394 



NATURE 



[December iS. 1919 



comes to 2-3x10-' dpjdx radians, where p is the sur- 

 face pressure in dynes cm.-' and dx is in cm. 



Lewis F. Rich.ardson. 

 Benson Observatory, Wallingford, 

 December 12. 



1, OF COURSE, admit the force of the remarks in the 

 letters which appeared in Nature of December 11. 

 But the problem of air refraction during a total eclipse 

 is a verv complicated one. The air is not in equili- 

 brium. There is, I imagine, a downward rush of cold 

 air in places deprived of the sun's radiation, as well 

 as a lateral motion of the air from all sides towards 

 such places. The whole refraction effect depends on 

 the shape of the changing surfaces of equal density, 

 and the gradient of density perpendicular to these 

 surfaces. The effect observed would be about equal 

 to the ordinary refraction effect caused by the atmo- 

 sphere at ij° from the zenith, and then the rays of 

 light are nearly perpendicular to the surfaces of equal 

 density. 



It is well to remember that, perhaps unfortunately, 

 the stars in the neighbourhood of the sun during a 

 total eclipse must be viewed through air of which t'lc 

 distribution of density must not be assumed to be the 

 same as that of the atmosphere in its normal state. 



.\lexr. Anderson. 



EINSTEIN'S RELATIVITY THEORY OF 

 GRAVITATION.^ 



III. — The Crucial Phenomena. 



IN the article last week an attempt was made to 

 indicate the attitude of the complete relativist 

 to the laws which must be obeyed by gravitational 

 matter. The present article deals with particular 

 conclusions. 



As Minkowski remarked in reference to Kin- 

 stein's early restricted principle of relativity : 

 " From henceforth, space by itself and time by 

 itself do not exist ; there remains only a blend of 

 the two " (" Raum und Zeit," 1908). In this four- 

 dimensional world that portrays all history let 

 (.-Vj, x^, x^, X4) be a set of co-ordinates. Any par- 

 ticular set of values attached to these co-ordinates 

 marks an event. If an observer notes two events 

 at neighbouring places at slightly different times, 

 the corresponding points of the four-dimensional 

 map have co-ordinates slightly differing one from 

 the other. Let the differences be called 

 (d.Vj, dx^, dx^, dxf). Einstein's fundamental 

 hypothesis is this : there exists a set of quantities 

 §„ such that 



gndxi^+2giidx^dx^+ . . . + g^dx^^ 



has the same value, no matter how the four- 

 dimensional map is strained. In any strain g^^ 

 is, of course, changed, as are also the differences 

 dx.* 



If the above expression be denoted by (dsf, 

 ds may conveniently be called the interval between 

 two events (not, of course, in the sense of time 

 interval). In the case of a field in which there 

 is no gravitation at all, if dx^ is taken to be dt, it 



^ Previous articles appeared in Nature of December 4 and 11. 



* The gravitational field is specified by tlie .set of quantities ^ri. When 

 the gravitational field is small, thes; are all zero, excrpt for ^44, which is 

 approximately ths ordinary Newtonian gravitational potential. 



NO. 2616, VOL. 104] 



is supposed that d.^^ reduces to the expression 

 dx{- + dx^ + dx^-c''dt^, where c is the velocity of 

 light. If this is put equal to zero, it simply ex- 

 presses the condition that the neighbouring events 

 correspond to two events in the history of a point 

 travelling with the velocity of light. 



Einstein is now able to write down differential 

 equations connecting the quantities g^ with the 

 co-ordinates (xj, Xo, -Vg, x^), which are in com- 

 plete accord with the requirement of complete 

 relativity.^ These equations are assumed to hold 

 at all points of space unoccupied by matter, and 

 they constitute Einstein's law of gravitation. 



Planetary Motion. 



The next step is to find a solution of the equa- 

 tions when there is just one point in space at which 

 matter is supposed to exist, one point which is a 

 singularity of the solution. This can be effected 

 completely ' : that is, a unique expression is ob- 

 tained for the interval between two neighbouring 

 events in the gravitational field of a single mass. 

 This mass is now taken to be the sun. 



It is next assumed that in the four-dimensiona; 

 map (which, by the way, has now a bad twist in 

 it, that cannot be strained out, all along the line 

 of points corresponding to the positions of the 

 sun at every instant of time) the path of a particle 

 moving under the gravitation of the sun will be 

 the most direct line between any two points on it, 

 in the sense that the sum of all the intervals 

 corresponding to all the elements of its path is the 

 least possible.'' Thus the equations of motion are 

 written down. The result is this : 



The motion of a particle differs only from that 

 given by the Newtonian theory by the presence of 

 an additional acceleration towards the sun equal 

 to three times the mass of the sun {in gravitational 

 units) multiplied by the square of the angular 

 velocity of the planet about the sun. 



In the case of the planet Mercury, this nev^' 

 acceleration is of the order of lo-^ times the New- 

 tonian acceleration. Thus up to this order of 

 accuracy Einstein's theory actually arrives at New- 

 ton's laws : surely no dethronement qf Newton. 



The effect of the additional acceleration can 

 easily be expressed as a perturbation of the New- 

 tonian elliptic orbit of the planet. It leads to the 

 result that the major axis of the orbit must rotate 

 in the plane of the orbit at the rate of 429" per 

 century. 



Now it has long been known that the perihelion, 

 of Mercury does actually rotate at the rate of about 

 40" per century, and Newtonian theorv has never 

 succeeded in explaining this, except by ad hoc 

 assumptions of disturbing matter not otherwise 

 known. 



Thus Einstein's theory almost exactly accounts 



'■i These equations take the place of the old I aplace equation r-\'=o. Just 

 as that equation is tlie only differential equation of the second order which is 

 entirely independent of any chanqe of ordinary spice co-ordinates, so 

 Einstein equations a-^e uniquely determined hv the condition of relativity. 



'■> The result is that the invariant interval </j is given by 



A2 = (, _j„,/,.)(^/2 _rf,-2)_ ,2(a'»2+sin2 erfifJ), 



the four co-ordinates being now iiilerpreted as lime and ordinary spherical 

 po'ar co-ordinates. 



* This corresponds to the fact that in a field where there is no acceleration 

 at all the path of a particle is the shortest distance between two points. 



