CIVIL UNIVERSITY OF CALIFORNIA DEPARTMENT OF CIVIL. ENGINEERING BERKELEY. CALIFORNIA CORRIGENDA Page 4, Line 3 cancel "Meanwhile" "5 "3 from bottom, for R* read R~* "16 "3 from bottom, for (3) ' (4) ^ " 31 " 5 " " " " " & ' 47 " 6 from top' " N b k " N* b " 86 " 5 from bottom, " Appropriate read approximate " 93 " 12-14 from top, insert equation number (76) " 99 " 1, for X read L "ill " 3 " F.-F- " 112 " 10 " four-current read four-potential "120 " 9 " FVB readvWB "121 "10 " F K * " F" x " 122 " 11 from bottom, for (47 d) read (42a) 2 2 " 126 " 11 for - 2 ad # " 140 " 1 " -star " -stars " 141 " 16 from bottom, for 50 read 56 The Theory of GENERAL RELATIVITY and GRAVITATION Based on a course of lectures delivered at the Conference on Recent Advances in Physics held at the University of Toronto, in January, 1921 BY LUDWIK SILBERSTEIN, Ph.D. \ NEW YORK D. VAN NOSTRAND COMPANY EIGHT WARREN STREET 1922 . '-- Library COPYRIGHT, CANADA. 1922 BY THE UNIVERSITY OF TORONTO PRESS PREFACE At the Conference on Recent Advances in Physics held in the Physics Laboratory of the University of Toronto from January 5 to 26, 1921, a course on Einstein's Relativity and Gravitation Theory, consisting of fifteen lectures and two colloquia, was delivered by the author. The first six of these lectures were devoted to what is known as Special Relativity, and the remaining ones to Einstein's General Relativity and Gravitation Theory and to relativistic Electromagnetism. In view of the time limitations only the essentials of these theories were dealt with, due attention, however, being given to the critically conceptual side of the subject. The Univer- sity was kind enough to undertake the publication of that part of the course which dealt with general relativistic ques- tions, on the express understanding that my prospective readers should be assumed to be already familiar with the special theory of relativity. In this connection it was sug- gested by Prof. McLennan that those unacquainted with the older theory should be referred to my book of 1914 (The Theory of Relativity, Macmillan, London) and that it would therefore be desirable to make the present volume, as much as possible, uniform in exposition and style with that work. With such requirements in view this little book was shaped, only a few pages at the beginning having been used in re- calling the essentials of the special relativity theory. The treatment, as compared with the Toronto lectures, has been made somewhat more systematic and the subject matter has, here and there, been considerably extended. In this respect the author has been partly influenced by a larger course on Relativity, Gravitation and Electromagnetism delivered, in the time of writing, during the last Summer Quarter at the University of Chicago. Such is especially the case with Chapter III in which care has been taken to give the readers a systematic exposition of the calculus of generally covariant beings called Tensors. The exposition follows here mainly upon Einstein's own presentation of the subject, with the difference, however, that due emphasis has been laid upon the distinction between metrical and non- metrical properties of tensors. But even in this chapter 494261 technicalities have been avoided, stress being laid upon the guiding principles of this new, or rather newly revived, and most powerful mathematical method. It seems hard to say whether Einstein's admirable theory has or has not a long life before it in the domain of Physics proper. But indepen- dently of its fate the time applied for studying the Tensor Calculus and acquiring some skill in handling it will be well spent. The plan of the remainder of the book will be sufficiently clear from the titles of the chapters and sections arrayed in the table of Contents. Such matter as seemed for the present too speculative and controversial has been relegated to the Appendix where, however, also some points concerning the curvature properties of a manifold have found their place, not only as a preparatory to Einstein's cosmological specu- lations but perhaps as a useful supplement to Chapter III. The book is felt to be far from being complete. But as it is, it is hoped that it will give the reader a good insight into the guiding spirit of Einstein's general relativity and gravita- tion theory and enable him to follow without serious diffi- culties the deeper investigations and the more special and extended developments given in the large and rapidly growing number of papers on the subject. Some of my readers will miss, perhaps, in this volume the enthusiastic tone which usually permeates the books and pamphlets that have been written on the subject (with a notable exception of Einstein's own writings). Yet the author is the last man to be blind to the admirable boldness and the severe architectonic beauty of Einstein's theory. But it has seemed that beauties of such a kind are rather enhanced than obscured by the adoption of a sober tone and an apparently cold form of presentation. My thanks are due to Sir Robert Falconer and to Prof. J. C. McLennan for promoting the cause of this publication, to Prof. R. A. Millikan and Prof. Henry G. Gale of the University of Chicago for reading part of the proofs, and to the University of Toronto Press for the care bestowed on my work. L. S. Rochester, N.Y. November 1921. CONTENTS CHAPTER I Special Relativity recalled. Foundations of General Relativity and Gravitation Theory PAGE 1. Inertial reference systems , 1 2. Special relativity principle 2 3. Principle of constant light velocity 2 4. Lorentz transformation. Galilean line-element. Minimal lines and geodesies representing light propagation and motion of free particles 4 5. Transition to general relativity and gravitation theory. Infini- tesimal equivalence hypothesis and local coordinates < 9 6. Gaussian coordinates, and the general line-element 14 7. Light propagation and free-particle motion expressed by the general null-lines and geodesies 17 CHAPTER II The General Relativity Principle. Minimal Lines and Geodesies. Examples. Newton's Equations of Motion as an Approximation 8. Principles of general relativity; general covariance of laws. ... 22 9. Local and system- velocity of light 25 10. Developed form of geodesies. Christoffel symbols 26 lOa. First example: galilean system 29 lOb. Second example : rotating system 30 11. Geodesies, and Newtonian equations of motion as an approxi- mation 35 CHAPTER III Elements of Tensor Algebra and Analysis 12. Introductory. Gaussian coordinates 39 13. Contravariant and covariant tensors of rank one or vectors ... 40 14. Inner or scalar product of two vectors. Zero rank tensors, invariants 42 15. Outer product. Tensors of rank two, symmetrical and anti- symmetrical. Mixed tensors 43 16. Tensors of any rank v . 46 17. Contraction. Intrinsic invariants 46 18. Inner multiplication. Differentiation of tensors 48 19. Tensor properties in a metrical field. Quadratic differential form or line-element . . .^-. ; .r^ .^ 50 20. Fundamental tensor. Metrical properties of tensors. Norm and size. Conjugate tensors 52 21. Supplement. Reduced tensor 55 22. Angle and volume. Sub-domains 56 PAGE 23.*Differentiation based on metrics. Covariant derivative or ex- pansion; contra variant derivative. Rotation of a vector. Antisymmetric expansion of a six-vector. Divergence of a six-and of a four-vector 59 24. The Riemann-Christoffel tensor. Riemannian symbols and curvature. Lipschitz's theorem 62 CHAPTER IV The Gravitational Field-equations, and the Tensor of Matter 25. Contracted curvature tensor. Einstein's field-equations outside ^ of matter. Bianchi's identities 69 26. Laplace's equation, and Newton's law, as a first approximation 72 27. The tensor of matter. Einstein's field-equations within matter. Laplace-Poisson's equation as a first approximation. Mean curvature and density of matter. Example 74 28. Equations of Matter. The principles of momentum and of energy. Remarks on conservation 81 29. Hamiltonian Principle 88 30. Gravitational waves. Einstein's approximate integration of the field-equations 90 CHAPTER V Radially Symmetric Field. Perihelion Motion, Bending of Rays, and Spectrum Shift 31. Radially symmetrical solution of the field-equations 92 32. Perihelion of a planet. Mercury's excess 95 33. Deflection of light rays. Results of the Sobral Eclipse Expedi- tion 100 34. Shift of spectrum lines. The atoms as 'natural clocks' 102 CHAPTER VI 35. Generally covariant form of the equations of the electromag- netic field 106 36. The four-potential Ill 37. Orthogonal curvilinear coordinates 113 38. Propagation of electromagnetic waves in a gravitational field .. 115 39. Ponderomotive force and energy tensor of the electromagnetic field 119 APPENDIX A. Manifolds of Constant Curvature 124 B. Einstein's New Field-Equations and Elliptic Space 129 C. Space-Time according to de Sitter 135 D. Gravitational Fields and Electrons 137 Index . . 138 CHAPTER I. jcial Relativity recalled. Foundations of General Relativity and Gravitation Theory. In accordance with the purpose and the origin of this volume* its readers are assumed to have already made them- selves familiar with the essentials of Einstein's older or special Relativity. It will be enough, therefore, to recall here very concisely what of that theory may be conducive to, and even necessary for, a thorough grasping of the structure and the aims of the more general theory, and of the spirit pervading it. 1. First of all, then, out of all thinkable reference-frame- works, the special relativity is concerned only with a certain privileged class of frameworks or systems of reference, the inertiol systems. Of these there are < 3 . If 5, say the 'fixed-stars' system, is one of them, any other rigid system S r of coordinate axes moving relatively to S with any uniform and purely translational velocity v, in any direction whatever, is again an inertial system or belongs to the same privileged class. And the systems thus derived from 5, or from one another,** exhaust the class. Since the size or absolute value of the relative velocity implies one scalar datum, and its direction two more such data, all independent of one another, there is just a triple infinity of inertial systems,! as already stated. Not that the special relativity theory abstains from considering accelerated, i.e. non-uniform motion of particles within any of these systems; but it does not contemplate any frameworks other than the inertial ones as systems of refer- *Cf. Preface. **If S f moves uniformly with respect to 5, and 5" w ; th respect to S f , so does S" with respect to S. If the reader so desires, he may consider this as a postulate. ' fThe purely spatial orientation of the axes, implying further free data, is irrelevant in the present connection. 1 * T 3 O^ *-J a* f % RELATIVITY AND GRAVITATION ence, and cannot, nor does it propose to deal with them. It is unable, for instance, to transform the course of phenomena from the S system to the spinning Earth or to an accelerated carriage as reference systems. 2. Keeping this in mind, the first main assumption of the older theory, known as the Special Relativity Principle, can be briefly stated by saying that it requires the laws of physical phenomena to be the same whether they are referred to one or to any other inertial system. In short, the maxim of the 1905 Relativity was: Equal laws for all inertial systems. The italicized words are, mathematically speaking, at first somewhat vague. In fact, they are intended to stand for 'the same form of mathematical equations expressing the laws.' Now, since this implies the use of some magnitudes, such as the coordinates and the time, or the electric and the magnetic vectors (forces), in each of the said systems, the requirement of mathematical 'sameness' remains cloudy until we are told what dictionary is to be used to translate the language of one into that of any other inertial system, or technically, to transform from the non-dashed to the dashed variables. This vagueness, however, soon disappears, giving place to precision, in the next fundamental step of the theory as will be seen presently. The attentive reader might here object by saying that 'sameness of laws' means absence of difference, absence of observable different behaviour (of moving bodies or of electric waves) in passing from an 5 to an S', and that, therefore, to begin with, no mathematical magnitudes or equations are required. But actually we are, perhaps forever, confined to one (approximately) inertial system, our planet, and are thus unable to observe directly the permanence of behaviour in passing tp another system of reference. The only way open to us is to proceed, through more or less long chains of abstract reasoning, from the principle of relativity to some observable prediction, and such processes are scarcely practicable without the use of mathematical symbols and equations. 3. The second assumption, called the Principle of Constant Light- velocity, apart from its own importance, provides for the need just explained, its true office in the structure of the theory being to set an example of a 'physical law' which is postulated to satisfy rigorously the first assumption. It CONSTANT LIGHT VELOCITY 3 runs thus: Light is propagated, in vacua, relatively to any inertial system, with a velocity c, constant and equal for all directions, no matter whether the source emitting it is fixed or moving with respect to that system. This is shortly referred to as uniform and isotropic light propagation in any inertial system. The light velocity, in empty space, plays the part of a universal constant, which role, however, it will readily give up in generalized relativity. The reader is well acquainted with the mathematical expression of the consequence of these two assumptions (together with a tacit requirement of formal equivalence of any two inertial systems S, S f ), to wit, the invariance of the quadratic form c*P-x*-y*-&, (a) where x, y, z are the cartesian co-ordinates and / the time of the 5-system. That is to say, if x', y', z 1 ', t f be the cartesian co-ordinates and the time used in any other inertial system 5', (a) should transform into cW-x't-yV-z'*. (a') As a matter of fact, what was originally required was that the equation (a) should transform into (a') = 0, and this would be satisfied by putting (a')=\ . (a), where X is inde- pendent of x, y, z, t but might be some function of v, the relative velocity of 5", S f . This, however, would amount to a dis- tinction between the two systems, at least a formal one, unless X=l. If, therefore, equal rights are claimed not only physically but also formally, mathematically, for all inertial systems, we have (a) = (a'), that is to say, the quadratic form (a) is raised to the dignity of an invariant. There is, certainly, nothing to object to in such a procedure, especially as it carries simplicity with itself. Yet these remarks did not seem superfluous, especially as there is among the relativists a strong tendency to a certain kind of hypostasy of the said quadatic form* (by declaring it to be more Intensified more recently in the case of the more general (differ- ential) quadratic form playing a fundamental r61e in the newer relativity theory, as will be seen hereafter. 4 RELATIVITY AND GRAVITATION 'objective, real or intrinsic' than space-distance or time) just because it "is" invariant, and forgetting that we have deliberately made it invariant. 4. MawMPtfciie, returning to the quadratic expression (a), let us write it down for a pair of events infinitesimally near to one another in space and time. Thus, writing # lf x%, ac 3 , x 4 for x, y, z, ct, the statements made above can be expressed by saying that the quadratic differential form ds* = dxf-dxi*-dx> i *-dxt (1) should be invariant with respect to the passage from one inertial system 5 to any other such system S'. The differ- ential foim is here preferable to the original one, as it will be helpful in paving the way for general relativity. As is well known, this requirement of in variance gives the rule of transformation of the variables x t into those x\ of the S'-system, called the Lorentz transformation. If both the Xi and the x'\ axes are taken along the line of motion of 5' relatively to 5, with the velocity v = fic, if further the x z , x s axes are taken parallel to those of #'2, #'3, and if the convention x'i = x f i = Q for XI = XI = Q is adopted, the Lorentz transforma- tion assumes the familiar form *'i = 7(*i 0* 4 ), x f 2 = x 2 , x'z = X3, x' 4 = y(xi-pxi) (2) where y = (1 2 ) ~^. Vice versa, we have, by solving (2), showing the complete (including the formal) equivalence of the two systems. Let us keep well in mind, for what is to follow, that this transformation is a linear one, with constant co- efficients, and that special relativity, concerned with inertial systems only, does not contemplate any other space-time transformations. Every tetrad of magnitudes X L (i = l to 4) which are transformed as the x t , is called a four-vector or, after Min- kowski, a world-vector of the first kind. Such four-vectors are, in addition to dx t or x t itself, their prototype, the four- velocity dxjds and the four-acceleration of a moving particle, the electric four-current, and so on. To every vector X l LORENTZ TRANSFORMTAION 5 belongs a scalar or invariant XfX^X'f^-X^, its only invariant with respect to the Lorentz transformation. But we need not stop here to reconsider the properties of the four- vector and other world-vectors, such as the six- vector, which constituted the only lawful material of the older relativity for writing down laws of Nature, especially as we shall soon return to these mathematical entities as particular cases of tensors of various ranks which are indispensable to the general theory of relativity. On the other hand we may profitably dwell yet a while upon the quadratic form (1) itself, the square of the line- element, as ds is called. Granted the assumptions of special relativity, this expression becomes the fundamental quadratic differential form of the four-manifold, the world or space - time, in exactly the same way as is the fundamental form of a flat two-space or surface, and more generally, da 2 = Edu*+ 2 Fdudv + Gdv 2 that of any surface, and da 2 = dr 2 +R 2 sin 2 ^ (sin 2 d8 2 +d 2 ) the fundamental differential form of any three-space of con- stant curvature R~ 2 .~\ Now, it is enough to open any book on differential geometry to see that, with the usual assump- tions of continuity, etc., the whole geometry, i.e., all metrical properties of the two-space or the three-space in question are completely determined by the corresponding differential forms. Their geodesies or, within restricted regions at least, their shortest lines, the angle relations, and their whole trigometry, all this is fully determined provided the co- efficients of the differentials, such as E, etc., appearing in fAccording as R* is positive, zero or negative, we have an elliptic, lidean (or parabolic) or hyperbolic s becomes R sinh (r/R), where R? = R?. euclidean (or parabolic) or hyperbolic space. In the latter case R sin - JK. 6 RELATIVITY AND GRAVITATION the fundamental form are given functions of the variables.* This deterministic mastery of the quadratic differential form has been, as far back as 1860, technically extended to spaces or manifolds of four and, in fact, of any number of dimensions, although, not being sufficiently sensational, it never attracted the attention of anybody beyond a few specialists. In much the same way all the metrical properties of the four-dimensional world of the special relativist sjiould be, and are, derivable from the fundamental form (1) belonging, or rather allotted to it. This is, from the point of view of the poly-dimensional differential geometer, but a very special, in fact, the most simple quadratic form in four variables. For it contains but the squares of their differentials, and the coefficients of these are all constant, which in view of the sequel it may be well to bring; into evidence by writing (1) ds 2 = g^dx.dx., (la) to be summed over i, ic = l, 2, 3, 4, tabulating the co- efficients, thus -1000 0-100 (Ib) 0-1 0001 and calling this array of special coefficients the inertial or the galilean g lK . We shall denote them in the sequel by g t(C . To give this array is as much as to give the form (la), and herewith the properties of the world, for it is manifestly irrelevant how we call or denote the four corresponding variables. The values of the g tK being given, the properties *To be rigorous we should have said ' all properties of a restricted region of the contemplated manifold'; for certain properties of the manifold as a whole are still left free. The choice, however, is limited to a small number of discrete possibilities. Thus, for example, there are two kinds of elliptic space, the spherical or antipodal, and the polar or elliptic proper. In the former the total length of a straight line (geodesic) is 2irR, and in the latter TrR; the planes are two-sided, and one-sided, respectively, and so on. MINIMAL LINES AND GEODESICS 7 of the x will follow by themselves. There is no need to declare beforehand that they are cartesian coordinates of a place and its date. Further, the circumstance that these coefficients are of different signs, three being negative, and one positive, creates for the general geometer no difficulty. This circumstance brings only with it the important feature that there are in the world real minimal lines* as the geometer would put it, that is to say, lines of zero-length, or These special world lines represent the propagation of light or, apart from physical difficulties, the uniform motion of a particle with light velocity c. As a matter of fact the very first step of the theory consisted in writing ds = Q as the ex- pression of light propagation in vacuo. In the next place consider the equally fundamental con- cept of the geodesies of the world. These are defined by the limits of the integral being kept fixed. To derive from this variational equation the differential equations of a geodesic, proceed in the well-known way. If u be any inde- pendent parameter, and if dots are used for the derivatives with respect to it, we have 8fsdu =/&> .du = Q, where, by (1), s 2 = (xf+x-\-xJ-)-\-x?, and therefore, ds = - XtdXi *Whereas on any (real) surface all the 4 minimal lines ' (known also as null-lines), which play in the surface theory an important analytical r61e, are always imaginary. The reader will do well to consult on this and allied topics a special treatise on differential geometry. 8 RELATIVITY AND GRAVITATION By partial integration, and remembering that all 8x t vanish at the limits of the integral, I - x t dx t .du=- \(- x\x t . du. j s J du\ s / Thus, the dx t being mutually independent, the required differential equations are If the geodesic does not happen to be a null-line (light pro- pagation) we can as well take u = s, when $=1, and the equations become whence dx t d t = = a = const. ds dx ds The fourth of these equations is dx*/ds = const. , and therefore, the first three, dt dt ' dt and these represent uniform rectilinear motion, which is the motion of a free particle. Let us, therefore, keep well in mind these two properties of the line-element ds of special relativity: I. The minimal lines of the world, ds = 0, (I) represent light propagation in vacua. II. The world geodesies, defined by d/ds = Q, (II) with fixed integral limts, represent the motion of a free particle. MINIMAL LINES AND GEODESICS 9 A special emphasis is here put on these two properties because they will be carried over to the general relativity and gravitation theory, and because these and principally only these two properties constitute the connection of the otherwise purely analytical differential form ds 2 = g uc dx l dx K with physics. In other words, (I) as the equation of light propagation, and (II) as that of the motion of a free particle impart physical meaning to the mathematical form which is the 'line-element' ds. Without this all the properties of the quadratic form, though interesting, perhaps, in them- selves, would have nothing to do with the world of physical phenomena. It is scarcely necessary to say that the law (II) of the motion of free particles is, as well as (I) for light, invariant (thus far) with respect to the Lorentz transformation. For it is, by its very structure, independent of the choice of a reference system S. Since ds is invariant, so isfds, extended between any two world-points. Thus also the developed form of (II), the system of differential equations d z xJds* = Q, is transformed in S f into d*x, f /ds 2 = Q. And in fact, uniform motion of a particle relatively to 5, means also (originally by an assumption) its uniform motion in any other inertial system S'. In short, the Lorentz transformation leaves the uniformity of motion of a particle intact. 5. We are now ready to pass to Einstein's theory of general relativity and gravitation. Not that our task is an easy one, but we are somewhat better prepared to embark upon it. Why equal form of physical laws, why equal rights for the inertial systems only? Why not equal rights for all (systems)? Such would be the urgent, and yet vague, ques- tions naturally suggesting themselves after what was said in the preceding sections. Yet it is not with these questions, nor with an attempt to answer them, that we will begin our journey across this new and revolutionary country. For, even if answered, these questions would remain physically barren were it not for the existence of gravitation and 10 RELATIVITY AND GRAVITATION especially of a certain peculiarly simple property of this universal agent. This, therefore, will first occupy our attention for a while. The cardinal feature of gravitation just hinted at is the pro- portionality of weight to mass, in other words, the proportion- ality of heavy (gravitating) and inert mass. First tested by Newton in his famous pendulum experiments with bobs of different material, and carried to further precision by Bessel, this proportionality has been more recently shown by Roland Eotvos to hold to one part in ten millions. It is reasonable, therefore, to assume, with Einstein, that it holds rigorously,* at least until proofs to the contrary are forthcoming. In our present connection it is better to express this property more directly by saying, even with Galileo, that all bodies, light or heavy, fall equally in vacuo. All particles, that is, acquire at a given place of a gravitational field equal accelerations independently of their own mass or chemical nature, etc., and no matter how much of their inertia is due to the energy stored in them and how much of other origin. This remark- able property distinguishes the gravitational field from other fields. Take, for instance, an electric field given by the vector E. The force on a particle of rest-mass m, carrying the electric charge e, and starting from rest, is eE, and the accelera- tion eE/m. Now, in general, there is no relation between m and e, and even if the mass is purely electromagnetic, when m is proportional to e z /a, the acceleration will vary from particle to particle inversely as its charge and directly as its average diameter, 2a. We have disregarded, of course, the dielectric properties of the particle which would make its behaviour in a given electric field still more complicated. The same remarks would hold, mutatis mutandis, for the behaviour of different bodies placed in a magnetic field. In short, gravita- tion is, in this respect, unique in its simplicity. *In a theory of matter and gravitation proposed by G. Mie, Annalen der Physik, vols. 37, 39, 40 (1912 and 1913), the proportionality between weight and mass does not hold rigorously, though to an order of precision much exceeding that stated by Eotvos. EQUIVALENCE HYPOTHESIS 11 This very circumstance enabled Einstein to undertake his mental experiment with the falling or ascending elevator, now so familiar to the general public. In fact, consider a homogeneous or a quasi-homogeneous gravitational field such as the terrestrial one in a properly restricted region. Let a lift or elevator, small compared with the earth, yet ample enough for a physical laboratory and for those in charge of it, descend vertically with the local terrestrial acceleration g. Then all bodies placed anywhere within the elevator and left to themselves will float, in mid-air or better in vacuo, and particles projected in any direction will move uniformly in straight paths relatively to the elevator. Moreover, all objects, including the physicists, standing or lying about will cease to press against the floor or the tables, as the case may be. In short, all traces of gravitation will be gone,* and the inmates of the lift, assumed to have no intercourse whatever with the outer world, will declare their reference system to be a genuine inertial system, so far, at least, as mechanical phenomena are concerned. For an unbiassed judge cou Id not tell beforehand whether it will be also optically inertial, that is to say, whether the law of constant light velocity will hold good for the lift. Einstein thinks it will, or rather assumes it, more or less implicitly. If this be granted, we can say that the elevator will be an inertial reference system in every respect. The possibility of thus undoing a gravitational field is manifestly based on the said equal behaviour of all bodies placed in it. For otherwise the artificial motion of the elevator could not be adapted to all bodies at the same time, each of them requiring a different acceleration. Next, pass to any, non-homogeneous gravitation field, which in the most general case may also vary with time. This certainly cannot be undone, as a whole, by a single elevator as reference system. But you can imagine an ever increasing number of sufficiently small elevators, each appropriately accelerated, fitted into small regions of the field, and each, *Vice versa, in absence of a gravitational field, a lift in accelerated ascending motion would give us a faithful imitation of such a field. 2 12 RELATIVITY AND GRAVITATION perhaps, to do its duty for a very short interval of time, and to be replaced by another in the next moment. These minute elevators will do their office at least in the mechanical sense of the word. Einstein assumes that they will act as inertial systems also in the optical sense of the word, as explained above. This process of subdividing a gravitational field, in space and time, and fitting in of appropriate small elevators can be carried on to any required degree of approximation. In fine, passing to the limit, let us make, with Einstein,* the explicit assumption: With an appropriate choice of a local reference system (i, u 2 , Us, u^) special relativity holds for every infinitesimal four -dimensional domain or volume- element of the world. That is to say, at every world-pointf a system of space- time coordinates u\, u%, u 3 , u can be chosen in which the line- element assumes the galilean form ds 2 = du? - dui* - du^ - du^. (3) These four coordinates are called local coordinates. With respect to this local system there is then no gravitational field at the given world-point, and in accordance with special relativity ds 2 has^ there a value independent of the 'orienta- tion ' of the local axes; that is to say, the quadratic form (3) is invariant with respect to the Lorentz transformation (2) . It is this assumption which can now be properly referred to as the infinitesimal equivalence hypothesis, for it grew out of Einstein's original equivalence hypothesis applied to finite regions when, in his first attempt at a theory of gravitation (1911), he was confining himself to a homogeneous field. Whatever the origin of this hypothesis or assumption, it is certainly not difficult to adhere to it. For it scarcely amounts to anything more than to assuming, in the case of a curved surface, say, the existence of a tangential plane at any of its *A. Einstein, Die Grundlagen der allgemeinen Relativitatstheorie . Annalen der Physik, vol. 49, 1916, p. 777. fWith the possible exception of some discrete points, such perhaps as those at which the density of matter acquires enormous values. INFINITESIMAL WORLD FLATNESS 13 points, or to declare the surface to be (in Clifford's termino- logy) elementally flat. And it will, perhaps, be well to restate shortly Einstein's hypothesis by saying that it assumes the four-dimensional world to be, in presence as well as in absence of gravitation, elementally flat. It will not be forgotten, how- ever, that this geometric term is nothing more than a synonym of elementally galilean, i.e., satisfying special relativity in- finitesimally.* To avoid the danger of any misconception let us dwell upon this subject yet for a while. The coordinates u t with their corresponding galilean line-element (3) were set up only for a local purpose, their real office being confined to a fixed world-point P, say Xi, Xz, x 3 , #4 (in any coordinate system). If we so desire, we may think of a whole galilean world [/determined throughout, to any extent, by the simple form (3) . But as a tangential plane has something in common with a surface only at the point of contact and then diverges from it, ceasing to represent any intrinsic properties of the surface itself, so has the auxiliary and fictitious world U anything to do with the actual world W (complicated by gravitation) at the world point x t only. The fictitious world U is tangential to the actual world W at that point, and parts company with it beyond the point of contact. At other world-points the role of U is taken over by other and other fictitious galilean worlds. One more cautious remark. The contact of U and W is one of the first order, i.e., such as the contact between a surface and its tangential plane or between a curve. and its tangential line, but not as the more intimate contact bewteen a curve and its circle of curvature (which is of the second order). This circumstance may acquire some importance later on. *As to the concept of elementary flatness of a surface or a more-dimen- sional space, it is beautifully explained in W. K. Clifford's ' Philosophy of Pure Sciences', published in his famous Lectures and Essays (Macmillan, London). Notice that in Clifford's sense every regular surface, no matter how curved, is elementally flat, with the exception of some singular points, such as the vertex of a cone. 14 RELATIVITY AND GRAVITATION 6. Having thus made clear the local character of the w t coordinates, let us now introduce any coordinate system x t whatever, to be used as a reference system of coordinates for the whole world, i.e., throughout the gravitational field and through all times. Then, if x t be the reference coordinates of P, and x t + dx t those of a neighbour-point u t + du t , the differentials du t will in general be linear homogeneous func- tions of all the dx t , say 4 du L = S a u dx K , K=l or with the conventional abbreviation, du t = du,. dx K , (4) where the coefficients a^ will in general be functions of all the x t . It is of importance to note that that the relations (4) will, generally speaking, be not-integrable, or borrowing a name from dynamics, non-holonomous, that is to say, the a^ will not necessarily be du t /dx K , the differential expressions on the right of (4) will not be total differentials of functions of the x t , and there will be no finite relations between the local and the general or reference-coordinates. Substituting (4) into (3), collecting the terms and calling gut the coefficient of the product of differentials dx t dx K we shall have, for the line-element in the general reference system, ds 2 = g lK dx t dx K , (5) where g uc = g lct will, in the most general case, be functions of all the x t . But since ds*, as defined originally by (3), was in- dependent of the orientation of the local system of axes, so also will the ten different coefficients g tK , though functions of the coordinates x t , be manifestly independent of the orienta- tion of the local system. The line-element will thus be represented, in any reference coordinates x t whatever by the most general quadratic differen- tial form of these four variables, such in fact being the form (5). As before the summation sign is omitted; the sum- mation is to be extended over t, K from 1 to 4, each of these GENERAL TRANSFORMATIONS 15 suffixes, i and /c, appearing twice. Thus, 2gi 2 dxidx 2 + . . . -\-gudxf. The reader will soon learn to handle this abbreviated and very convenient symbolism. Suppose now we introduce instead of x t any other set of space-time coordinates x t ', any functions whatever of the x t , such, that is, that between the two sets exist any given holo- nomous relations #i = & (*i'i #2', #3', xi), (6) the t being any functions whatever, but continuous together with their first derivatives and such that their Jacobian, the well-known determinant xi dxi dx\ J = (7) d* 4 ' does not vanish. Under these circumstances we have and vice versa, dx K (8) (8a) and, as it may be well to notice in passing, //' = !, where /' is the inverse Jacobian dx! dx K Now, substituting (8) into (5), gathering again the terms, and denoting the coefficient of dx t 'dx K ' by g lK r , i.e., putting ' = d^q dffff /g\ we shall have which is (5) reproduced in dashed letters. Not that the &/ will be functions of the #/ of the same form as were the g u of 16 RELATIVITY AND GRAVITATION x lt but only that the quadratic differential form remains quadratic. There is certainly nothing surprising in this kind of permanence.* Yet this and this only is justifiably meant when we say that the line-element ds 2 is invariant with respect to any transformations whatever. If the relativist sees anything more in "the invariance of ds", namely that ds is something belonging to a pair of world-points (*, and jc t + dx t ) inherent in that pair independently of the choice of a reference system, it is what he puts into it at a later stage by ascribing to it certain physical properties, or by inter- preting it physically in certain ways. The meaning of these remarks will gradually become more intelligible. Before passing on to the two cardinal virtues conferred upon the line-element, one more mathematical remark about it may not be out of place just here. Suppose the line- element (5) is actually given with some determined and more or less complicated functions as the g lK . By trying, in succes- sion, other and other new variables x t f we would arrive at a great variety of new forms of functions g lK f . The natural question arises: Are there not among all these sets of co- ordinates just such as would convert (5), throughout the world or a finite world-domain, into a galilean line-element, i.e., one with constant coefficients? The answer is, in general, in the negative. A given form ds 2 = g tK dx t dx K is equivalent, that is to say, can be reduced by holonomous transformations, to a form with constant coefficients and thus also to the galilean line-elementf when and only when certain differential expressions formed of the g w , their first and second deriva- tives, all vanish.* These expressions, of which more will be *Notice that the case is different in special relativity, where we require the form to reappear with all its original coefficients, three 1, and one+1. fThe circumstance that three of the coefficients of this form are negative and one positive imposes on the original g dx dx to be thus transformable certain further conditions in connection with the so-called 'law of (alge- braic) inertia', due to Sylvester. *The restriction to ' holonomous transformations ' is of prime importance. For by means of non-holonomous or non-integrable relations, such as every g^ dx t dx K can be transformed into a quadratic differential form with constant coefficients. RIEMANN'S SYMBOLS 17 said in the sequel, are known in general differential geometry as Riemann's four-index symbols. Of these symbols there are in the case of any number n of dimensions, 2 (w 2 1) linearly independent ones. Thus an ordinary, two-dimensional, surface has but one Riemann symbol and this is its Gaussian curvature, multiplied by gng 2 2 gi2 2 , the determinant of the g lK . Any three-dimensional manifold has six, and our, or rather Einstein-Minkowski's world has as many as twenty linearly independent Riemann symbols. Thus any finite domain of the world is equivalent to a galilean domain when and only when all these twenty symbols vanish in that domain, i.e., when the ten different g llc satisfy within it a system of twenty partial differential equations of the second order. (It will be useful to keep in mind the last italics.) By what has just been said it is manifest that if all the Rie- mann symbols vanish in one system of coordinates x it they will vanish also in any other xj obtained from the former by any holonomous transformations whatever. But enough has for the present been said on the symbols of that great geometer. Later on they will be seen to play an all-important role in Einstein's gravitation theory. It is now time to return to the physical aspect of our subject. 7. Having assumed, after Einstein, that special relativity holds for every infinitesimal domain, or that the world is elementally galilean, we wrote down the simple form (3) in local coordinates u t . Then, passing to any coordinates x t by means of the non-holonomous relations (4) we obtained for the line-element of the world the general quadratic differential form (5), with variable coefficients g^, functions of the x t . But what is the physical meaning of this general ds with all its ten different g llc ? What are they to represent physically? The answer is that we are still to a certain extent the masters of the situation, and can make them have that physical meaning which we will put into them. For thus far we know only the physical meaning of the galilean element belonging to a world U, and that (in virtue of an assumption) the world 18 RELATIVITY AND GRAVITATION W as a seat of or deformed by gravitation is galilean in its elements, or that at each of its points a /-world tangential to it can be constructed. At this stage then we are entitled only to say that (since W without gravitation is /, and since to U belong the constant coefficients ~g lK ) the essential differences* in the coefficients of the two worlds, g^ and ~g tK , are due to, or better, are somehow connected with gravitation. But exactly how, we cannot, thus far, say. For our position is somewhat like this : Suppose we know that a surface cr, which is not a plane as a whole, is elementally flat and thus has a tangential plane TT at each of its points. Suppose further we know the physical properties of certain lines (straights, or circles, etc.) drawn on any TT. Does this alone enable us to say what the physical properties of similarly defined lines will be when drawn on o-? Clearly not. For the 7r-lines have but a single point of contact with cr, and that only of the first order, and deviate from the surface or become extra-a beings all around the point of contact. Now, in the case of space- time, we fixed the physical meaning of the line-element of the /-world by declaring its minimal lines, ds = Q, to be the law of light propagation, and its geodesies, dfds = 0, to represent the motion of free particles. Does this, and the existence of a tangential U at every point of the actual world W, entitle us to assert that the minimal lines and the geodesies of W will again represent the optical and the mechanical laws in this world? This is by no means a superfluous question. For the auxiliary tangential world U leaves the actual world beyond the point of contact and becomes at once fictitious or extra-mundane, so to speak. Now, the minimal lines of /,f defined by a differential equation of the first order, are also, at P, minimal lines of W, so that at least the starting elements of these lines are identical. At the next element the role of U is taken over by another *i.e., those, at least, which cannot be abolished by holonomous co- ordinate transformations. fWhich fill out only a conic hypersurface (of three dimensions) with the contact point P as apex. GEODESICS AND LAW OF MOTION 19 galilean world; yet the reasoning can be repeated, so that we can say that every element of a minimal line of W represents light propagation, and thence deduce that such a HMine possesses also as a whole the same physical property. But the position is altogether different with the geodesies. For these world-lines are defined by differential equations of the second order.* so that the mere contact of U and W (being of the first order) does not at all entitle us to transfer any properties of the geodesies of U upon those of W, not even at their very starting point P. If, however, the said physical property of the PF-geodesics does not follow logically from the previous assumptions, yet we are free to introduce it as a further explicit assumption. In fact, while thus generalizing the physical significance of the geodesies Einstein is well aware that this is a new assumption,! although one that easily suggests itself. Nor is there any inconsistency in thus transfering a property from the galilean to the more general world-geodesies. For, as we shall see later on, the developed form of the equations of the geodesies contains only the g^ and their first derivatives with respect to the x tj whereas the conditions characterizing a world as galilean (the vanishing of the Riemann symbols) are equations between the g lKJ their first and second derivatives, and there are no relations at all between the g lK and their first derivatives alone. But even with this new assumption, the total number of assumptions of Einstein's theory is remarkably small. And as to the advisability of making the one just discussed, we may say that Einstein's theory owes to it the greater part of its power. The property of the geodesies being thus assumed, and that belonging to the minimal lines being deducible from what preceded, we are now in the positibn to sum up definitely and very concisely, if not the whole, yet the most fundamental part of Einstein's theory. For this purpose we have only to *A geodesic issues from P in every direction whatever in the four- manifolds U and W. fA. Einstein, loc. cit. t p. 802. 20 RELATIVITY AND GRAVITATION repeat the previous statements I. and II. without their restrictions, replacing the galilean ds by the general one and adding a few explanatory words: Thus: The world-line element, in any system of coordinates, and whether gravitation be absent or present, is given by ds* = g lK dx t dx K , (10) where g = g are some functions of, in general, all the Jour coordinates, but of these alone. If these ten functions be given, all metrical properties* of the world are determined, and among these its minimal lines, ds = 0, (I) and its geodesies, 0. (II) The physical significance of these world-lines is that the former represent propagation of light in vacuo, and the latter the motion of a free particle. By a 'free' particle is meant one which, having received any initial impulse is left to its own fate, whether in absence or in proximity of other lumps of matter (absence or presence of 'gravitation'), but not colliding with them, and in absence of, or better not immersed in, an electromagnetic field. One strives in vain to enumerate all the attributes of a concept which can become clear only a posteriori, through the concrete applications of the theory. Suffice it to say that ' free particle ' may as well stand for a projectile, in vacuo, or a planet circling around the sun. Their laws of motion are given by the corresponding world-geodesies. The developed form of the equations of the geodesies, as well as of light propagation, will be given later on. Since the g^ are to determine, through (II), the fall of projectiles and the motion of celestial bodies, it is scarcely necessary to repeat that they are intimately connected with gravitation. These ten coefficients will replace the unique scalar potential of newtonian mechanics. They will influence *Apart from some properties of the world as a whole, of which more later on. FREE PARTICLES AND LIGHT 21 also, through (I), the course of light in interplanetary and interstellar spaces, and finally, by their very appearance in the line-element, they will mould the geo- and chrono-metrical properties of our world. These latter properties thus appear intimately entangled with gravitation and optics. It remains to explain how these all-powerful coefficients are, in their turn, determined in terms of other things such as the density of 'matter'. This is the office of Einstein's 'field-equations' which will occupy our attention in the sequel. CHAPTER II. The General Relativity Principle. Minimal Lines and Geodesies. Examples. Newton's Equations of Motion as an Approximation. 8. Most readers will perhaps be surprised to find in the first chapter almost no mention of the general principle of relativity which claims equal rights for all systems of co- ordinates, and which in all publications on our subject is given the most prominent place. Instead of this we insisted on the general form of the line-element (10), on the null-lines and the geodesies of the world metrically determined by that line-element, and still more upon the physical meaning of these two kinds of world-lines as representing light propa- gation and the motion of free particles. The reason for adopting this plan is that, as far as I can see, these things are most important from the physical point of view, nay, they are perhaps* the only relevant constituents of the new theory looked upon as a physical theory. This is particularly true of the optical and mechanical meaning attributed to the said two kinds of lines, thus giving what the logicians call a concrete representation of what otherwise would be only a purely mathematical or logical science, an abstract geometry of a manifold of four dimensions deter- mined by that quadratic differential form. It is exactly this physical interpretation which invests the theory with the power of making statements of a phenomenal content, of predicting the course of observable events. On the other hand, the much extolled Principle of General Relativity which, in Einstein's wording, f requires The general laws of Nature to be expressed by equations valid *Apart from ' the field equations ', yet to come. \Loc. cit. t p. 776. 22 GENERAL RELATIVITY PRINCIPLE 23 in all coordinate systems, i.e., covariant with respect to any substitutions whatever (generally covarianl), is by itself powerless either to predict or to exclude anything which has a phenomenal content. For whatever we already know or will learn to know about the ways of Nature, pro- vided always it has some phenomenal contents (and is not a merely formal proposition), should always be expressible in a manner independent of the auxiliaries used for its descrip- tion. In other words, the mere requirement of general covariance does not exclude any phenomena or any laws of Nature, but only certain ways of expressing them. It does not at all prescribe the course of Nature but the form of the laws constructed by the naturalist (mathematical physicist or astronomer) who is about to describe it. The fact that some phenomenal qualities are technically (with our inherited mathematical apparatus) much more difficult to put into a generally covariant form than some others does not in the least change the position. To make my meaning plain, let us take the case of plane- tary motion. For the sake of simplicity let there be but a single planet revolving around the sun. It is well-known that according to Newton the orbit of the planet should be a conic section, say an ellipse with fixed perihelion.* It is, in our days, almost equally well known that according to Einstein's theory the perihelion should move, progressively, showing a shift at the completion of each of its periods. And so it does, at least to judge from Mercury's behaviour. At the same time Einstein's equations are generally covariant, while Newton's 'law' or Laplace- Poisson's equations are not.f What of this? Does it mean that fixed perihelia are excluded or prohibited by the principle of general covariance? Cer- tainly not. Provided that 'fixed perihelion' and 'moving perihelion' have, each, a phenomenal content, and this they do, both kinds of planetary behaviour should be expressible in a generally covariant form. Newton's inverse square law and his equations of motion certainly do not express it so, *Fixed, that is, relatively to the stars. fNot even with respect to the special or the Lorentz transformation. 24 RELATIVITY AND GRAVITATION and it may be difficult to find a covariant expression for a strictly keplerian behaviour. But if it were urgently needed, some powerful mathematician would, no doubt, succeed in constructing it. If, as actually is the case, Einstein's theory excludes a fixed perihelion, and other newtonian features, it does this not in virtue of the said principle alone (nor even in part), but pre-eminently owing to the physical meaning ascribed to the world-geodesies, and to the choice of his field equations which again are physically relevant since they determine the g iK influencing essentially the form of those world-lines. That the principle of general relativity turned out to be helpful in guessing new laws (by limiting the choice of formulae) is an altogether different matter. It may prove an even more successful guide in the future. f But here its role ends, always taking the Principle only as a mathematical requirement of general covariance of equations. And so it is at any rate enunciated (and interpreted, cf, p. 776, loc. ciL) by Einstein himself, although some of his exponents put into it a physical meaning. In fact, as we shall see later on, the sameness of form of the equations (of motion, say) in two reference systems, as in a smoothly rolling and a vehemently jerked car, does not at all mean sameness of phenomenal behaviour for the passengers of these two vehicles. So much in explanation of the absence of the general principle of relativity in all our preceding deductions. It will be noticed, however, that although no explicit mention of this principle has been made in Chapter I, yet the fundamental laws (I) and (II) there given do satisfy this principle. In fact, both the null-lines and the geodesies of the world were defined without the aid of any reference system. And as to the line-element itself, its invariance was seen to be automatic. Thus, in what precedes we have, without insisting upon it, been faithful to the formal principle of general relativity. Nor is it our intention to depart from it in what will follow. fOr it may become sterile to-morrow, as is the fate of almost all our Principles. LIGHT VELOCITY 25 As was already mentioned at the close of the first chapter, to make the exposition of the fundamental part of Einstein's theory complete, it remains to add to (10), (I), (II), together with their optical and mechanical meaning, a set of equations determining the ten coefficients g^ of the quadratic form. But before passing to these differential equations, Einstein's field-equations, it will be well to discuss somewhat more and to develop those already given. Some explanations and examples concerning the transformation of coordinates will also be helpful at this stage. 9. First, concerning the law of propagation of light (in vacuo), to obtain its developed form it is enough to sub- stitute the line-element (10) into the equation (I) of the minimal lines. Thus the fundamental optical law will be g^dx.dx^O. (11) It gives the velocity of light for every direction of the ray, i.e., of the infinitesimal space-vector dx\, dx 2 , dx s , if dx^/c be the time element of the reference system. In general the light velocity will differ from c and have different values at different world points and for different directions of the ray. This "light velocity" which has nothing intrinsic about it is to be distinguished from the local velocity of light (that corresponding to a local, galilean system of coordinates) which is the same for all directions. To avoid confusion the former may be called the system-velocity of light or, according to some authors, the 'coordinate velocity' of light. It is a kind of velocity estimated from a distant standpoint. If we write it, in a given reference system, da _ da ~Jt~ C dx~t' the very concept of such a light velocity, whose value is to be derived from (11), presupposes that 'the length' da of the infinitesimal space-vector dx it dx^, dx 3 has been defined in some way for that system in terms of these differentials and the coefficients g llc . We shall have the best opportunity of 26 RELATIVITY AND GRAVITATION explaining how this is done technically in deducing physical results, when we come to speak of the bending of rays of light around a massive body such as the sun. Then also the question will be mentioned under what circumstances the law of Fermat, giving the shape of the rays, is applicable. In the meantime it is advisable to look upon (11) as the equation of the infinitesimal wave surface at the instant t-\- dt corresponding to a light disturbance started at Xi, x z , x z at the instant /, the differential dx^ being treated as a constant parameter. From the local standpoint this surface is, of course, a sphere, but from the distant (or system-) standpoint it may have a variety of more complicated shapes. It would, perhaps, be rash to say that it will be a quadric. But, being locally closed, it may also be expected to be a closed surface from the system-point of view. 10. Next for the geodesies of the world. The developed form of their differential equations is easily derived from their original definition (II), As in the case of a galilean world, let u be any parameter, and let dots stand for derivatives with respect to it. Then fds .du = Q, where, in the most general case, s 2 = *.*. (12) The variation of s can be written 8s = B Xl + dx t , dx t dx t to be summed over t = 1 to 4. Thus, by partial integration of the second terms, the limits of the integral being fixed, d,_ /j9s \ _ ds_ = 0, du V dx t / dx t and by (12), with 5 itself taken for u, GEODESICS 27 \ _ dg a(i dx a / " d# Js or dx ds ^_l_ dg^^t _ i ^^ = Q ds 2 d#x ds ds dx t ds ds Introducing the expressions, known as Christoffel's symbols, fan = /a^ a as, _ ^ = r^n LTJ V^ ajc a dx y / LTJ we can condense the last set of equations into dx a dxp _ = These are four linear equations for the four d?x K /ds z . Let us solve them for these derivatives. Denoting the second term by a lt and writing g for the determinant of the g l(e , we shall have 0, etc. or, if g" c = g Kt be the minor of g, corresponding to g lK , divided by g itself, -f- dig 11 H~ Q-zg 12 ~f~ &3g 13 ~f~ #4& 14 ~ 0, etc., i.e., t ap = Q ds 2 ' L J ds ds ~ Here we will write, after Christoffel, -3 (14) 28 RELATIVITY AND GRAVITATION Thus, ultimately, the differential equations of the geodesies or the equations of motion of a free particle will be, in any system of coordinates, ** (o01^ dx, ds* I O ds ds These are four equations. But since we have, identically, dx t dx K &IK - ' - := 1> ds ds one of these equations of motion is a consequence of the remaining three, a feature already familiar to the reader from special relativistic mechanics. Since these differential equa- tions are only the developed form of dfds = 0, they will mani- festly be generally covariant, that is to say, in any new coordinates xj the equations (15) will be d 2 xj , | a/3) ' ^r + W ds ds If the coefficients g lK are all constant, all the Christoffel symbols \ f vanish and the equations (15) reduce to d 9 xjds* = 0, which represent uniform rectilinear motion. And since the general equations (15) represent the motion of a free particle in any gravitational field and in any system, the symbols < >, built up of the g iK and their first derivatives, can be said to express the deviation of the motion from uniformity due to gravitation, and partly due to the peculiari- ties of the system of reference. In view of this property, and disregarding any distinction between gravitation proper and the effects of the choice of the coordinate system,! Einstein *This form of the equations of a geodesic of a manifold, of any number of dimensions, has been used by geometers for a long time. See, for instance, L. Bianchi's Geometria differ enziale, vol. I, Pisa 1902, p. 334. fOr between permanent acceleration fields and such that can be trans- formed away. CHRISTOFFEL SYMBOLS 29 proposes to call these Christoffel symbols ' the components of the gravitational field'. Notice, however, that if all < > vanish in one system of reference they do not necessarily vanish in other systems* (even if obtained from the former by holonomous transforma- tions). In view of this circumstance the name proposed by Einstein seems utterly inappropriate and misleading, even if one agreed not to distinguish between permanent fields and such that can holonomously be transformed away, as for instance the 'centrifugal force'. 10a. In fact, consider for example the galilean line-element in three dimensions, i.e., for $ = const. = 7r/2, taking c/, r, as x 4 , #1, # 2 respectively. Calculate the corres- ponding Christoffel symbols. Since gn= 1, 22= f 2 , 44= 1, and all other g tK vanish, we have, for instance, the non-vanish- ing symbol (22] But who would call it a ' component of the gravitational field ' ? This case is a particularly drastic one, for the world-geodesies corresponding to our line-element do represent uniform recti- linear motion. The appearance of non-vanishing Christoffel symbols is simply due to the use of polar instead of cartesian co-ordinates. In short, gravitation certainly contributes to the Chris- toffel symbols, but so does also a mere transformation of space-coordinates, although it has nothing whatever in common with 'gravitation' of the permanent or the non- permanent kind. This criticism does not in the least diminish the value of the general equations of motion (15). It is given here only to prevent misconceptions which have seemed particularly likely in the case of beginners. *In the terminology of the tensor calculus, to be explained later on, the Christoffel symbols are not the components of a tensor. 30 RELATIVITY AND GRAVITATION lOb. Let us take yet another simple example, this time not for the sake of criticism but because of its instructiveness. Consider the line-element arising from the galilean one, just quoted, (') ds 2 = dx'S - dr' 2 - / W 2 , by the transformation 0' = 8+i-* r > i = -^ again seven in number. Substituting these Christoffel symbols into (15), with i = l, 2, 4 (for r, 0, xi = ct), we have the equations of the world-geodesies, i.e., the equations of motion of a free particle in the system S, ROTATING SYSTEM 31 r = (l-2coV)(0+co* 4 ) (17) In where the dots stand for derivatives with respect to s. virtue of the identical equation s= 1, i.e., (l-r 2 co 2 )* 4 2 - r 2 r 2 2 -2cor 2 0* 4 = 1, (18) one, say the third of (17), should be a consequence of the remaining two.* Thus, the proper equations of motion in the S- system being the first two alone, we can use (18) to eliminate from them # 4 , and to replace d/ds by d/dt. In the first place, to see the approximate meaning of these equations of motion, consider the case of small velocities dr/dtj rdd/dt (as compared with c), and of small values of cor. [Notice that, by (16), co = coc is an angular velocity, in its dimensions at least, so that cor = ur/c is a pure number.] Thus ds=^dx^ = cdt, 4 =Fl, and the approximate equations of motion of a free particle in 5 are dr z dt* = -2 dr_ dt j + dt In Cartesians, # = rcos0, y = identical with dy /, these equations are (b) d*y A - dx = coy 2, co dt* dt The reader will recognize at once in the right hand member of equation (a) or in the first terms of (b) the purely radial centrifugal acceleration (or 'force' per unit mass), provided, *The verification may be left to the reader as an exercise. 32 RELATIVITY AND GRAVITATION of course, that he is at all willing to interpret o>, in accordance with the transformation 0' = 0-fo>/, as the angular velocity of the system S (say, plane disc) relatively to the galilean S f . The second terms of (b) express then the Coriolis acceleration. If we so desire we may, with Einstein, reckon these accelerations to the gravitational ones, especially if we are confined to the (rotating) system S. The centrifugal acceleration, at least, is radial, though away from the origin. The Coriolis acceleration, however, is perpendicular to the velocity and, therefore, generally oblique. Certainly we have in (17) a field of acceleration, but the only feature this has in common with a gravita- tional field is that all bodies placed in it will behave alike. But unlike gravitational fields they cannot be deduced from the distribution of matter. Yet Einstein would not like to have us distinguish them from gravita- 1 22 i 1 24 1 1 44 1 tional fields. If so, then 1 1 f i | f "i 1 f contribute to the centrifugal, ( ) ( ) and I 2 ( ' I 9 ( t0 the Coriolis field - But unti l we are to l d now to derive these non-permanent 'fields' as gravitational effects of all the masses of the universe turning around S* all this will be an idle question of pure nomenclature. We may leave it here for the present. In the second place, returning to the rigorous equations (17), consider a particle, placed (by an 5-inhabitant) at any point r , 6 of the disc S and left there, at the instant / = 0, to its own fate. If it is nailed down it will, of course, remain there for ever, being simply part of this reference system. But let it be a free particle from / = onwards. In short, let r= = 0, for / = 0. Then, by (17), we shall have, for that instant, 6 =0 so that the particle will not evince any tendency of moving transversally, and ds* d /. dr\ = -1*4 - I = dx 4 \ dt / By (18), 4 2 = (1 r 2 cu 2 ), and since r = 0, the last equation will become, rigorously, and always for / = 0, dt 2 *This was tried by H. Thirring but not very successfully. ROTATING SYSTEM 33 In fine, our particle will initially experience the familiar centrifugal acceleration.* It will fly off, for an S'-observer at a (straight) tangent, but from the 5-standpoint at a spiral-shaped orbit. This is perhaps the clearest way of stating the relation of our system 5 to the galilean S'. The reader need not, however, think of 5 at this stage as a material rigid disc rotating uniformly with respect to the fixed stars, although a uniform rotation is just one of the possible motions of a relativistically rigid body (Born, Herglotz). Notwithstanding that S was called, in passing, a disc, it will be safer to treat it here simply as a system derived from S' by the trans- formation (16) with co as constant. As to the orbit of a free particle relatively to 5, its equation could be derived, not without some trouble, from the differ- ential equations (17). This, however, can be done much easier by transforming the orbit from S' to 5. In fact, the former being a galilean system, a free particle describes in it, uniformly, a straight line. Its equation can be written r' cos 6' = r f = const., where r f is the shortest distance of the straight orbit from the origin. Transformed by (16) the orbit in 5 will be ! = cos r and since v't'=Vr 2 r 2 , where v' is the constant S'- velocity of the particle, we shall have ultimately, as the orbit of a free particle in 5, ^-V --i < 19 > v a *One of Einstein's most vigorous exponents, de Sitter, sees herein a particularly extravagant property of the rotating system. Thus in Monthly Notices of the Roy. Astron. Soc., vol. 77 (1916), p. 176, de Sitter says: 'For ro> = A v A tt = : A *tp If N IK = N Ki , for all i, K, we have an antisymmetrical (or skew) tensor. Since N KK = N KK means N KK = Q, a whole diagonal of components vanish, and thus only %n(n+l) n = \n(nl) non-vanishing and independent components are left, the surviving ones being oppositely equal in pairs. Thus an antisymmetric tensor in a four- world consists of six independent components, and is therefore called a six- vector, in the present case a covariant six- vector. With such six-vectors the reader is already acquainted from the special relativistic treatment of the electromagnetic field. We shall see them at work in a similar duty in general relativity later on. As the symmetry so also the antisymmetry is an invariant property, i.e., N iK = N Kl is transformed into N' LK = N' Kt . Any tensor N iK can be split at once into a symmetrical and an antisymmetrical one. For we have identically and the first term represents a symmetrical, the second an antisymmetrical tensor. Similarly to (27), and starting from the special tensor A 1 B", any array of n 2 magnitudes which are transformed by the rule 7V m = ^L dx -- N afi (27 a) dx dx MIXED TENSORS 45 v . is called a contravariant tensor of rank two. If N UC =N K \ it is a symmetrical, and if N uc = TV" 1 , an antisymmetrical tensor. (A tensor N need not be the product of two contravariant vectors.) Lastly (starting from A L B K ), any array of n 2 magnitudes N* which are transformed by the mixed rule - dxj dx a (276) is called a mixed tensor of rank two, covariant with respect to its lower suffix t, and contravariant with respect to its upper suffix or index K* Special cases of symmetry and anti- symmetry as before. A new feature, however, offered by the mixed tensor is this. With any N\ make I = K, getting N K K and, by the usual convention, sum over all K. In other words add up all the components of the chief diagonal (slanting down from left to right) of the mixed tensor. The result will be a single magnitude. Now, the important thing is that this magnitude is a general invariant. In fact, by (276), AT/*: / UXft vX 1 ATd . N * = I f jMff ' \dx' K dx a / but (as mentioned before) the bracketed expression is zero for all a?^j8 and one for a = /3. Thus N' K K =N:=N K KJ which proves the proposition. Thus, equalling the upper and the lower index and summing over it degrades the mixed tensor by two ranks giving, in the present case, a tensor of rank zero or an invariant (scalar). In other words, N K K =N *It seems inappropriate to call 'suffix' (from sub, under) an upper mark or sign. I propose .therefore, to call such signs by the more general name index. Since all English writing authors accepted the ' three-index symbols ' and the 'four-index symbols' (of Christoffel and Riemann), they will per- haps not object to calling t, K indices. 46 RELATIVITY AND GRAVITATION is an invariant of the tensor N K t . We shall see presently that this procedure of equalling an upper to a lower index, called contraction (German ' Verjungung') can be applied, with equal success, to a mixed tensor of any rank whatever. Notice, however, that this process is not applicable in the case of (purely) covariant or contravariant tensors. Thus, for instance, M KK = Mi\~\-Mm-{- ... is not invariant, as a glance on (27) will suffice to show. In short, the diagonal sum of M IK has no intrinsic meaning. Similarly, in the case of a four- vector, say, AI+ . . . -\-A 4 is not an invariant. 16. The next step, leading to tensors of rank three, and so on, is obvious. Generally, any system of n r (in our world, 4 r ) magnitudes N\\', with r\ lower and r% upper indices, which are transformed by the rule dx.' dx K ' dx a dx is called a mixed tensor of rank r = n+f2, covariant with respect to its fi lower, and contravariant with respect to its r 2 upper indices. If all the components of such a tensor vanish in one system they will also vanish in any other system of coordinates. Any tensor, therefore, can be used for writing down generally covariant laws.* In particular, if ri = 0, the tensor (28) is con- travariant, of rank r 2 ; and if r 2 = Q, covariant of rank r\. The sum of any number of tensors of the same rank and kind, each multiplied by any scalar, is again a tensor of the same rank and kind, the numbers n, r 2 retaining their significance. 17. Contraction. This process, already illustrated on the simplest example, can now be generally explained. Let a be any upper and i any lower indexf of a mixed tensor of any rank r whatever. Put a = t and sum over a. Then the result will be a tensor of rank r 2, with r\ 1 covari- ant and r z 1 contravariant indices. *In the less technical sense of the word. fThe place of a among the upper, and of t among the lower indices is irrelevant. CONTRACTION OF TENSORS 47 The proof follows at once from (28). For the process gives us in the coefficients of transformation a term dx a dx t ' dx t ' dXi which vanishes for all a 7^1 and equals one for a = i, thus reducing (28) to . dx K ' dx k which proves the statement. This process of contraction can obviously be applied again and again, degrading the tensor each time by two ranks until there will be no upper or no lower indices left. In fine, the mixed tensor can be degraded until it becomes purely covariant or purely contra variant or (if ri = rs) until it is reduced to a scalar or invariant. Thus, for example, the tensor A^ of rank five gives rise to which is denoted by A^ , and this tensor of rank three gives rise to A' A = A-, which is a (covariant) tensor of rank one or a vector. Again (as an example of r\ = r z ) , the tensor A jf of rank four gives by contraction A% , and this tensor of rank two gives a scalar. We may as well write at once A = A, the meaning and the value of A being the same as before. This final invariant may be considered as a property of the original tensor A* . In general every such half-and-half tensor (r\ = r^) will have the final scalar (A) as its intrinsic* invariant. And, as far as I can see, this is its only intrinsic invariant. **.. an invariant of its own, independent of any extraneous form such as ds* (or any auxiliary tensor, such as g lK ) determining the metrics of the manifold. 48 RELATIVITY AND GRAVITATION On the other hand a purely covariant or contravariant tensor or an unequally mixed one (fi^fz) cannot be contracted to an invariant. It seems that it has no intrinsic invariant at all, that is to say, that there are no processes which would lead to an invariant combination of the components of the original tensor itself (without using other tensors). 18. The inner multiplication, already mentioned in con- nection with vectors, can now be considered as an outer multiplication followed by a contraction. Consider two tensors, generally mixed, one of rank r = fi+ r z , the other of rank 5 = Si+s 2 . Combine (by ordinary multi- plication) each of the n r components of the former with each of the n s components of the latter. The n r+s magnitudes thus obtained will be the components of a tensor of rank r+s with n+5i covariant and TI+SZ contravariant indices. That the entity thus arising is a tensor follows at once from (28). Thus the outer product of two vectors is a tensor of rank two, A.B K = M IK , A 1 B K = M K L . Similarly A afi B tK is a covariant tensor of rank four, M a(3lK , and A afi B" = N"p y is a mixed tensor of rank five, and so on. The outer multiplication combined with contraction (when there are indices to contract) gives the inner product. Thus the inner product of A t and B K is A K B K =M K K = M, an invariant.* The inner product of A" and B a p is their outer product Mrf degraded by contraction, i.e., M^ = M a , a covari- ant vector. The inner product of A^ and B IK is their outer product A a 0B uc =M% i degraded (to the utmost) by two contractions, M? K = M, i.e., a scalar or invariant. Vice versa, if A a p be any array of ri* magnitudes such that A^B is an invariant for any con- travariant B, then A^ is a covariant tensor of rank two. This criterion of tensor character, already mentioned in con- nection with A t B K , can be easily proved by writing down the There is no inner product of A t , B K . TENSOR DIFFERENTIATION 49 transformation formula of the given factor (tensor). And it can be extended to any rank and kind, no matter whether the inner product is a scalar or a tensor of any rank higher than zero. As we already know, the differential operators D t = d/dx t have the character of the components of a covariant tensor of rank one. Therefore, the 'product' of this tensor into a scalar or scalar-field/=/(xi, x 2 . - .) that is to say, the result of operating with Di upon/, will again be a covariant tensor of rank one or a covariant vector, (29) But we cannot go further than that. That is to say, an iterated application of the operation D K does not give a tensor. Thus d 2 f/dx t dx K is not a tensor. Nor do, in the more general case of any vector B t , the n 2 derivatives D K B l = dBJdx K constitute a tensor. The different behaviour of D K B L and of products of magnitude-tensors lies herein that the operational tensor D K acts also on the coefficients dxjdxj of the transformation formula of B t . In fact, we have a*' and* this is not the same thing as - D a Ba- The same dx K ' dx/ remark applies, a fortiori, to higher derivatives of scalars and of tensors of any rank. In fine, the only tensor derivable by simple differentiation, unaided by other auxiliaries (cf. infra), is the covariant vector (29) yielded by a scalar. The vector or vector-field df/dx t is called the gradient of /. In the case of space-time it consists of four components. *Unless the coordinate transformations are linear as in the special relativity theory. 50 RELATIVITY AND GRAVITATION 19. Tensor properties in a metrical manifold. Having sufficiently acquainted ourselves with the properties of tensors in themselves, let us now consider them in relation to the fundamental quadratic form ds 2 = g lK dx t dx K which converts the hitherto amorphous world into a metrical or riemannian* manifold. It is of the utmost importance to grasp well this distinction between a riemannian and a non-metrical manifold and to understand the true role of ds 2 in converting the latter into the former. Let us place ourselves yet for a while upon the non-metrical standpoint. Of all the tensors described in the preceding sections let us confine our attention upon the prototype of all (contravariant) vectors, the infinitesimal position-vector dx t . Any such vector represents ultimately but an ordered pair of points, 0(X) the origin, and A(x i +dx i ) the end-point of the vector. Imagine a whole bundle of such infinitesimal vectors OA, OB, OC, etc., all emerging from the same world-point O as origin. Now, from the non-metrical point of view, all these vectors have (apart from their origin) nothing in common with one another. That is to say, if two of them, say OA and OB, are at all distinct from one another, and if their components dx L do not happen to be proportional to one another (in which case we can say that the vectors have a common 'direction'), there is in either of them nothing, no property, with respect to which they could be compared. They are, as it were, perfect strangers to one another. Similarly, if we call 'angle' a vector-pair a = OA, OB, there is nothing to base upon a comparison of two non-overlapping covertical angles a and /3 = OC, OD. In short, neither vectors nor angles (or other derived entities) have 'sizes'. There is, in fact, in the manifold itself nothing which could fix the mere meaning of suqh a concept. Of two vectors OA , OB nothing more can The name ' riemannian ' manifold or w-space is being often used in this connection in view of the historical fact that Riemann was the first to base the general geometry of an w-space upon its line-element given by such a differential form, although Gauss was his great predecessor in the case of surface theory. THE LINE-ELEMENT 51 be said than that they are either identical (or co-directional, collinear) with or distinct from one another. The origin being the same,* the points A, B are either identical or dis- tinct, and no other significant statement can be made about their relation. But while there is nothing in the manifold itself to base a comparison of distinct infinitesimal vectors upon, we are at liberty to provide for it at our will if we so desire. This is done by introducing a standard or fundamental entity such as the quadratic form called the line-element. In other words, we surround the world-point 0(x t ) by a hypersurface, a three-dimensional (generally an n I dimensional) quadric and declare all vectors emerging from O and ending in any point P (x t -\-dx t ) of this surface to be equal in size or in absolute value, or in 'length', the usual name in the case of our three-space. It is precisely this metrical surfacef which is expressed by gudx^Xt = ds 2 = const., the numerical value of ds being the 'size' common to all these infinitesimal vectors or point-pairs. J The part played by this quadratic form is essentially the same as that of Cayley's 'absolute' or standard quadric (a real quadric leading to lobatchevskyan or hyperbolic, an imaginary quadric leading to elliptic, and the intermediate degenerate quadric leading to euclidean geometry), the only important difference being that Riemann's treatment is much more general. It covers *We have limited the discussion to coinitial vectors solely for the sake of simplicity. All our remarks apply a fortiori to distant, non-coinitial bundles of vectors. fThe German geometers call it Eichflache. Jin Riemann's own treatment this r61e of the fundamental form im- pressed upon the manifold extends into distance, over all the manifold. That is to say, if O'(y^ be any other point and if a quadric g lK dydy K = const, be drawn around it with the same value of the constant as before, all the vectors of the bundle O' terminating upon this quadric are again said to have the same size as those of the bundle 0. In this respect a somewhat more general standpoint was recently proposed by Weyl, in connection with his ideas on electromagnetism. 52 RELATIVITY AND GRAVITATION all metrical spaces (in technical language, of variable and anisotropic curvature), whereas Cayley's device gives us only a space of constant isotropic curvature, negative, zero, or positive. This fully corresponds to his starting point, which was that of projective geometry. Yet, and this is of particular interest in the present connection, Cayley recog- nized thoroughly the true r61e of all such standard entities. In fact, he tells us plainly that geometrical figures have no metrical properties in themselves. Their metrical properties such as those of the foci of a conic, etc., arise only by relating them to other figures, as the 'absolute' conic in the plane, or quadric in three-space. The kind of metrics thus impressed upon a continuous manifold being essentially arbitrary, the utility of the metrical manifold thus obtained will, of course, from the physicist's standpoint, depend upon the interpretation which is given to the said 'size' of a position-vector, and to special lines of that metrical manifold, such as the geodesies, in terms of measuring rods, clocks, moving particles or light phenomena, and so on. But without dwelling here any further upon such questions of a concrete representation let us turn to consider the purely mathematical consequences of the introduction of g dx t dx K as a fundamental differential form fixing the metrics of the manifold. 20. As in Cayley's case the geometrical figures in relation to his 'absolute', so here the tensors acquire some new pro- perties in relation to the fundamental form or better, to its coefficients g lK . In fact, what determines the form are these coefficients, and we may look upon the matter in the following way. Instead of declaring the fundamental quadratic form at the outset as an invariant, let us better say that the symmetrical array of 16 (generally n 2 ) magnitudes g lK is being introduced as & fundamental tensor, symmetrical, of rank two and of the covariant kind, as defined in the preceding sections. Combined with this fundamental tensor all other tensors of the previously amorphous manifold will acquire some METRICAL PROPERTIES 53 new properties. These and only these will now be their metrical properties. To begin with the prototype of contravariant vectors, the infinitesimal vector dx L has had thus far no invariant of his own. But it will acquire one with the aid of the fundamental tensor. In fact, dx t being contravariant, denote it for the moment by X 1 . Form the outer product which will be a mixed tensor A^ . Contract it with respect to t, a, getting A^A^. Contract this again. Then the result will be A" = A, a scalar or invariant. Or perform both contractions at once, and write ds z for A, returning to the original notation, thus gu dx t dx K = ds z = invariant. In short, the inner product of the tensor dx a dx$ into the funda- mental tensor g tK is an invariant. There is no objection to calling it the invariant of dx t as a short name for its metrical or associated invariant. Thus, thanks to g iK , the vector dx t has acquired an invariant. And it can now be compared through it with other vectors, no matter what their com- ponents. The value of ds 2 may be called the norm, and the absolute value of dbds 2 the size of the vector dx,.. Thus we can speak of two vectors dx t and dy t being equal in size, or one having twice the size of the other, and so on. In application to the four- world, a vector dx, of no size will be a light vector, a vector of negative norm a space-like, and one of positive norm a time-like vector. Similarly, any other contravariant vector A 1 will have the metrical invariant g lK A>A K =A\ say.* (30) *Of course, even in the amorphous manifold an invariant could be built up from A 1 by the aid of any covariant tensor N tK , but the choice of N IK being entirely free, such an invariant would not have a fixed value. We fix it by introducing once for all a special tensor g lK to serve for all other tensors. 54 RELATIVITY AND GRAVITATION In much the same way, if B t be any covariant vector, we shall have in g^B.B^B 2 (30o) an invariant, the norm of B. . From a more general point of view we may call A 2 , in (30) , the tensor, of rank zero, metrically associated to A 1 , similarly, in (30a), B 2 toB t . Moreover, we can easily construct associated tensors of a rank other than zero, and differing also in kind from the original tensor. Thus, to dwell still upon vectors, &.A--A. (31) will be the covariant vector metrically associated with the contra variant vector A K . We may call A, shortly the conjugate of A 1 . Similarly, starting from a covariant vector A^ we shall have the contravariant vector g Kt A K = A l (31a) conjugate to A t . Two questions naturally suggest themselves: Will the con- jugate of the conjugate be the original vector? Have two conjugate vectors the same size or the same norm? In order to answer these questions as well as for the sake of what will follow, let us first note a simple property of the tensors g lK and g lK . By definition, chap. II, g lli is the minor of the determinant g = | g tK \ , corresponding to its t, K-th element, divided by g itself. But g is equal to the sum of the products of the elements of its first column, say, into the corresponding minors, i.e., g = g a i gg ai , whence g a ig ai== 1. Similarly for any other column (or row). Thus, underlining the index over which an expression is not to be summed, This is valid for every v. Thus g MJ ,g M ", summed over both indices, has the value 4 for our world, and n for an w-fold. Again, taking two different columns (or rows) of g, we shall easily prove that CONJUGATE TENSORS 55 Both properties can be united in a single formula &."-;-*!!, (32) where 5 is the conventional symbol for 1 or according as a = /3 or a 5^/3. This symbol is itself a mixed tensor. We are now able to answer our two questions. First, the conjugate of the conjugate of the vector A t is, by the definitions (31), i.e., the original vector. Similarly if we started with A". Thus, the conjugate of the conjugate is the original vector. Second, if A l be the conjugate of A t we have for the norm of the former vector, by (30) and (3 la), Thus any two conjugate vectors have equal norms. The norm of A t and of A' can also be written A t A l , for this is again equal to g llf A 1 A". Thus, for instance, if d^ be the conjugate of the contravariant vector dx L , their common norm or the squared line-element can be written ds* = dx t d&. (33) 21. In much the same way we can treat the metrical properties of tensors of any higher rank. To explain the method it will be enough to take up in some detail the second rank tensor A t/c . Its conjugate or supplement (Erganzung) will be the contravariant tensor defined by A" , or also g^g^A'A^ . (34) The tensor g itself is easily proved to be the supplement of the tensor g u . The scalar or invariant of A IK will be fA u =Al-A. (35) A single contraction of g" A aff will give g" A m . = A', , a mixed tensor metrically associated with the covariant A,,. 56 RELATIVITY AND GRAVITATION The supplement of the supplement (or the conjugate of the conjugate) is again the original tensor, for The tensors A LK and A" have the same scalar A , (35). In fact, the scalar of A" is a. A" -a. g" f'A^-P. fA. t = f'A t .~A. Since g v Ap V is an invariant, ^ = g lK * A^ v is again a tensor; Einstein calls it the reduced tensor belonging to A^. Notice that neither a covariant nor a contravariant tensor has an invariant independent of the metrical tensor; only a mixed tensor, B"^ has such an invariant, to wit B = 5\ This is a privilege of mixed tensors of even rank with ri=rz, and of these tensors only. The investigation of other metrical properties of tensors of the second and higher ranks may be left to the reader. Exercises of such a kind will soon make him familiar with this broad and powerful algorithm. 22. Angle and volume. Consider any two coinitial in- finitesimal vectors dx t , dy t . These are contravariant vectors. Therefore, as we already know, the inner product g tK dx, dy K will be an invariant. It will remain invariant when divided by the sizes of both vectors. By an obvious generalisation of the familiar cosine formula this invariant is used to define the angle e made by the two vectors, thus cos t i il^i^l , ( 36 ) as da where ds 2 = g lK dx t dx K , da 2 = g LK dy t dy K , The two vectors are said to be orthogonal or perpendicular upon one another if g tK dx t dy K = 0. Generally, the angle between any two vectors A 1 , B\ whose norms as defined by (30) are A 2 and B 2 , will be determined by ANGLE AND VOLUME 57 COS = AB (37) and the vectors will be orthogonal if g u ^4M' ( = 0. Similarly for covariant vectors, with the only difference that g tK is replaced by g 1 * . Let A lt B t be the conjugates of A 1 , B l ; then and since A t , B t have the same norms as A 1 , B l , we see that the angle between the conjugates is the same as between the original vectors. The integral fdxi dx 2 . . . dx M extended over a domain of the manifold is, by a well-known theorem, transformed into i dx z ' . . . dx n ', where J is the Jacobian dx t , as in (7). On the other hand the determinant g of the fundamental tensor (called also the discriminant of the fundamental quadratic form) is transformed into dx n dxa a*.' dx," the last step being based on the multiplication rule of deter- minants. Thus g' = J*g. (38) Consequently, the integral i dx 2 . . . dx n (39) is a scalar or an invariant of the n dimensional domain of integration. In the case of the four-dimensional world the determinant g is always negative.* Thus the invariant expression *In a galilean domain and in Cartesians g= 1, by (16), p. 6. By (38), therefore, it is also negative, always for a galilean domain, in any other system of coordinates derived from the Cartesians by a holonomous trans- formation. Now, although a non-galilean domain cannot be made galilean by a holonomous transformation, yet we know that in all practical cases the g LK differ but very little from the galilean coefficients. Thus g will also in general be negative. UNIVERSITY OF CALIFORNIA DEPARTMENT OF CIVIL. ENGINEERING 58 RELATIVITY AND GRAVITATION dtt = V-g dxidx 2 dx z dxi (40) will be real. This is taken as 'the local measure' of the size or volume of an infinitesimal world-domain. For in the local (cartesian) coordinates u lt for which g= 1, this expression becomes duidu 2 dMzdu^ = cdtdxdydz. The latter product is called by Einstein 'the natural' volume-element. Apart from names, the important thing to notice is the general invariance of the expression (40) as such or when integrated over any world-domain. Consider any sub-domain of the world, of three, two or one dimension. This can be represented by expressing the x t as functions of three, two or one parameter respectively. The differentials dx t will be homogeneous linear functions of the differentials dp a of these independent parameters. Thus the line-element within the sub-domain will be of the form ds* = h a pdp a dpp , h a p = hp a , and the sub-domain, therefore, will again be a metrical manifold (a three- space, surface or line) in Riemann's sense of the word, and if /* = | dQ^V h dpidp, .. . will (apart perhaps from a factor V 1) again be an invariant measure of an element (volume, area, length) of the sub-domain. Thus, in the case of a one-dimensional sub-domain or line, dp dp In this case h = hn and, therefore, dtt=V~h n dp, which is ds itself, as it should be. For a two-dimensional sub-domain or surface we have ds* = An dpi*+2h l2 dpi dpi+hn dp?, dx t dx K where h a b=g iK -T - -r - . Thus, dfi = V A d/>! d/> 2 , where djc t 5jc K d# t ^^ K /" d.r t == f* lK O . J . SlK f\ f\ \ &IK " n Opi Opi Op2 Opz \ Opl COVARIANT DERIVATIVES 59 23. Differentiation based on metrics. We have already seen (p. 49) that if / be a scalar or invariant, df/dx t , the gradient of/, is a covariant vector. This is independent of the metrics of the manifold. But, as was then pointed out, the iterated application of the operation d/dx t would not lead to tensors; nor would its application to a vector A, or another tensor yield by itself, unaided by auxiliaries such as g lK , a tensor. But the introduction of the metrical tensor opens in this respect new and important possibilities. It was remarked by Christoffel as long ago as 1869 that if A t be a covariant tensor, so is namely covariant, of rank two. Similarly if B^ be a covariant tensor of rank two, B..- j ^ | .. is again a covariant tensor of rank three; similarly *r (42a) is a mixed tensor of rank three, and so on. But it will be enough to consider here at some length the first case (41) only, especially as the other cases can be derived from it. The operation indicated in (41) is called covariant differentia- tion, and its result A IK the covariant derivative or the expansion (Erweiterung) of A t . If 5 l be a contra variant vector, is a contravariant tensor of rank two, the contravariant derivative of Z?'. But for our purposes it will suffice to consider only the covariant differentiation. That (41) represents a covariant tensor can be proved in a variety of ways. The most instructive of these is perhaps 5 60 RELATIVITY AND GRAVITATION that given by Einstein, since it makes immediate use of the equations of geodesies, and the r61e of the Christoffel symbols* appearing in (41) is thus far known to us only in connection with these world-lines. Einstein's reasoning is as follows: Let/ be a scalar or better a scalar field (i.e. an invariant function of position within the world). Differentiate it twice along any world-line. Then d 2 f df d 2 x t d 2 / dx a ds 2 dx t ds 2 dx t dxp ds ds will again be an invariant. Let the line be a geodesic. Then = \ ( , and the invariant will assume ds 2 I <- ) ds ds the form &L = r a* ds* Ldx a dx ft ( ) d# t J ds ds Since the contravariant tensor (of rank two) dx a dx# is arbitrary (for from a given point a geodesic can be drawn in any direc- tion, i.e. with arbitrary ratios of dxi t etc.) and its product into the bracketed term is invariant, the latter, i.e. (43) dx a is a covariant tensor of rank two. This proves the proposition for the special vector A t df/dx t . To prove it for any covariant vector, notice that any such vector A t can be repre- sented by the sum of four (generally n) terms of the form ifrdf/dXt, where \f/ and / are scalars. Thus it is enough to prove that dx K \ dx t is a tensor. But this is equal to *Notice in passing that < ! '* is not a tensor. XK), (iM> *cX) = (/cX, i/z), so ttyat the number of essentially different, i.e. linearly independent symbols is reduced to 12 For a proof see, for instance, Killing, loc. cit., p. 228. In the case of a one-dimensional manifold, a line, there is no such non -vanishing symbol. In fact, although a line may be ' curved ' from the standpoint of two- or more-dimensional beings in whose space it is imbedded, yet it has no intrinsic properties of its own to distinguish it from other lines, nor one of its parts from another. Take, for instance, a plane curve. If Aco be the angle between the tangents at two points separated by the arc As, the curvature of the line is defined as the limit dco/ds. Now, this curvature is often called an intrinsic property of the line, because (unlike the sloping of the line) it is independent of a coordinate system laid in *Cf. for instance L. Bianchi, 1902, loc. cit., p. 72, where it is denoted by | ta, X/c|. The geometrical applications of the Riemann symbols are fully treated in vol. I of Bianchi's work. See also W. Killing's Nicht-Euklidische Rawnformen, Leipzig (Teubner), 1885. RIEMANN SYMBOLS 65 that plane, yet it is entirely meaningless if the line is not conceived as a sub-domain of the plane. For so is the angle Aco. And from the bidimensional standpoint every curve is developable upon every other. In the case of a surface, n = 2, there is, by (51), essentially just one Riemann symbol, namely (12, 12), (21, 21) being equal, and (12, 21), (21, 12) oppositely equal to it, and all others being zero. This unique symbol divided by the discriminant g is a differential invariant of the surface (or of its metrical form ds 2 >=gndxi 2 -}-2gttdxidx2-\-g2, and herewith the vanish- ing of the whole tensor of Riemann symbols. For a three-space there are, by (51), six, and for the world or space-time as many as twenty independent Riemann symbols. A five-space has fifty independent symbols, and so on. But, no matter what the number of dimensions, the 66 RELATIVITY AND GRAVITATION Riemann symbols always represent the curvature relations of the manifold, and their vanishing continues to form the con- dition of an important property of the metrical form of the manifold. To begin with the latter, suppose all g tK are constant over a domain of the world. Then all (tju, X/c), and therefore also all the components of the tensor B" KX vanish throughout the domain. This then is the necessary condition for a domain of the world to be galilean, i.e., for the line-element to be holonomously transformable into ds 2 = c^dt ? dx 2 dy 2 dz 2 . It was proved by Lipschitz that this, i.e. ix = 0, is also the sufficient condition for the said reducibility (to a form with constant coefficients). In the second place, concerning the curvature relations, consider a surface or or a two-dimensional sub-domain of the world, or of any metrical manifold. More especially, let , so that S IK (which is not a tensor) is symmetrical, and such being also the first two terms in (55o), G M is seen to be a symmetrical covariant tensor, of rank 'two ; G IK = G Kl . Thus Einstein's field equations (III ), valid outside of matter, are ten in number, and such is exactly the number of the metrical tensor components g llc . The field equations would then give us a system of ten differential equations of the second order for ten unknown functions g llc of the co- ordinates. As a matter of fact, however, there exist between the covariant derivatives G t/cX of the G IK and the derivatives 3G/dx L of the invariant G = g lK G IK four identical relations (based upon certain identical differential relations discovered by Bianchi), to wit G, = g" x G^ = \ - - , t = 1, 2, 3, 4. (56) dx t Owing to these four identities, to which we shall have to return later on, only six of the field equations are mutually independent, leaving therefore four of the g^ or any four functions of the g lK free or undetermined. Such, however, should from the general relativistic standpoint be the case. 72 RELATIVITY AND GRAVITATION In fact, from this point of view one would expect beforehand the field equations or any differential laws to be such as to leave us a perfectly free choice of the system of coordiantes. Einstein himself, for instance, makes use of this freedom by putting in most of his formulae V g = l, which, by (55a), reduces his field equations to and leaves him still a threefold freedom of choice. The latter can often be used with advantage by making g 14 = g 2 4 = g34 = 0. It will be kept in mind, however, that the equations, such as (57), thus simplified do not retain their form under general transformations. They are only useful as technical devices offering some advanatges in the treatment of special problems. The generally covariant form of the field equations is only that obtained by equating to zero the complete or general value of G^ , such as (55) or (55a). 26. In order to see the relation of Einstein's field equa- tions to the more familiar Laplace equation, let us evaluate the curvature tensor G IK for the case of a 'weak' field, i.e. differing but litle from a galilean domain.* Thus, using a quasi-cartesian system of coordinates, let the fundamental tensor differ but little from the galilean tensor g lK , i.e., as in (21), let g = g + 7 K , where all the y tlc are small fractions. Then the products of the Christoffel symbols in (55) will be small of the second order, and the tensor in question will be reduced to dx K <* dx a Here, up to second order terms, *Notice in passing that all gravitational fields known from experience are 'weak' in this sense of the word. FIELD EQUATIONS 73 and since ^ n = g 22 = !:33= 1, 1*44= 1, while all other "g" vanish, (LK\ r t K~l < . > = - . . * = 1, 2, 3; \* } L i J Thus, using the index i for 1, 2, 3 and summing every term in which i occurs twice over 1, 2, 3, we have the approxi- mate curvature tensor In the present connection the only interesting component is that corresponding to i = K = 4. This is, by (58), and on sub- stituting the values (13) for the Christoffel symbols, (680) oXi 0X4 oxf a 2 . a 2 a 2 a 2 where V 2 = - is the well-known Laplacian -- + -- h dxf dxi 2 a* 2 2 d*s 2 If the field is stationary, the second and the third terms vanish and Einstein's last field equation, 6^4 = 0, reduces to the familiar equation of Laplace V ? 4 4 = 0. (59) At the same time, as we saw before (p. 36), the equations of motion assume, in absence of #4,-, the form of Newton's equations j = -IT' (236) dt 2 dXi c 2 where the potential fi = - 44, differing only by a constant factor from 44, again satisfies Laplace's equation. The complete contents of Newton's law of gravitation, thus far outside of matter, appear as a first approximation to Einstein's field equations and his equations of motion of a free particle. 74 RELATIVITY AND GRAVITATION 27. The ten field equations G^ = are valid outside of 'matter', i.e., as is expressly stated by Einstein, in such domains of space-time in which there is not only no matter in the ordinary sense of the word but also no electromagnetic field, or, in fact, no distribution of energy of any origin other than gravitational. Following Einstein's example the word 'matter' will be used to cover all such cases. This will har- monise with the property of energy already familiar to us from special relativity,* namely of possessing inertia, an amount of energy U being equivalent to an inert mass U/c~ t which, by the law of proportionality, is also its heavy or gravitational mass. As we saw before, the role of the newtonian gravitation potential 12 is, in a first approximation, taken over by the tensor component g 44 multiplied by c 2 / 2 . The vanishing of G 44 was approximately equivalent to Laplace's equation V 2 12 = which holds outside of matter. Within matter Laplace's equation is replaced in the classical theory of gravitation by the more general equation of Laplace-Poisson, where p is the density of mass in astronomical units. f Now, since G 44 reduces approximately, in a stationary field, to iV 2 g44== V 2 fl, the idea easily suggests itself to make , (60) and to consider this as the equation or at least as one of the field equations within matter. But, needless to say, such a single equation would not by itself serve any relativistic purpose. What is required is a system of ten equations, of *And partly even from pre-relativistic considerations, such as in Mosengeil's investigations on an enclosure filled with radiation or those made in connection with Poynting's light-pressure experiments. fit will be kept in mind that a mass m in astronomical units is defined by = force, so that its dimensions are \m] = [length X (velocity) 2 ]. TENSOR OF MATTER 75 which this should be one. In other words, the tensor G IK has to be made equal or proportional to a symmetrical co- variant tensor of rank two somehow associated with 'matter' and having for its 44-component the density p or what ap- proximately reduces to the usual mass density and therefore, apart from a constant factor, to energy density. Now, such a tensor was familiar from the special relativity theory under the name of stress-energy tensor often abbreviated to energy tensor. The merit of having introduced this concept into modern physics is chiefly due to Minkowski and Laue, pre- ceded in non-ielativistic physics by Max Abraham. The energy tensor made its first appearance in electromagnetism, in connection with the ponderomotive properties of an electro- magnetic field,* as the symmetrical array or matrix /ll /12 /13 Pi /21 /22 /23 p2 /31 /32 /33 p3 Pi Pi pz U f P p -u consisting of the six components fi k =/, of the maxwellian electromagnetic stress, of twice the three components pi of electromagnetic momentum (or Poynting's energy flux) and of the density u of electromagnetic energy. The physical significance of this tensor or matrix was that its product into the operational matrix lor dx dy dz dt gave the ponderomotive force P per unit volume and its activity Pv, -lorS = |P lt P 2 ,P 8 ,Pv|. Later on its r61e was generalized for a stress, momentum and energy density of any origin, not necessarily electromagnetic, *Cf. Theory of Relativity, Macmillan, 1914, Chap. IX, especially p. 238. In reproducing it here, with pi written for gi f I drop the imaginary unit and put c = l. 76 RELATIVITY AND GRAVITATION provided only that the force and its activity could be repre- sented in the form dt dt From the special relativistic standpoint this array of ap- parently heterogeneous physical magnitudes was important as it transformed from one inertial system S to another S r as a whole, to wit by the operator A( )A, where A is the funda- mental Lorentz transformation matrix of 4X4 elements and A the transposed of A. The developed form of the trans- formation equations of stress-momentum-energy need not detain us here.* The important thing in our present connection is that the said stress-momentum-energy array is a symmetrical tensor of rank two. And since such also is the contracted curvature tensor G tK , the idea naturally suggests itself to make G IK pro- portional to a symmetrical covariant tensor T tK , of which the first nine components 7\i, T\z, . . . T^ are of the nature of stress or equivalent to it, the components 2V(*=i, 2, 3) replace the momentum, and the last component T^ is, or approxi- mately reduces to, an energy- or mass-density. But then it is by no means necessary (nor is it possible) to fix beforehand the exact physical meaning of the several components of such an energy tensor or tensor of matter (as it is often called by Einstein). Their significance has to be fixed a posteriori, through physical applications of the field equations aimed at. If T IK is a covariant tensor of rank two, then, as we already know, T = f T x (01) is a scalar, the invariant of 7\ K .f Such being the case, g lK T is again a symmetrical covariant tensor. Now, guided partly by guesses (originally at least) and partly by considerations *It will be found on p. 236 of my book quoted above. tSuch also was Laue's 'scalar' in relation to his Welttensor', i.e. the matrix S. FIELD EQUATIONS 77 of conservation of energy and of momentum,* Einstein wrote down as his general field-equations G IK = - ^ (T -i&.rj, (III) G the factor 87T/C 2 being so chosen as to give, in a first approxi- mation, the equation of Laplace-Poisson. In fact, as we shall see from the more definite form to be given pre- sently, in a first approximation, r 44 =:r=p, 4?r so that the last of (III) gives GU= p, as in (60). Einstein's own coefficient differs from ours by the gravitation constant which is here in- corporated into p, the density in astronomical units. The previous equations (III ), holding outside of matter, are a special case of these general equations, for T tK = 0, when also r = 0. To be exact, Einstein speaks first of ' matter ' as 'everything except the gravitation field' (loc. cit., p. 802) and writes G IK =0 outside of matter in this sense of the word. But later on (p. 808), trying to justify the exact form (III) of his general equations, he states expressly that ' the energy of the gravita- tion field' (if there is such a thing) has also to 'act gravita- tionally as every energy of any other kind', in short that gravitation energy too has mass and weight. Thus, rigorously speaking, there is 'matter' everywhere, and the equations (III ) are valid nowhere, unless there is no gravitation field, when they are superfluous. In other words, gravitation itself contributes also to the tensor T IK . Its contribution, however, is practically evanescent, and this circumstance makes the equations (III ) physically applicable. But even the contribution to T ik a,k=i, 2, 3) of stresses within matter in the ordinary sense of the word (tensions or pressures) is practically negligible, and so is the contribution to 7*44 of the energy proper outside of molecules, atoms or electrons, and we may as well omit it in 7" 44 , and take, for a first approximation at least, T 44 = p, where p is the density *Principles to which we may return later on. 78 RELATIVITY AND GRAVITATION of ordinary matter or approximately so, always in astro- nomical units. Thus the idea easily suggests itself to build up the tensor T tK for a theoretically continuous body (a fluid, liquid or solid) out of its local density and the velocity com- ponents of its motion. For although the gravitational effects of the motion of matter are exceedingly small, yet the mere desire of writing generally covariant equations, say, of hydro- dynamics,* prevents us from discarding velocities in this connection. Thus, neglecting stresses, etc., let us introduce, after Einstein, the scalar or invariant p as 'the density' of matter, and the four-vector of velocity ' . Then p " * ds ds ds will be a tensor of rank two, a contra variant one, however. Construct therefore, by the principles explained in Chapter III, the associated tensor dx a dxp T iK = P g ta g K p f- , (62) ds ds which will be covariant and, manifestly, a symmetrical tensor. This is Einstein's energy-tensor or tensor of matter to be used in (III) whenever tensions, pressures, etc., are negligible. As a matter of fact, in view of the limitations of even the most accurate methods of observations now available, this particular tensor will cover, presumably for many years to come, all needs of the physicist and the astronomer with regard to gravitation. f The value of the invariant T or the scalar belonging to this tensor, which is defined by (61), follows at once. Since f .* =&* =&* =g* , and g a pdx a dx ft =dst, we have, by (62), r=p, (62a) that is to say, the scalar of the tensor in question is the density of matter. *And this was undoubtedly one of the reasons by which Einstein was influenced. jThe case of hydrodynamics will be covered by subtracting from (62) the tensor pg lK , where p is the (invariant) hydrostatic pressure. FIELD EQUATIONS 79 On the other hand, in a local rest-system, in which only i/ds) 2 survives and is equal to 1/gu, we have 44 .e., for instance, r 44 = pg44, and therefore, by (III), rigorously, 47T Cz 4 4= -- g44 P- In a first approximation (g 4 4=rl) this gives (60) or the Laplace- Poisson equation, as announced before. The general field equations (III) may, independently of (62), be given a slightly different form. Multiply both sides, innerly, by g llc , and write G = g lK G IK . (63) Then, since g g tK = 4, G= ^L.T. (64) Substitute this value of T into (III). Then C- -J&.G-- ^fr K , (Ilia) C 2 the required form of the field equations. Notice that G, as defined by (63), is the invariant of the curvature tensor G^ . This invariant or rather one-sixth of it is called the mean curvature of the world, at the world-point in question. In fact, in the case of a three-space G would, apart from a mere numerical factor, be the arithmetical mean of the three principal riemannian curvatures, and this would still be the case for a manifold of any number of dimensions, at least if ds 2 be a definite, positive quadratic form. This justifies the name given to G above,* and equation (64), independent of the particular form (62) of T IK , teaches us that *For a certain special world to be treated later on G will be proved explicitly to be six times the smallest value of the (constant and isotropic) curvature of three-space which it is possible to choose as a section of that four-world. (Cf. Appendix, A.) 80 RELATIVITY AND GRAVITATION this mean curvature is proportional to the scalar of the tensor of matter and vanishes, therefore, outside of matter. More especially, if stresses, etc., be negligible, the tensor (62) conies to its right and we have, by (62a) and (64), G = ~ P. (626) The mean curvature of the world is thus proportional to the density of matter. Notice that p/c 2 has the dimensions of a reciprocal area, and such also are the dimensions of G, and of all G IK , since the g tK are dimensionless, and the G llc are linear in the second derivatives of the g lK with respect to the coordinates, each of which is a length. The same remarks hold good with respect to the field equations (III). It may be interesting to notice, even at this stage, that the mean curvature in familiar matter, say, in water under normal conditions, is, comparatively speaking, not insignifi- cant. In fact, remembering that the gravitation constant is 6'658.10~ 8 , in c.g.s. units, we have for water at normal density G = S " 6-658.10- 8 =r86.1(T 27 ciri.- 2 9.10 20 What is technically called the world-curvature is one-sixth of this. Thus the radius of mean curvature defined by R= V6/G will be, for water, ^ = 5'688.10 13 cm., i.e., about 570 million kilometers or only 3'8 astronomical length units. But it would be rash to conclude with Eddington* that a globe of water of about this radius 'and no larger, could exist'. In fact, what is known from geometry is only that the total length of every straight (closed) tine in a three-space of constant and isotropic curvature 1/-R 2 , of the properly elliptic or polar kind, is irR, so that the greatest distance possible in such a space is fyrR and the total volume of the space is 7T 2 jR 3 . But, unlike such *A S. Eddington, Space Time and Gravitation, Cambridge, 1920, p. 148. CURVATURE INVARIANT 81 a space, the world has a non-definite fundamental differential form, and its riemannian curvature depends upon the orientation of the geodesic surface element. Thus a direct transfer of the properties of an elliptic space upon the (watery) world is certainly illegitimate. Notice, moreover, that G, and therefore R, is remarkable as a genuine invariant of the four- world and not of a three-space laid across it as one of an infinity of possible sections. The best plan for the present is, therefore, to see in it only such an invariant of space-time, within the world-tube of a mass of water. The few numbers were here presented only to give an idea of the order of magnitude attainable by the curvature invariant in ordinary matter, con- sidered as continuous. To those who like to contemplate sensational results the best opportunity is perhaps afforded by the atomic nuclei. According to Rutherford the radius of the nucleus of the hydrogen atom is about one- two thousandth of that of the electron, i.e, |10- 16 cm., and its mass practi- cally equal to that of the whole atom. This would give for the density of mass a value 3.10 24 times the normal water density, and therefore a curva- ture radius within the nucleus V3 . 10 12 smaller, i.e., R = 32 cm. only! The moral would then be that nuclei of about this radius and no larger could exist, with the same density. Fortunately they are believed to be much smaller. But it is time to return to Einstein's equations of the gravitational field in order to see some of their further pro- perties. 28. Multiply the field equations (Ilia), identical with (III), by g" a . Then, denoting by G t a and T? the mixed tensors associated with G IK and T IK , i.e., writing and remembering that g" a g iK = g? = d" , we shall have, with If G, a x be the covariant derivative of Gf, and similarly for the energy tensor, we have, contracting with respect to X = a, and remembering the meaning of 5", t dx a dx t But, by (56), the right hand member vanishes identically. 82 RELATIVITY AND GRAVITATION Thus, as a consequence of the field equations we have the four equations ^=0 (, = 1,2,3,4), (65) concerning the energy tensor or the tensor of matter. Thus also, out of the ten field equations only six are left for the determination of the potentials g llc , as announced before. The matter-equations, so to call (65), as a consequence of the field equations constitute a most remarkable result.* Notice that they are entirely independent of any special form of the energy tensor, such as (62). They are manifestly general, i.e., valid for any covariant tensor T IK , merely in virtue of putting it equal, or proportional, to the curvature tensor G IK -igG, the left hand member of Einstein's equations. In order to see the significance of the equations (65) remember that T? a is the contracted covariant derivative of the tensor T a = * *> (66a) V ff dx a \ P ) o where the mixed tensor of matter T* is as in (66). The left hand member of (65a) is a four- vector T = T a * i * ta which is called the divergence of the mixed tensor T*. The equations (65a) can thus be read technically: The divergence of the mixed tensor of matter vanishes. This, of course, does not enlighten us as to their signifi - cance. To see their physical meaning take any coordinate system for which g= 1, so that TS , (656) and consider the case of a weak gravitational field, for which the g lK differ but little from the galilean values, i.e., in quasi- cartesian coordinates, gn= I+TH, etc., as in (21). In the expressions (66) for the energy tensor itself the y lK can be neglected altogether, so that 84 RELATIVITY AND GRAVITATION and r a_,a, ^4 dx a -I 4 -/>4 +P - ~ as as In the right hand member of (656) the y lK cannot be disre- garded without anihilating that member altogether. For the Christoffel symbols vanish for constant g llc . But since their values are taken to be small of the first order, it is enough to retain in the right hand member only and since { l | ) == [ 4 | ] = | Jfii , the four equations become, dx t if pf be negligible in presence of p, dx a dx t For small velocities, ds dx 4i = cdt, and if ,- be the cartesian velocity components dxi/dt, f etc. ; TS = ^ - - etc. , C and c c Thus, neglecting cpf and cp in presence of the momentum (per unit volume) pV of molar motion of matter, the first three equations will be ~ (p^-^H (puM-tpf) +... + - (pvi) =P -, etc., dxi dx 2 dt dX where 12= ic 2 g 44 plays again the part of the newtonian MATTER- EQUATIONS 85 potential, and the fourth equation d/ dt or, in obvious three-dimensional vector notation, a* c 2 dt and, with fi written for the three-vector C 2 (pi l , pi 2 , pi*), (pvi) 4- (pur) + (pvi z> 2 ) + - - (pvi v 3 ) = div fi+p , etc. dt dxi 8x2 dx s dxi The left hand member of the last written equation is equal to n *\ *\ (pv\) +^i -- (pvi) + . . . +v s dt dXi dx (pvi)+pvi . div V, dt or, if dr be the volume and dm = pdr the mass of an individual element of matter, equal to - - . Similarly we have dt. dr dp . ,. , , dp ,. d(dm) - +d^v ( P v) = - +pdw V= f - . dt dt dt . 6r Ultimately, therefore, the approximate equations of matter are, with^i, j, k as unit vectors along the coordinate axes, and =5m . V12 (A) dt d /t> N dfl dm (8m) = . (B) dt dt c 2 The first three equations embodied in the vector formula (A) are the equations of motion of a continuous medium* under internal stresses (tensions) f ik =0^, and under the action of *A deformable solid, liquid or fluid. 86 RELATIVITY AND GRAVITATION the gravitational field of which the newtonian potential is again, as in the case of the approximate equations of motion of a particle, 12 = \&g&. The fourth equation, (B), is, apart from the new term on the right hand, the familiar equation of continuity. In other words, the first three equations express the principle of momentum, the amount of momentum acquired by matter from the field per unit time being given by 8m . grad 12 which is the newtonian force on the mass element dm. And the fourth equation expresses the principle of energy or, equivalently, of matter, the amount of energy (c 2 5m) acquired by a material element per unit time being equal to the mass 8m of that element multiplied by the local time- variation of the potential, i.e., approximately equal to the decrease of the potential energy of that element. It is scarcely necessary to say that this gain or loss in energy or in mass of 'matter' placed in a gravitational field, according to the sign of d!2/d/, is immeasurably small. Its discussion in this place may have only a mere academic interest. If it be neglected, (B) gives at once the usual equation of continuity, and (A) assumes the perfectly familiar form pV i divfi . . . =pV 12 = gravitational force per unit volume. On the other hand it is interesting to notice that the equation (B) becomes, for v = 0, at once integrable and gives 8m = 8m e~^/ c where 8m is 8m for 12 = 0. A similar result followed from a gravitation theory proposed some time ago by Nordstrom (Phys. Zeits., 1912, p. 1126 1 . Its interpretation may be left to the care of the reader. Returning once more to the rigorous equations (65a) we now see that the terms / i a \ \fi-r represent in general the momentum and energy (or mass) acquired by 'matter' in a gravitational field. The four equations themselves express the principles of momentum and of energy, as was made plain above on their appropriate form. I avoid purposely to call them principles of 'conservation' of momentum and of energy. For although Einstein succeeded in giving them the form* *Sitzungsberichte of Berlin Academy, vol. 42, 1916, p. 1115, where the German T and / are the above V g T t vgt. ENERGY PRINCIPLE 87 in which they would -deserve the name of conservation, yet the /^ built up of the g M " and their first derivatives has not the character of a general tensor, but behaves so only with respect to a certain class of coordinate systems (for which g= 1). In view of this it has not seemed necessary to quote here the values of the t? . Suffice it to say that since, unlike T* f they contain only the gravitational potentials (and their first derivatives), Einstein calls v g t* 'the components of energy of the gravitational field', and v g T a those of matter, and reads the last set of equations: the total momentum and the total energy of matter and of the field are conserved. The point under consideration is after all but a formal one, and we prefer therefore to content ourselves with the original equations (65a), interpreting their second terms as momentum and energy gained (or lost) without attempting o locate them as such in the gravitational field before their passage to, or rather appearance in, matter. Historically, the position is this. In the special or restricted theory of relativity the principles of conservation of momentum and of energy were expressed by the vanishing of the 'lor' or, in Laue's nomenclature, of the Divergence of a world tensor, this ' Divergence ' being a four- vector whose components were transformed by the Lorentz transformation, the four equations themselves being thus invariant with respect to this kind of transformation. The tendency to imitate these principles of conservation in the generalized theory was but a most natural one. But the proper generalisation of that special Divergence in a theory admitting any trans- formations of the coordinates is the Divergence defined in the general tensor calculus, i.e., the contracted covariant derivative r,= l?. of the mixed tensor of matter 7** . This is a genuine four- vector, a covariant tensor of rank one, and the original generally covariant equations (65), r = o, are the only appropriate expression of the principles of momentum and energy. Their expanded form is (65a) and this cannot in general be given the form of 'conservation laws'. Only for constant g , that is in a galilean domain, does it reduce to r~ T ? = dx a which is identical with the vanishing of what was called the Divergence of r a in the restricted relativity theory. All attempts to squeeze the broader Divergence T* a into the narrower one seem artificial and useless. For conservation as an integral law, cf. Einstein, Berlin Sitzungsber., 1918, p. 448. 88 RELATIVITY AND GRAVITATION 29. That the gravitational field equations (together with the equations of motion and those of the electromagnetic field) can be deduced from a single variational principle or, as it is called, a Hamiltonian Principle, was first shown by H. A. Lorentz (Amsterdam Academy publication for 1915-16) and by D. Hilbert (Gottinger Nachrichten, 1915, No. 3), and later on by Einstein himself.* More recently Hilbert, Weyl and others have returned to this subject in a large number of publications, in some of which the importance of the Hamil- tonian principle seems to be unduly overestimated. Since this matter is, after all, of a purely formal nature, it will be enough to give here but a very brief account of Einstein's own treatment as developed in the paper just quoted. With dx as a short symbol for dx\ dx 2 dx z dx Einstein writes the Hamiltonian principle ^~g(G+M)dx = 0, (68) where G, M are invariants. Since v gdx, the volume of a world-element, is invariant, so also is the whole integrand. Let M be a function of the g^ and of q ,and dqjdx a , where q, are some space-time functions describing 'matter', while G is assumed to be linear in d 2 g^ / dx a dx$ with coefficients de- pending only upon the g^. Then, by partial integration, \ where da is an element of the boundary of the world-domain \dx (the particular value of the integrand F being irrelevant), and G* is a function of the g* 1 " and their first derivatives only. Let it be required that the values of f and of their first derivatives should be fixed at the boundary. Then d Fdcr = 0, and we can write, instead of (68), *A Einstein, Hamiltonsches Prinzip und allgemeine Relativitdtstheorie, Sitzungsberichte der Akad. der Weiss., Berlin, vol. XLII, 1916, p. 1111- 1116. HAMILTONIAN PRINCIPLE 89 = 0, (68a) where the whole integrand depends upon the f and q t and their first derivatives only. Thus the variation of the g* v gives at once the ten equations , (69) dp where and Now, let G be the curvature invariant, i.e., in our previous notation, Then, on performing the said partial integration, it will be found that [{7} {?}-{:} With this value of H* //?e equations (69) become identical with Einstein 1 s field equations as given above, if we put .e., or, in terms of the covariant tensor of matter, dM The verification of this statement may be left to the care of the reader who may confine himself to systems for which 90 RELATIVITY AND GRAVITATION 30. Gravitational waves. Let us close this chapter by briefly mentioning a method of approximate integration of the field-equations given by Einstein (Berlin Academy pro- ceedings for 1916, p. 688) which exhibits the propagation of gravitational disturbances. Let the g lK differ but little from the galilean values, in a cartesian system, say, or in our previous notation let where the y lK are small. Then Einstein's approximate solution of his field-equations is %. -T',, -KT'XX, (72a) where y f M is the retarded potential of 2/c T M , that is to say, the familiar particular solution - T IK (x, y,z,ct- r)dx dy dz (726) of ' the wave-equation ' ' =-2*7:,. me) In (72b) r is the three-dimensional distance of the point for which 7' l/c is required (for the instant f) from the integration element dxdydz at which the value of T IK prevailing at the instant t is to be taken. c This solution represents gravitation as being propagated with the normal light velocity c, the slight changes of the latter due to the gravitational field itself being manifestly neglected. In this approximation the rigorously non-linear field equations are replaced by linear differential equations of the form (72c), the usual wave-equations. In the sub-case of a stationary gravitational field, when the whole tensor of matter is reduced to r 44 = p, we have by (72b), as the. only surviving y\ K , , K \pdxdydz x!2 744= ~2;j \r ' 2T- APPROXIMATE INTEGRATION 91 where 12 is the ordinary newtonian potential of the gravitating masses, and, by (72a), the only surviving y w , *ft 2fi 711 = 722=733= 744= = - , 47T C 2 so that, as before, the role of the potential 0, is taken over in part by %c*yu. CHAPTER V. Radially Symmetric Field. Perihelion Motion, Bending of Rays, and Spectrum Shift. 31. In order to represent the motion around the sun of a planet as a 'free particle', of mass negligible compared to that of the central body, it is enough to find a radially sym- metrical solution of Einstein's field equations outside the sun, G IK = 0, (III ) considering the origin r = of polar coordinates r, <, 6 as a singular point. As a form of the line-element, sufficiently general for this purpose, let us assume ds* g! dr*~- r*atf - r 2 sin 2 dd*+gt c*dt\ (73) where gi, g, written instead of gu, g 44 , are functions of r alone, of which we shall thus far assume only that &(>) = -1, g4() = l, (74) i.e., that at distances r large compared with a certain length belonging to the sun (which will appear in the sequel) the line-element tends to its galilean form ds 2 = dr 2 r z (d 2 -\- Let us correlate the indices of the coordinates by putting *i, x 2 , x 3 , x 4 = r, 0, 0, ct respectively. Then the metrical tensor in question will con- sist of the components 2i = 2iW, &= -r 2 , 3= -r 2 sin 2 0, gi = g*(r), (73a) where g K has been written for g KK . In the more general case (j~1 _ x/ Y 2 ** SK UX'K ) in which the g K = g KK are any functions of all the variables, we have, for the only surviving associated tensor components, 92 RADIALLY SYMMETRIC FIELD 93 f- - , and therefore, recalling the definition of the Christoffel symbols, = , for all i, K , 2g K dx t >-L , for (75) while all other Christoffel symbols vanish. Applying these formulae to the more special tensor (73o) , writing and using dashes for derivatives with respect to r = #i, we have the rigorous values of the only surviving Christoffel symbols, altogether nine in number, 22 23 These values substituted into the general expressions (55) for the components G tK of the curvature tensor give zeros for all those having t^/c, while the remaining four diagonal com- ponents are, rigorously, J_ 6^ 22 (77) 94 RELATIVITY AND GRAVITATION Thus we have, according to the field equations for r>0, that is, outside of matter, for the two unknown functions gi, g 2 the three differential equations 9/7 ' (a) V+JJuW-V)- =0 r (b) (/*/ -&/)+!+ 1=0 (c) V+V = 0. The last of these equations gives hi+ h* = log (gigO = const., that is to say, by (74), gi g4 = const. = 1 . Equation (6) now becomes g 4 +rg 4 ' = l or so that r(g 4 1) = const. = 2L, say. Thus the rigorous, and the most general, solution of the field Aquations (6), (c) is ultimately, where L is any constant, soijie length, characterising the sun, i.e., here the singular point or centre of the gravitation field. As to the first field equation (a) it is satisfied identically by these values of gi, g 4 .* To express the constant L in terms of M, the sun's mass in astronomical units, we may apply the following reasoning: As we already know, in the approximate equations the newtonian potential ft = M/r is replaced by -- 744. Now, 2 *In fact, since gig*= 1, the left hand member of equation (a) becomes k"+fc*4W/r, and this is, by (78), -~^ [4l-2r+2fr-2L)], which vanishes identically for all r=*=2L. RADIALLY SYMMETRIC FIELD 95 in the present case, 744 = ^4 1= 2L/r. Thus M=c 2 L, whence . , : ,-,; = '.. (79) Ultimately therefore, the line-element (73) corresponding to a radially symmetrical field becomes, rigorously, V, g 4 = --- . r / g l The general equations of motion (15) with the values (76) of the Christoffel symbols become, for t = 2, 3, 4,* d 2 2 dr d(f> ds 2 r ds ds E2 dr d ~| dO ~r <& ' dTJds 4 , , dr dx* z 4 ^s ds *Instead of the first equation of motion (t = 1) it will be more convenient to take the identical equation g lK X t X K =1. 96 RELATIVITY AND GRAVITATION Lay the plane < = 7r/2, the equatorial plane of the coordi- nate system, through the direction of motion of the planet at some instant t . Then, at that instant, d/ds = Q and sin 20 = 0, and therefore, by the first equation, permanently < = ?r/2, that is to say, the planet will describe a plane orbit, and the remaining two equations, together with the identical equation i X K = 1> will become +*L = o ---V-r a # = l, (80o) #4 where 7z 4 = log g 4 and g 4 =l 2L/r. The first two of these equations can be written 4-log(r 2 l?)=0, logfe4*4)=0, as as and give r*'$ = p, g& = k, (81) where p and k are arbitrary constants.* With the values of # 4 and derived from (81) equation (80a) becomes ^ r* or, putting p = , r IA 1 O7" (82) P 2 P 2 The determination of the orbit is thus reduced to a quad- rature. As an alternative we may write down the differential equation of the orbit, by differentiating the last equation with respect to 6, *The first of (81) represents the slightly modified law of Kepler: areas swept out by the radius vector in equal proper times of the particle (s/c instead of t) are equal. PERIHELION MOTION 97 Either equation differs from the familiar equations of celestial mechanics, based on Newton's principles, only by the under- lined last term of the right-hand member. It is well known that in the absence of this supplementary term the orbit is a conic (an ellipse, a parabola or a hyperbola) p = L_ [i+; CO s(0-)] (83) with fixed perihelion, w = const. In fact, equation (82a) is identically satisfied by (83) ; and so is (82) if we put fh (83o) so that the orbit is an ellipse, a parabola or a hyperbola according as k 2 is smaller, equal to or greater than 1 . In general, for orbital velocities comparable with the light velocity, equation (82) gives as an elliptic integral of p, to which corresponds a complicated non-closed orbit. Its dis- cussion may be left to the care of the reader.* Here it will be enough to consider small velocities such as occur among the planets of the solar system. The supplementary term is then small compared with the newtonian ones, and the problem can be solved approximately by a conic (83) with slowly moving perihelion. If dp/ 68 is the derivative of p when d> is kept fixed, and if the term with (d&/d#)* is neglected, we have V = ( d J- + * / " \d0 do> de / \ee ' d& de J : \d0 / de d& de ' and since (dp/36) 2 itself accounts for the first three terms of the right hand member of (82), the perihelion motion will be determined byf *Cf. A. R. Forsyth, Proc. Roy. Soc., XCVII (1920), p. 145, also W. B. Morton, Phil. Mag., XLII (1921), p. 511. fThis reasoning, aiming at the secular motion of the perihelion, pre- supposes the knowledge of absence of a secular variation of the eccentricity . Cf. footnote on p. 99, infra. 98 RELATIVITY AND GRAVITATION dp dp dw , --- = JL,p . dO d de Here (83) can be used with sufficient accuracy for p and its two derivatives, so that du L 1 + 3e cos u + 3e 2 cos 2 pk snw where w = co. Integrating this from to 2?r over or, what for our approximation is the same thing, over u, we shall have the secular perturbation 5o>, the motion of the perihelion per period of revolution. The second and the third terms of the integrand, having in the second and the third quadrants values opposite to those in the first and fourth, contribute nothing to the secular perihelion motion, and the same is true of the first term, since this is the derivative of the periodic function cot u. We are thus left with 27T = _^L f P* \ 5co = cot 2 du t P 2 J o and since cot 2 w is the derivative of cot u+u, *~ /5M . This being essentially positive, the secular motion of the perihelion is progressive, that is, in the sense of the revolution of the planet. If the orbit be an ellipse (e 2 = const. 102 RELATIVITY AND GRAVITATION from a distant source (star) to the earth, if r be, approxi- mately, the shortest distance of the ray from the origin, e.g., from the sun's centre. In the latter case, if R be the sun's radius, we have 4L/^ = 5'88/6'97.10 5 radians = 1"75, so that A = l"75- , n> This is Einstein's famous formula for the displacement of star images seen in comparative angular proximity to the sun's disc. It can be considered as fairly well verified by the results of the Eclipse Expedition at Sobral, Brazil,* of May 29, 1919, which were ultimately estimated to give, when reduced to r = R, the value 1"'98 with a probable error of about six per cent. This is certainly more than a mere order-of-magni- tude coincidence, and speaks strongly in favour of Einstein's theory. The displacements according to Einstein's formula should, of course, be away from the sun and purely radial. The displacements measured on the Sobral plates deviated from radial directions, at least for four out of the seven stars, considerably, to wit by 35, 16, 8, and 6 for the stars numbered 11, 6, 2, and 10, whose distances from the sun's centre were about 8, 4, 2, and 5R respectively. These deviations or the presence of transversal displacement components may well be due to the distortion of the coelostat-mirror by the sun's heat, as pointed out by Prof. H. N. Russell. Yet a refined investigation of this point during the next eclipse seems very desirable, and, as I understand, will be taken special care of at the Eclipse Expedition of September 20, 1922, at which it is designed to avoid the use of a mirror. The field of stars near the sun, during totality, will then be almost as favourable as in 1919.f 34. Shift of spectrum lines. Consider an atom, say of nitrogen, placed in the photosphere of the sun, at rest or practically so. Then its line-element or the element of its 'proper time' will be, by (80), and writing for the present 5 instead of s/c, *The measurements of the Principe Expedition, made under un- favourable weather conditions, seem by far less reliable. fSome preliminary details will be found in Monthly Notices of the Roy. Astr. Soc. for May 1920, p. 628. SPECTRUM SHIFT 103 and any finite interval of its proper time R / \ R Let another nitrogen atom be placed in one of our terrestrial laboratories, at a distance r from the sun's centre. Then its proper-time interval will be In particular, let Ai be the terrestrial, and At the solar time period of one of the natural vibrations or spectrum lines of nitrogen. Now, encouraged by the traditional belief in the somewhat vague 'sameness' of atoms of a given kind, Einstein assumes, as he did already in other circumstances in the special rela- tivity theory, that the said two atoms are 'equal' to each other in the sense of the word that the proper times* of their vibration periods are equal to each other. Eddington in his Report (p. 56) simply says that an atom is "a natural clock which ought to give an invariant measure of an interval ds, i.e., the interval ds corresponding to one vibration of the atom is always the same". Weyl states the case in an apparently more profound way by saying that if the two atoms are "objectively equal to each other, the process by which they emit waves of a spectrum line, when measured by the proper time, must have in both the same frequency". In short, the founder of the theory, as well as his exponents assume, more or less implicitly, that As = A,yi. If so, then the ratio of the solar to the terrestrial period of vibrations is r / \ R or, since in our case R/r is but a small fraction, =1+- =l+2-109.1(T 6 . (88) A/i R *It is now usual to extend this name for ds/c from special to general relativity theory. 104 RELATIVITY AND GRAVITATION Einstein's conclusion then is that the lines of the solar spectrum, compared with those of a terrestrial one, should be shifted towards the red, the proportionate increment of wave- length being 5 ^ = L =2-109.10-, X R or equivalent to a Doppler effect due to a (receding) source velocity of 0*633 kilometers per second. This amounts, for violet light, to about O'OOS A. Now, although with the modern means one-thousandth of an A or even less can be well detected in comparing spectra, Dr. St. John of the Mount Wilson Observatory, who observed 43 lines of nitrogen (cyanogen) at the sun's centre, and 35 at the limb, was unable to detect any trace of the predicted effect. His observations were made and discussed in 1917, and his final conclusion then was that "there is no evidence of a displacement, either at the centre or at the limb of the sun, of the order O'OOS A". Since that time, however, in view of the entanglement of the Einstein effect with shifts of a different origin, and seeing that the results of other astrophysicists were not quite so definite, Dr. St. John suspended his final judgment and is now taking up a thorough discussion of the whole material of solar spectrum shifts from E. L. Jewell's first observations, made about 1890, up to the present. The natural impression now is that it would be premature to either assert or deny the existence of the gravitational spectrum shift. Einstein himself has, on more than one occasion, expressed the very radical opinion that, should the shift be absent, the whole theory should be abandoned. Yet, in view of the hypo- thetical nature of the sameness of atoms in the explained sense of the word, such an attitude, though personally in- telligible, is by no means necessary. It is true that the in- variability of an atomic ^-period of vibration in a gravitational field can, with the aid of the equivalence hypothesis, be re- duced to its invariability while the atom is being moved about, a property of atoms as 'natural clocks' already NATURAL CLOCKS 105 utilised in special relativity.* Yet we do not know whether the atoms actually possess even the latter property. Thus, Einstein's intransigent attitude proves only the strength of his belief that the atoms are or will turn out to be such natural, ideal clocks. But, after all, this is only a guess. A very reasonable one to be sure ; for if not among the atoms, then there is indeed but little hope to find such clocks among other 'mechanisms', natural or artificial. At any rate, a final astrophysical verification of Einstein's spectrum-shift formula, supported perhaps by repeated experiments on canal rays, would be an achievement of fundamental importance. Until then 'the natural clock' will remain a purely abstract concept. *It is this theoretical attribute of atoms which has led to the conclusion that moving hydrogen atoms (canal rays) will emit, in transversal direc- tions, waves (1 iP/c*)"^ times longer than atoms at rest. But even this shift effect, though tried experimentally, does not seem to have ever been detected. CHAPTER VI Electromagnetic Equations 35. Maxwell's equations of the electromagnetic field in empty space supplemented by the convection current pV, or the fundamental equations of the electron theory are, in three-dimensional vector notation, with x = ct, -f-curlE = 0, divM = dE , .. v j. _ --- h curl M = p , div E = p. 8x4 c They contain, apart from the velocity v of moving charges, but two vectors E, M which may be provisionally called the electric and the magnetic forces. As is well-known from the special relativity theory, these equations retain their form or are co variant with respect to the Lorentz transformation, i.e., in passing from one to another inertial system.* They are not, however, generally covariant, and thus not appropriate to the purposes of the general relativity theory. What is covariant with respect to any coordinate trans- formations is the somewhat broader system of equations, containing two more vectors D and B which may be called the electric and the magnetic polarizations,! r)"R +curlE = 0, divB = 0, (A) 8x4 - +curlM = p~, divD = p. (B) _ 6X4 C *Cf. for instance my Theory of Relativity, 1914, Chap. VIII, and, for the historical aspect of the subject, Chap. III. fOr the electric displacement and the magnetic induction respectively. 106 ELECTROMAGNETIC EQUATIONS 107 In a galilean domain or an inertial system D and B reduce to E and M respectively, but in general, in a gravitational field or a non-inertial system, the polarizations differ from the forces, being some linear vector functions of the latter. The general covariance of these two groups of electro- magnetic equations was first noticed and developed by F. Kottler as early as in 1912* and shortly afterwards, with due acknowledgement, incorporated by Einstein into the physical part of his general theory of relativity. Let F IK be an antisymmetric covariant tensor of rank two or a six-vector, which will embody in itself B and E, and thus may be called the magneto-electric six-vector. Then the group (A) of equations can be replaced by the equations (Ai) d# x dx, dx K which are generally covariant since their left hand members are, by (46), Chap. Ill, the components of a general tensor of rank three, the antisymmetric expansion of the six-vector F IK . To compare (Ai) with (A) and to see the simplest form of the correlation between B, E and the six components of F IK use cartesian coordinates or, in the presence of a gravita- tional field (always- 'weak'), quasi-cartesian coordinates and denote by 1, 2, 3 the rectangular components of B, E along the three axes. Then the group (A) of equations will be where 'etc.' means two more equations by cyclic permuta- tion of the suffixes 1, 2, 3 only. On the other hand, writing out (Ai) and remembering that F IK = F Kt , we have *Friedrich Kottler, Raumzeitlinien der Minkowski' schen Welt, Sitzungs- berichte Akad. Wien, vol. 121, section Ila, pp. 1659-1759. -8 108 RELATIVITY AND GRAVITATION 2S u 2 * _ - " - u, etc. 8x4 dx z dx 3 2 s , S i z _Q dXi dx z dx s and these four equations become identical with those just written if we put FM, F 3 i, Fiz Bi, B 2 , B s FU, F%4, 7*34 =1, 2 , ES respectively, or more compactly, if i, k be reserved for 1, 2, 3 only, F ik =-B-, *i 4 = E. (89a) This then is the required correlation for the case in hand. Non-cartesian coordinates will be dealt with in the sequel. Next, let F IK be the supplement of F^ defined, as in (34), by F'-ffF*. (90) Then the group (B) of the electromagnetic equations will be replaced by the four equations --(Vg F tK ) = C\ (50 Vg dx K where C l is a contravariant four-vector. Such also being the left hand member, the divergence of F tK , as in (47), the equa- tions (Bi) will be generally contravariant. To compare them with (B) and to find the correlation proceed as before. Thus, on the one hand, = p , etc. dx 2 dx 3 c i + i . - and on the other hand, remembering that F KK = and F 1 " = -F", ELECTROMAGNETIC EQUATIONS 109 Vo, etc., / etc. dx\ The required correlation is, therefore, l D or, in the previous abbreviated notation, Since F IK is thus seen to embody the electric polarization and the magnetic force, it may be distinguished from its supple- ment by the name of the electro-magnetic six-vector. At the same time we have, by comparing the right-hand members of the two forms of equations, V c c c or, more shortly, s ~ N. (91) V- exhibiting C K as the electric four-current. It is interesting to note that since we can put v i /c=dx i /dx 4: and the last correlation can also be written V g dx 4 Since dx K is a contravariant vector as well as the four-current, the factor of dx K will be an invariant, and since V gdxi dx^dxz dx 4 is also an invariant, the volume of a world-element, we see that the electric charge de=pdx\ dxzdxs is again an invariant. Then, however, not p itself but p divided by the determinant \gik\ will be the system-density of electricity. 110 RELATIVITY AND GRAVITATION It may be well to illustrate the general transformation formulae of by writing them out for the simplest case of two inertial systems S, S' in uniform translational motion relatively to each other. The transformation is in this case the familiar Lorentz transformation, i.e., in cartesian co- ordinates and with the Xi axis along the direction of motion, where p=v/c and 7 = (1 (&} ~ # f if v be the velocity of S' relatively to S. First of all, since in this case the g tK have their galilean values (in both systems), we have B = M, D=E, so that there is no need to consider the supplement of F tK ; it is enough to treat F tK itself. Next, since x 2 , x 3 depend only on x 2t x 3 ', being equal to them respectively, we have Similarly, i.e., Af 2 ' and so on. Thus we get the transformation formulae familiar from the special relativity theory. The corresponding transform- ation of the four-current may be left as an exercise for the reader. It will be kept in mind that the correlations of the forces, the polarizations and the current and charge density to the two conjugated six-vectors and the four-current given in (89a), (896), (91) are valid only for the particular case of a cartesian or quasi-cartesian coordinate system. With other systems, such for instance as the polar coordinates, even in a galilean domain, the correlation formulae are more compli- cated, and contain besides the determinant g the several components g llc of the metrical tensor or (in a non-galilean domain) parts of them, as will be seen later on. It is important ELECTROMAGNETIC EQUATIONS 111 to understand that there is nothing general about these correlations, apart from the fact that F IK embodies somehow the three- vectors B and E, and F^ the vectors D and M, and C" the convection current and the charge density, everything being entangled with the metrical tensor and through it also with gravitation. From the standpoint of general relativity the master equations are henceforth no more the broadened maxwellian equations (A), (B) but the set of generally covariant or contra variant equations (Ai), (Bi) with the metrical link (90) between the two six-vectors. It will be well to gather here these somewhat scattered equations; the whole generally covariant electromagnetic set is thus dx t dx K _L A (Vg>)= V dx K (IV) This will read as follows: the expansion of the magneto- electric six-vector vanishes; the divergence of the electro- magnetic six-vector, the supplement of the former, is equal to the electrical four-current. 36. The four-potential. Manifestly, the first of the equa- tions (IV) will be identically satisfied if we put ^ = ^ _ a*^ dx K dx t where t is a covariant vector. If this be substituted, the six terms destroy themselves in pairs, and the covariant nature of 0, ensures the required tensor character of F tK , the rotation of t (cf. p. 61). The latter, which is seen to embody Maxwell's vector potential and the electrostatic potential, is called the four-potential. With the correlation (89a) the six equations (92) become 112 RELATIVITY AND GRAVITATION or B=curlA, E=- -V0, cot exhibiting the three-dimensional vector A= (0i, 02, 0s) as Maxwell's vector-potential and = 04 as the electrostatic potential. The first group of equations (IV) being thus satisfied by (92), the second group gives -*)>, (93) d* tt AJ which, assuming g tK to be known, are four differential equa- tioijs o the second order for as many components of the four- ' current. Since the four-potential enters only through its rotation, we can without loss to generality subject its com- ponents to a kind of solenoidal condition, as follows. If 0* = g* B 0o De the associated four-potential, a contravariant vector, then its divergence defined by (48) is a general in- variant or scalar, and the condition in question can be written -(Vg>)=0. (94) In a galilean domain the equations (93), (94) become -- V 2 A= P v. -^L-v 2 4> = p c*dt\ c c*dP div A+ 1 * = 0, c dt the familiar equations of the electron theory for the vector potential A = ( K dx K (95) is an invariant. This invariant linear differential form plays the same role with respect to electromagnetism as the quad- ratic differential form with respect to gravitation. As the latter determines, inter alias, the gravitational field, so does the former determine the electromagnetic field. This is only a different way of stating that the K , the coefficients of dl, determine the electromag- netic, similarly as the g tK determine the gravitational field together with the riemannian metrical properties of space- time. Recently a differential geometry somewhat broader than Riemann's was proposed by Weyl who goes deep into the matter and attributes to the linear differential form an equally fundamental metrical (gauging) function as to the quadratic differential form. But reasons of space prevent us from entering here into this subject, and the interested reader must be referred to Weyl's own book* for further information. Moreover, these new physico-geometrical speculations, although undoubtedly attractive, are still being debated between Weyl and Einstein, f and may therefore be appro- priately omitted in a book of the present type. 37. Let us once more return to the electromagnetic equa- tions (A), (B) in order to compare them with the tensor equations (IV) for the case of a non-cartesian system of space coordinates. As a good example of this kind we may take any orthogonal curvilinear coordinates x i} x 2 , # 3 . It is well known that if the space line-element in these coordinates be given by (96) w z Ws Wi *H. Weyl, Raum-Zeti-Materie, 3rd ed., Berlin 1920, 16 and 34. fCf. Einstein's remarks to Weyl's paper, with Weyl's reply, in Berlin Sitzungsber., 1918, and Einstein's recent paper, ibidem, 1921, pp. 261-264. 114 RELATIVITY AND GRAVITATION and if R t be the components of a three-vector R tangential to the Alines of the network, (97) / & A d / R 2 \ d / Ra VI 1 - - I + - I - - I + - 1 I i \W2W3/ dx 2 \w^Wi/ dXt\WiWt/-l and the curvilinear components of curl R are (curl H) t -* J~- ( - 3 ) - I/ * yi etc. (98) Ld# 2 \ w 3 / dx 3 \W2/-} With these expressions the group (A) of equations becomes, provided of course that the w f are independent of time, dxi dx 2 \W 3 Wi and similarly for the group (B) of electromagnetic equations. These equations are to be compared with the first and second of the tensor equations (IV). To find the required correlation in terms of the g tK notice that if the domain is assumed to be a galilean one,* we have ds 2 = g llc dx L dx K = dx the last two equations give '- ^E+Fn(^~ 1 FnE)=0, Cf for every direction of E. Here the operator K " l is the inverse of K. If KI, etc., be the principal values of K and n\, etc., the components of n or the direction cosines of the wave-normal with respect to the principal axes, the last equation gives at once i.e., a propagation velocity independent of the orientation of the light vector, which proves the statement. If gi> 2, 3 are the principal values of the vector operator gut i2> 33, the inverse of -co, then the principal values of -co itself are 1/gi, etc., and we have, by (103) and since , inf - + + (105) gi g2 Such being the formula for the velocity of propagation on the electromagnetic theory of light, it is interesting to com- pare it with the light velocity v yielded directly by Einstein's fundamental equation ds = Q. This velocity is taken along 'the ray' instead of the wave-normal. Thus, by (100), if u be a unit vector along the ray, and Ui its direction cosines, *L. Silberstein, Annalen der Physik, vol. 26, 1908, p. 751 and vol. 29, 1909, p. 523, or Theory of Relativity, London, 1914, p. 56. PONDEROMOTIVE FORCE 119 c- dxi dx k 44 = gik - = gikUiU kj v 2 da dor and especially if HI be the direction cosines with respect to the principal axes of the operator gn, gi 2 , . . . 33, . (106) V 2 g44 Formula (85), used in connection with the bending of rays around the sun, is only a special case of (106). In that case the principal axes are along the radial and all the transversal directions, while the principal values g44 4 and Ui z = cos 2 r), 2 2 +3 2 = sin 2 r7, so that~(106) reduces to (85). If the wave-normal n coincides with a principal axis, say with the first one, we have, by (105), tf/c z = gu/&j an d by (106), c 2 /v z = gi/g44; that is to say, v = b, as it should be. For then the ray falls into the wave-normal. But in general the ray does not coincide with the wave-normal, and so does v differ from *) The question whether the null-line equation (106) is always compatible with the electromagnetic equation (105) may be left to the care of the reader. If the ray be defined, as usual, by the Poynting flux of energy, its direction will be that of the vector product FEM, and all questions concerning the light ray will follow from (104) with K = p as given by (103). 39. Ponderomotive force, and energy tensor of the electro- magnetic field. The general tensors corresponding to these were easily suggested by the results already known from the special relativity theory. The inner product of the magneto-electric six-vector and the four-current, i.e., the covariant vector PI-FC; (V) gives the ponderomotive force on a charge, per unit volume, together with its activity or, in other words, the momentum and the energy transferred, per unit volume and unit time, from the electromagnetic field upon the electric charges. 120 RELATIVITY AND GRAVITATION In fact, using for instance cartesian coordinates and g = 1, we have for the first three components of P t , by (89) and (91), Pi = P f (v*B 3 -v 3 B z ) +EiJ, etc., or if PI, P 2 , Pa be condensed into the three-vector P, which is the familiar formula for the ponderomotive force, while the fourth component becomes p 4 = - - (EM+E^+EM) = - -*- (Ev) c c or, sinceVFvB = 0, which, apart from the factor l/c, is the activity of P. Somewhat more generally, the same formulae will hold with p replaced by p/V g . But it will be understood that from the standpoint of general relativity the master formula for the electromag- netic momentum and energy transfer is again (V), as were before the electromagnetic field equations, all generally covariant. By means of (IV) and (V) the four-force P t can be repre- sented as the covariant derivative of a second rank tensor, a generalization of the array of maxwellian stresses, momen- tum and energy density. Following Einstein's example it will be enough to give here the required form of P t for such co- ordinates for which g= 1, and therefore, by the second of (IV), Thus, by (V), ENERGY TENSOR 121 The second term is, by the first of the equations (IV), K \ , ^^x t dx K dx. dx t dx K But the bracketed expression vanishes. In fact, since the summation is to be extended over all K, X, and since both F- tensors are antisymmetric, this expression can be written dx t dx K dy. to be summed only over K < X. But the third term of the bracketed factor is +dF lK /d# x , so that the whole factor of F0 vanishes, by the first of (IV). Thus j E" r) 7? f) p? <9v r)v r)T O^X OJi i ox t and since here K, X can be replaced by a, /3 and vice versa, The last term can be transformed into J F Kr F K ^ g xp dg pr /dx t , so that P. = -^- (P M F*)- } (F. x F^J-JPA/^^ ^- r . d# x 5af t drc t Finally, if we denote by F the invariant F K ^ 7 r G=--g,,, (HI,) 126 RELATIVITY AND GRAVITATION thus verifying (111) for the case w = 3, whence also G = 6/R 2 , as above. Manifestly, if we took for d ^ s~* *-* "^' G22= ^7 G33= >V G == ]p*. and since g\ 1, g z = ^ 2 sin 2 (r/.R), and all components with IT^K vanish, C.. = -|a., (116') which, apart from the changed sign of the constant factor, agrees with (111) for n = 4. On the other hand, substituting into (115) the alternative solution g4 = l we have, for the line-element (114), = J- 2 &* (=1.2.3); ^44 = 0. The best way of stating the properties of the two solutions is to write the corresponding contravariant tensors which in our case reduce to G tt =G u /g u . These are, for the line- element (114), Gii = G2 = G 3.8 = ^ f G 44 = 0, and for the line-element (116), Thus the time-space defined by the line-element (116) behaves, apart from the common sign change, as an orderly four-fold of constant and isotropic riemannian curvature. This is its characteristic difference from the manifold defined by (114) which is deprived of isotropy and is a rather loose, uneven melange of time and space. Such at least would be EINSTEIN'S NEW EQUATIONS 129 the comparison of Einstein's line-element (114) with de Sitter's, (116), from the standpoint of general geometry. Their physical merit must, of course, be judged by other standards. B. Einstein's New Field-Equations and Elliptic Space. About two years after the publication of the original form of the gravitational field-equations, (III), Chapter IV, Einstein found weighty reasons for slightly modifying them.* Without attempting an exhaustive discussion of all his reasons for that change or amplification we shall give here a brief account of his new field-equations and of some of their consequences. The tensor of matter T M being given, the metrical and at the same time the gravitation tensor components g lK are not, of course, determined by the field-equations alone, as indeed would be the case with any other set of differential equations in infinite space (and time). A necessary supple- ment of the data consisted, exactly as in the case of Laplace- Poisson's equation, in prescribing the behaviour of the g w at infinity. Now, as may best be seen from the example of the radially symmetrical field treated in Chapter V, the g llc were assumed to tend 'at infinity', that is, for ever growing r/L, to their galilean values g tK , say in cartesian coordinates, -1000 0-1 0-1 0001 But such boundary or limit conditions, not being independent of the choice of the coordinate system, have seemed 'repugnant to the spirit of the relativity principle'. In fact, to remain generally invariant the limit tensor would have to be an array of sixteen zeros. Moreover, the adoption of the galilean or inertial tensor at infinity would be tantamount to giving up the requirement of the relativity of inertia. For whereas the inertia or mass of a particle generally depends upon the *A. Einstein, Kosmologische Betrachtungen zur allgemeinen Relativitdts- theorie. Berlin Sitzungsberichte, 1917, pp. 142-152. 130 RELATIVITY AND GRAVITATION g w and these are even at the surface of the sun but slightly different from g lK , the mass of the particle at infinity would differ but very little from what it is near the sun or other celestial giants. In fine, the bulk of its mass would be inde- pendent of other bodies, and if the particle existed alone in the whole universe, it would still retain practically all its mass. As a matter of fact we do not know whether such would not be the case.* But somehow, not uninfluenced by Mach's older ideas, Einstein inclines to the belief that every particle owes its whole inertia to all the remaining matter in the universe. Yet another reason against the said conditions at infinity is given which is based on considerations borrowed from the statistical theory of gases and which would equally apply to Newton's theory. But for this the reader must be referred to Einstein's original paper (/. c., 1). In conclusion Einstein confesses his inability to build up any satisfactory conditions at infinity, in space that is.f But here a way out naturally suggested itself. The conditions at infinity being hard or perhaps impossible to find, let the world or universe be closed in all its space extensions. If this be a possible assumption, no such conditions were needed. Thus Einstein comes to assume space to be a finite, closed three-fold of constant curvature, in short an elliptic space, either of the antipodal (spherical) or of the polar, properly 'elliptic', kind. But, as we saw before, the curvature proper- ties of space-time are modified by the presence of matter, the invariant G, for instance, being proportional to the density of matter. Thus the curvature of space, as a section of the four-fold, can only be approximately constant and isotropic, and Einstein assumes therefore that space is elliptic or very nearly so on the whole, deviating here and there, within and near condensed matter, from the average value of its curvature "L/R 2 and from isotropy, somewhat as, in two dimensions, a *Provided, of course, we had some massless phantoms to serve us as a reference system and thus to enable us to state the lonely particle's perse- verance in uniform motion. t'Fiir das raumlich Unendliche'. There is nowhere a mention of the behaviour at infinite past or future, no doubt, because such questions with regard to time are not urgent in the usual (stationary) type of problems. EINSTEIN'S NEW EQUATIONS 131 slightly corrugated or wrinkled sphere. As we know, the line- element of such a three-space is R and Einstein constructs the line-element which is to determine the four- world ' on the whole ' by simply subtracting dv 2 from In short, far enough from condensed matter, stars, planets, and so on, his line-element, in polar coordinates, is -R 2 sin 2 (d 2 + sintyft*), (114) R a differential form treated in Appendix A.* Now, this line-element is incompatible with Einstein's older field-equations (III). In fact, the corresponding curva- ture tensor consists of the only surviving components G,,= -g,-,, G 14 = 0; G=, (114') *From the four-dimensional point of view, the assumption that three- dimensional space is elliptic is, of course, as unsatisfactory as the older assumption of galilean g tlc at infinity. For although the space properties as defined by dff z are invariant for transformations of the Xi, x^, Xz alone into any x'i, x'z, x' 3 , they cease to be so when all four coordinates are freely transformed. What is then invariant are the curvature properties of the four-fold of which the three-space is an arbitrary section. If at least the four-fold (114) were isotropic, Einstein's elliptic space could be invariantly defined as that of its sections to which corresponds the minimum mean curvature, and this is the mean curvature of the four-fold itself (cf. W. Killing, loc. cit., pp. 79-83). But the four-fold defined by (114) is by no means isotropic, as was explained in A. Figuratively, and with some licence, it resembles not a sphere but rather the surface of a circular cylinder. By (114) not only the value of the curvature of three-space remains unsettled but even its property of being at all a closed space. In fine, the assumption that three-space is elliptic should be as 'repugnant to the spirit of relativity' as was the older condition at infinity. But as a matter of fact it did not appear to Einstein in that light. The clearest way of stating Einstein's new assumption is to say that, outside of condensed matter, it is possible to choose a coordinate system in which the line-element ds* assumes the form (114). 132 RELATIVITY AND GRAVITATION and if these values be introduced into the field-equations (Ilia), which are identical with (III), the result is But 'on the whole', that is, outside of condensed matter, T n , TM, T 33 are to vanish (though the value of T 44 and T = p need not be prejudiced), and since actually gn= 1, etc., the in- compatibility of (114) with (III) is manifest. Such being the case, Einstein is driven to modify his original equations (III) by subtracting from their left-hand members the terms Xg w with a constant X. Thus his new field-equations are G u -Xg = - - (T x - i &. T), (117) 2 and since these give, obviously, G-4\= T, (117a) c 2 they can also be written =~T m . (1176) Since the supplementary term \g tK is itself covariant of rank two, the general covariance of the new equations is manifest. It remains to evaluate the constant X in terms of the curvature 1/R 2 -. Now, if we assumed that outside of 'con- densed matter ' there is no matter at all, i.e., T IK = for all L, K, we should have, by (114 1 ) and the first of (1176), X = l/# 2 , clashing with (117a), through (114 1 ) which would require \ = 3/R 2 . But, as Einstein expressly states, his new theory is to be associated with the approximate concept that all the matter of the universe is spread uniformly over immense spaces. In other words, Einstein substitutes for the granular structure of the universe (the grains being not only planets but stars, nebulae and similar giants) a macroscopically homogeneous distribution of matter, exactly as for many purposes a con- EINSTEIN'S NEW EQUATIONS 133 tinuous homogeneous medium is substituted for an assem- blage of molecules or atoms. The total mass contained in the universe being M and its volume V, Einstein's homogeneous density, prevailing on the whole, is M Only here and there, within the celestial bodies, the density p exceeds p considerably, and is perhaps somewhat larger in interstellar spaces within our galaxy than half way between one star cluster or 'island universe' and another, a million or more light years apart. Moreover, basing himself upon the known fact of the small relative velocity of stars as compared with the light velocity, Einstein makes the approximate assumption that there is a coordinate system, relatively to which matter is on the whole permanently at rest, and in which therefore the tensor of matter is reduced to its 44-th component which is then also its invariant T = p. In fine, we have outside of condensed matter as the only surviving component, and therefore, by (114 1 ) and (1176), \ 1 47T A = = - p . R* c* Thus, Einstein's new field-equations (117) become ulti- mately .=- 8 7 ;r. (U7c) At the same time we see that the curvature of space on the whole is proportional to the average density of matter, The whole volume of elliptic space of the polar or properly elliptic kind being 134 RELATIVITY AND GRAVITATION the total mass of the universe, in astronomical units, will be M=R, (119) 4 which moved some authors to the enthusiastic exclamation: 'the more matter, the more room'. The corresponding 'gravitation radius', or better, the mass in bary-optical units, which is a length, would be L=^ = ^, (lift,) c 2 4 or just one-quarter of the total length of an elliptic straight line.* According even to our coarse knowledge of the average density of matter (some thousand suns per cubic parsec) , and in view of the formula (118), it is impossible to believe in a curvature radius much smaller than 10 12 astronomical units or, say, R IQ 20 kilometers. This would mean, by (119a), a total mass amounting again, in bary-optical units, to almost 10 20 kilometers. To this tremendous total our own sun contri- butes but 1| kilometers, and our whole galaxy not more than 10 10 kilometers. The total would thus require 10 10 such galaxies or Shapley 's ' island universes ' . All these stellar systems may perhaps be found among the spirals. But if Shapley's esti- mate (Bull. Nat. Res. Council, 1921, No. 11, The Scale of the Universe) be materially correct, these island universes are from 500 thousand to 10 million light years apart, and then it remains to be seen whether the last mentioned space would be ample enough. Yet it would certainly be foolish to deny the possibility of a much larger R and of the existence of many more island universes. That Einstein's requirement, at least in the present state of astronomical knowledge, can at any rate be satisfied, is perhaps best seen from its form (118) which is compatible with as small an average density as we please. *The total length of a straight line (geodesic) in the polar kind of space is irR, and in the antipodal or spherical kind of space 2wR. The total volume of the latter space is 2TT 2 R 3 , which would give the double mass, as in Einstein's paper. The space in question being thus far defined only differentially, the choice between the polar and antipodal kind remains free. . SITTER'S SPACE-TIME 135 Further details concerning these cosmological speculations will be found in de Sitter's third paper on Einstein's Theory of Gravitation,* where the role played by elliptic space in astronomy since the time of Schwarzschild (1900) is discussed. The light rays corresponding to Einstein's line-element (114) turn out to be straight lines in elliptic space, and these lines, described with uniform velocity, are also the orbits of free particles. Planetary motion would undergo some modi- fications due to the finite value of R] but these are, for the present, too small to be detected. Nor does Einstein's 'cosmological term', as the supplement gJR 2 to his original field-equations is called, lead to any other predictions verifi- able in our days by experiment or observation. C. Space-Time according to de Sitter. Returning to Einstein's amplified field-equations (117) let us assume, with de Sitter, that there is outside of 'con- densed matter' no matter at all, so that in such domains all the components of T^ , including 7*44, vanish. Thus we shall have, in free space, so to speak, G* = Xg u for all i, K. Now, as we saw in Appendix B, these equations > which are of the form of (111), can be satisfied by the line- element (116), and give G = 12/R 2 . And since, on the other hand, G = g"G lK = Xr^ = 4X, we have 3 ~ R 2 ' This is the solution of the cosmological problem proposed by de Sitter in his last quoted paper. Thus, de Sitter's free space-time is defined by the line-element 2 +sm 2 dd 2 ] (116) R R and is therefore, as we saw, a manifold of constant isotropic *W. de Sitter, Monthly Notices of R.A.S,, November 1917. 136 RELATIVITY AND GRAVITATION curvature. Within matter Einstein's new equations, with X = 3/7? 2 , are valid, i.e., Gu-%=-"*(^- J&.r). (120) J\. C" The isotropy of de Sitter's space-time, expressed by Gu = G 22 = G = G** = , 2> as in (116 2 ;), distinguishes it characteristically from Einstein's space-time. This goes hand in hand with p = outside of matter proper. De Sitter's line-element differs from Einstein's by g44 = COS 2 R instead of 44 = !. Consequently, if the permanency of atoms be assumed as in Chapter V, the spectrum lines of distant stars should be displaced towards the red. If r be the distance of a star from an observer placed at the origin of coordinates, the observed wave-length should be increased from 1 to 1 : cos , becoming infinite for r = R, the greatest distance R 2 possible in a properly elliptic space. Manifestly, everything is at a standstill at the equatorial belt, i.e., all along the polar of any observing station as pole. This, though sounding strangely, entails no actual difficulty at all. As to the spec- trum shift of less distant celestial objects, de Sitter quotes the helium or .B-stars which show a systematic displacement towards the red such as would correspond to a velocity of 4*5 km. per sec. If, as de Sitter suggests, one-third of this is considered as a gravitational Einstein-effect, the remainder may be accounted for by the decrease of 44, and since the average distance of the .B-stars is believed to be r = 3.10 7 astronomical units, we should have R and therefore a curvature radius ^ = |10 10 . But there is, for the present, nothing cogent in the attribution of the said GRAVITATION AND ELECTRONS 137 remainder of spectrum shift to the dwindling of g 44 with mere distance, and it would certainly be premature either to reject or to accept the results of this attractive piece of reasoning. Other consequences of the theory and a more thorough comparison with Einstein's solution will be found in de Sitter's paper. Here it will be enough to mention still that according to de Sitter's line-element the parallax of a star should reach a minimum at r = %irR, the greatest distance in the polar kind of space (which de Sitter prefers to the anti- podal). This minimum, of the semi-parallax, is equal to p=a/R, if a be the distance of the earth from the sun. On the other hand, Einstein's line-element gives, for r = %irR, a vanishing parallax. Since de Sitter's minimum is at least as small as p = 10" 10 = 0". 00002, one cannot reasonably hope to discriminate between the two proposals by direct observations of parallaxes, while indirect ones contain too many assump- tions to be considered as crucial. Soon after the publication of de Sitter's paper Einstein raised some objections to his form of the line-element. For these, however, not altogether crushing, the reader must be referred to Einstein's own paper (Berlin Sitzungsberichte , March 1918, pp. 270-272). D. Gravitational Fields and Electrons. The problem of the equilibrium of electricity constituting the electron as the structural element of matter proper, already attacked by G. Mie and others, has been taken up by Einstein in a paper of April 1919 (Berlin Sitzungsber., pp. 349-356). The result of the investigation is that this tempting question cannot be completely answered by means of the field-equations alone. For details of the reasoning the reader must be referrred to the original paper. It will be enough to mention that the fixed relation between the universal constant X in the amplified field-equations and the total mass of the universe, as related in Appendix B, is here given up. Space continues to be considered as closed but the curvature radius R and, therefore, the volume of the universe appears as independent of the total mass contained in it, though its macroscopic density p is still treated as uniform. INDEX (The numbers refer to the pages) Abraham, M 75 Absolute, Cayley's 51, 52 Absolute differential calculus. . 39 Absolute value, or size, of vec- tor 51 Angle.. . ... 56 Antipodal kind of elliptic space 6 Antisymmetrical tensors 44 Associated invariant 53 vectors 54 Astronomical unit of mass .... 74 Atoms, as natural clocks 104 1 Bary-optical unit of mass 134 Bessel 10 Bianchi, L 28,39,64,71 Boundary conditions 129 Canal rays 105 Cayley 51,52 Centrifugal acceleration 31,37 Christoffel 59 symbols 27 Clifford 13 Closed space 130 Coelostat distortions 102 Compatibility conditions 118 Componens of a tensor 41 Conjugate vectors 54 Conservation laws 87 Constant curvature 67 Constant light-velocity, prin- ciple 2 Contraction, of mixed tensors. 46 Contravariant devriative 59 tensor, defined. 41 Cosmological term ; . 135 Covariance of natural laws 23 Covariant differentiation ..... 59 tensor, defined 41 Current, four- 109 Curvature, gaussian 17, 65 riemannian . . 67 Curvature invariant 79, 125 Curvature tensor . 63, 68 contracted 70 Cyanogen spectrum lines 104 Deflection of light rays 100-102 Density of matter 134 Differential geometry 5 quadratic form ... 14 Differentiation of tensors . . 59 et seq. Divergence of mixed tensor. . . 83 of a vector 62 of a six- vector .... 62 Eclipse expeditions 102 Eddington, A. S 39, 80, 103 Einstein, 1, 10, 11, 12, 19, 22, 24, 28, 38, 39, 56, 58, 61, 70, 77, 82, 86, 88, 104, 107, 113, 129, 130, 137 Electro-magnetic six- vector . . . 109 Electromagnetic equations .... 106 et seq. stress, momentum, and energy 75 Electrons 137 Electrostatic potential 112 Elementary flatness 13 Elevator 11 Elliptic space 6, 130 Energy, principle of 86-87 Energy tensor 75, 122 EotvOs, R 10 Equation of motion, of a free particle 28 Equivalence hypothesis 12 Equivalent differential forms . . 16 Expansion, of a vector 59 of a six- vector .... 62 Fermat's principle 100-101 Field equations, gravitational 70,77,89,132 UNIVERSITY OF CALIFORNIA DEPARTMENT OF CIVIL ENGINEEI rERKELEY. CALIFORNIA 140 INDEX Fixed-starssystem 1 Four-current 109 Four-index symbols 17, 64 Four-potential Ill Four- vector 4, 40 Fundamental quadratic form . . 52 tensor 52 Free particle motion, and geo- desies 8, 20 Galaxies 134 Galilean coefficients 6, 129 Galileo 10 Gaussian coordinates 39 General relativity principle ... 22 Geodesies 7, 26-28 Gradient, of a scalar field 49 Gravitation, and Christoffel symbols 29 Gravitation law, Newton's. ... 73 Gravitation radius 95 Gravitational field equations, 69 et seq., 89, 132 waves 90 Hamiltonian principle 88 Heavy and inert mass 10 Helium or B-stars 136 Hilbert, D 88 Holonomous transformations. . 16 Homaloidal, or flat, manifold . . 67 Hydrogen nucleus 80 Indices, upper and lower 45 Inertia, induced 130 of energy 74 Inertial systems 1 Inner product, of tensors 42 Invariance, of line-element 16 Invariants 42, 46, 47 metrical 53, 57, 62, 79, 88, 125 Island universes 134 Isotropic curvature 67, 125 Jacobian. . . Jeffreys, H.. Jewell, E.L. 15 99 104 Keplerian laws 96, 98 Killing, W 64,124,131 Kottler, F 107 Laplace-Poisson's equation . .69, 73, 74,79 Laue, M. v 75, 76 Law of (algebraic) inertia 16 Levi-Civita, T 39, 41,101 Light propagation, and mini- mal lines 8, 20 Line-element 5 Linear differential form 113 Lipschitz's theorem 66 Local coordinates 12 Lor, matrix 75 Lorentz, H. A 88 Lorentz transformation 4 Mach, E 130 Magneto-electric six- vector . . . 107 Mass, astronomical unit of. ... 74 Matter 74,75,77 equations 82, 85, 123 Maxwell's electromagnetic stress 122 equations 106 Mean curvature 79, 125 Mercury's perihelion motion . . 99 Metrical manifold 50 properties of tensors . 53 Mie, G 10,137 Minimal lines 7 Minkowski 4,75 Mixed tensor, defined 45 Momentum, principle of 86-87 Mosengeil, K. v 74 Natural clocks 104 volume 58 Newcomb 99 Newton 10 Newton's equations of motion. l?6 Node motion, of Venus 99 Non-holonomous transforma- tions 14 Norm, of a vector 53 INDEX 141 Outer product, of tensors 43 Orthogonal coordinates 113 vectors, defined. . . 56 Parallax 137 Perihelion motion 95 et seq. Permanent field 70 Perturbations, secular 99 Planetary motion 95 et seq. Polar kind of elliptic space ... 6 Ponderomotive force, in elec- tromagnetic field 119 Potential, electrostatic 112 four- Ill newtonian 36 retarded 90 vector- 112 Poynting 74, 75 Principe, eclipse expedition . . . 102 Principles of momentum and energy 86-87 Product, inner 42, 48 outer 43 Propagation of gravitation .... 90 Proper time 103 Radially symmetrical field. 92 et seq. Radius, gravitation- 95, 134 of world-curvature . . . 80-81 Rank ,of a tensor 40 et seq. Rankine, A.O., and Silberstein, 117 Reduced tensor 56 Retarded potential 90 Reference frameworks . .l t et passim Relativity principle, general. . . 22 special 2 Ricci, G 39, 41 Riemann 17,51,64 Riemannian manifold 50 Riemann-Christoffel tensor. .62, 63 Rotating system 30-35, 101 Rotation, of a covariant vector 61 Russell, H. N 102 Rutherford 81 Scalar, tensor of rank zero .... 42 Scalar product, of tensors 42 Schwarzschild, K 95, 135 Seeliger 99 Shapley 134 Shift of spectrum lines. 102-105, 136 Sitter, W. de. .33, 99, 101, 127, 135, 136, 137 Six- vector 44 Size, of a vector 51 Sobral, eclipse expedition 102 Space-like vector 53 Special relativity, recalled .... 1-9 Spectrum shift, due to gravita- tion 102-105 due to distance 136 Spherical space. 6 St. John, C. E 104 Stress-energy tensor 75 Sum of tensors 41 Sun, gravitation radius of. . .95, 134 Supplement of a tensor 55 Sylvester 16 Symmetrical tensors 43 Tangential world 13 Tensors SQetseq. Tensor character, criterion of. . 48 Tensor of matter 76, 78 Thirring, H 32 Time-like vector 53 Universe, mass and volume of . 134 Vector 40 Vector potential 112 Velocity of light 25, 118 Venus, motion of nodes 99 Volume 58 Wave, electromagnetic, in gra- vitation field 118 Waves, gravitational 90 Wave surface . 26 Water, curvature in 80 Weight and mass 10 Weyl, H 39, 88, 103, 113 World curvature 80 vector 4 Wright, J. E 39 Zodiacal matter . . 99 D.VAN NOSTRAND COMPANY are prepared to supply, either from their complete stock or at short notice, Any Technical or Scientific Book In addition to publishing a very large and varied number of SCIENTIFIC AND ENGINEERING BOOKS, D. Van Nostrand Company have on hand the largest assortment in the United States of such books issued by American and foreign publishers. 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