September 23, 1922] 



NA TURE 



433 



may be interpreted as the conservation of momentum 

 and of energy, provided that the momentum and 

 energy of the electrons are denned as above, and that 

 the momentum and energy of the field are included, 

 the momentum of the field per unit volume being 

 defined as il/c", where II is Poynting's vector. Ob- 

 servations on the spectral lines of hydrogen, and 

 Guye and Lavanchy's experiments on cathode rays, 

 confirm these results. 



III. Fundamental Assumptions of Einstein's 

 Generalised Theory (1915). — (1) For an infinitelv 

 small region of space and time, axes may be chosen so 

 that the restricted theory is true in that region. This 

 implies that for two events there exists a certain 

 absolute quantity, the interval ds, which, by a suitable 

 choice of co-ordinates, may be expressed as before, but 

 which in a general system of co-ordinates, x 1 x 2 x 3 x 4 

 (these being arbitrary functions of x y z t) , take the form 

 s ! (Z r Z s g ri dx r dXg), where r and s take all values from 

 1 to 4, and the g's are functions of x x x 2 x s x t . 



(2) All physical laws must be expressible by means 

 of equations which are valid for all co-ordinate 

 systems. That is to say, the equations are covariant, 

 or unaltered in form, for the most general transforma- 

 tion (not necessarily linear). Newton's law of 

 gravitation and all other laws that do not satisfy this 

 condition are to be modified so as to conform with it. 



(3) The Principle of Equivalence. — A gravitational 

 field of force at a point or infinitely small region is 

 exactly equivalent to a field of force introduced by a 

 transformation of the co-ordinates of reference, so 

 that by no possible experiment can we distinguish 

 between them. (Eddington pointed out that the 

 assumption is made for phenomena which depend on 

 the g's and their first differential coefficients, and in 

 general it will not apply to those involving second 

 differential coefficients.) 



(4) The path of a particle in a gravitational field is 

 such that Sfds = o. (For the case when there is no 

 gravitation this reduces to Newton's first law of 

 motion.) This assumption reduces particle dynamics 

 to something like the geometry of geodesies on sur- 

 faces, except that we have four independent variables 

 instead of two. 



(5) Although the coefficients in the expression for 

 ds 2 are capable of infinitely many forms, according to 

 the system of co-ordinates used (just as in measure- 

 ments on a surface the square of the shortest distance 

 on the surface between two points can be similarly 

 expressed in many forms corresponding to the choice 

 of the independent variables), yet these g's are not 

 quite arbitrary functions of the co-ordinates, but 

 satisfy a set of partial differential equations (analogous 

 to those which for a surface express intrinsic pro- 

 perties of that surface). These differential equations 

 are assumed to be of a certain particular form, known 

 as those expressing the vanishing of the contracted 

 Riemann-Christoffel tensor. (A tensor may roughly 

 be defined as a generalised vector. If all its com- 

 ponents vanish in one system of co-ordinates, they 

 all vanish in any other system.) This assumption is 

 not quite as arbitrary as it looks, for it is the second 

 simplest set which is of the covariant form required 

 by assumption (2). The simplest set of all corre- 

 sponds to the absence of any gravitational field. 



(6) The energy of a gravitational field exerts 

 gravitating action just like ordinary masses. This 

 assumption leads to equations which may be inter- 

 preted as implying the conservation of momentum 

 and energy, including contributions due to the 

 gravitational field (and to the electromagnetic if 

 present). 



Mathematical Deductions from these Assumptions. — 

 (a) Formula? for the Interval. — By solving the differ- 

 ential equations the g's may be obtained. The 



NO. 2760, VOL. I io] 



number of solutions is infinite. For a single heavy 

 mass, choosing the units so that c and the gravita- 

 tional constant are unity, 

 Schwarzschild gave 



ds 2 = ( 1 - 2 -y ) dP - ( 1 - 2 " Z ) *dr 2 - r 2 de 2 - r 2 sin W0 2 . 

 F. W. Hill and G. B. Jeffery gave 



ds 2 



irl 



dt 2 -\ i + - 



(dr 2 + r 2 de 2 + r 2 s:n 2 ed</> 2 ). 



and Painleve has given a great variety. 



(b) Perihelion of Mercury. — From any of these 

 forms and assumption (4) we can by the "Calculus of 

 Variations determine the orbit of a planet. The 

 orbits so deduced differ very little from those cal- 

 culated on the Newtonian laws. The only difference 

 big enough to be observed is that for Mercury. 

 Leverrier estimated that the older theory differed 

 from observation by about 43" per hundred years. 

 Einstein's theory accounts for these 43". (But 

 Grossmann (1922) has recalculated the old discrep- 

 ancy as 38", not 43".) 



(c) Deflection of Ray of Light bv Sun's Gravitational 

 Field. — The rays should be slightly curved, as if the 

 gravitational field round the sun were a converging 

 lens, thus making stars on opposite sides of the sun 

 appear farther apart than when the sun is in another 

 part of the sky. The result of the measurements 

 made during the solar eclipse of May 29, 1919, 

 agreed very closely with Einstein's predictions. This 

 is strong evidence in support of Einstein's modifica- 

 tion of the Newtonian law, as on the old law the 

 deflection should be only half the amount predicted 

 by Einstein and actually observed. 



(d) Spectral Shift. — Einstein believes that the 

 formula for ds 2 implies that the spectral lines in the 

 light coming to us from the surfaces of big stars 

 should appear shifted towards the red end of the 

 spectrum. Eddington and others think it possible 

 that this argument may be founded on an assumption 

 which may be rejected while the rest of the relativity 

 theory is retained. Grebe and Bachem (Bonn) claim 

 to have observed the predicted effect, and so do 

 Perot and Buisson and Fabry ; St. John claims to have 

 shown that it does not occur, but his results have 

 been doubted. The experimental difficulties are 

 enormous. 



(e) Apparent Contraction of a Rod placed radially in 

 a Gravitational Field. — Einstein deduces this from the 

 formula for ds 2 and also deduces that there is no such 

 tangential effect. Painleve (1921) strongly objects 

 to these deductions and points out that by taking 

 other forms of ds 2 we can reject these conclusions, 

 while retaining all the verifiable results of the theory. 

 If Einstein's views are correct, Euclidean geometry 

 (e.g. Pythagoras's theorem) is not exactly true for 

 measurements made in a gravitational field. It will 

 be replaced by Riemann's geometry. 



IV. Einstein's Cosmological Theory (1917). — 

 The leading feature of this is that our universe, as 

 measured by material rods or light rays, is finite, so 

 that a ray of light will never get more than a certain 

 distance from its starting-point. However, he is 

 willing to admit that other universes may exist out- 

 side this limit, but such that their light can never 

 meet ours. Eddington and others regard this theory 

 rather unfavourably. 



V. Einstein's Views on the JEther (1920). — 

 Space is endowed with physical qualities. In this 

 sense, therefore, there exists an "aather." Without 

 it there would be no propagation of light. But this 



