a lot of points close together ; it is a lot of distances interlocked. Accordingly our fundamental hypothesis is that— Everything connected with location which enters into observational knowledge —everything we can know about the configuration setup of events—is contained in a relation of extension between pairs of events. This relation is called the interval, and its measure is denoted by ds. If we have a system 8 consisting of events A, B, G, D, ..., and a system S/ consisting of events A', B', C, D', . .., then the fundamental hypothesis implies that the two systems will be exactly alike observationally if, and only if, all pairs of corresponding intervals in the two systems are equal, AB = A'B' } AC = A'C, .... In that case if 8 and 8'. are material systems they will appear to us as precisely similar bodies or mechanisms ; or if 8 and 8' correspond to the same material body at different times, it will appear that the body has not undergone any change detectable by observation. But the position, motion, or orientation of the body may be different ; that is a change detectable by observation, not of the system 8, but of a wider system comprising S and surrounding bodies. Again let the systems 8 and 8' be abstract coordinate-frames of reference, the events being the corners of the meshes ; if all corresponding intervals in the two systems are equal, we shall recognise that the coordinate-frames are of precisely the same kind—rectangular, polar, unaccelerated, rotating, etc.

The fundamental quadratic form.

We have to keep side by side the two methods of describing the configurations of events by coordinates and by the mutual intervals, respectively —the first for its conciseness, and the second for its immediate absolute significance. It is therefore necessary to connect the two modes of description by a formula which will enable us to pass readily from one to the other. The particular formula will depend on the coordinates chosen as well as on the absplute properties of the region of the world considered ; but it appears that in all cases the formula is included in the following general form— The interval ds between two neighbouring events with coordinates (x1} sc2 , x3 , x4) and (a\ + dxx , x2 + dx2 , x3 + dx3 , x4 + dx4 ) in any coordinate-system is given by ds2 = gu dx^ + g22dx 2 + g33dx3 2 + g44dx4 2 + 2gi2dx1dx2 + 2fg13dx1 dx3 -f 2g14dxl dx4 + 2g23dx2 dx3 + 2g24dx2dx4 + 2g3i dx3dx4 --(21),

fundamental quadratic form

where the coefficients gn , etc. are functions of xlf x2 , x3 , x4 . That is to say, ds2 is some quadratic function of the differences of coordinates. This is, of course, not the most general case conceivable ; for example, we might have a world in which the interval depended on a general quartic function of the dx's. But, as we shall presently see, the quadratic form (2'1) is definitely indicated by observation as applying to the actual world. Moreover near the end of our task (§ 97) we shall find in the general theory of relationstructure a precise reason why a quadratic function of the coordinatedifferences should have this paramount importance. Whilst the form of the right-hand side of (2'1) is that required by observation, the insertion of ds2 on the left, rather than some other function of ds, is merely a convention. The quantity ds is a measure of the interval. It is necessary to consider carefully how measure-numbers are to be affixed to the different intervals occurring in nature. We have seen in the last section that equality of intervals can be tested observationally ; but so far as we have yet gone, intervals are merely either equal or unequal, and their differences have not been further particularised. Just as wind-strength may be measured by velocit}', or by pressure, or by a number on the Beaufort scale, so the relation of extension between two events could be expressed numerically according to many different; plans. To conform to (21) a particular code of measure-numbers must b< adopted; the nature and advantages of this c^de will be explained in th^uext section. The pure geometry associated with the general formula (2'1) was studied by Riemann, and is generally called Riemannian geometry. It includes Euclidean geometry as a special case.

Measurement of intervals.

Consider the operation of proving by measurement that a distance AB is equal to a distance CD. We take a configuration of events LMNOP..., viz. a measuring-scale, and lay it over AB, and observe that A and B coincide with two particular events P, Q (scale-divisions) of the configuration. We find that the same configuration* can also be arranged so that C and D coincide with P and Q respectively. Further we apply all possible tests to the measuring-scale to see if it has "changed " between the two measurements : and we are only satisfied that the measures are correct if no observable difference can be detected. According to our fundamental axiom, the absence of any observable difference between the two configurations (the structure of the measuring-scale in its two positions) signifies that the intervals are unchanged ; in particular the interval between P and Q is unchanged. It follows that the interval A to B is equal to the interval C to D. We consider that the experiment proves equality of distance; but it is primarily a test of equality of interval.

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