March ii, 1920] 



NATURE 



Letters to the Editor. 



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Gravitational Deflection of High-speed Particles. 



Mr. Leigh Page has given a very simple method of 

 treating the motion of high-speed particles in a 

 gravitational field on Einstein's theory (Nature. 

 February 26, p. 692). In one respect his results differ 

 from those which have been obtained by more 

 laborious methods, and I think that some error must 

 have crept in, either through a failure of his ap- 

 proximation or from some other cause. He finds 

 that a particle travelling with the velocity of light 

 would be undeflected, whereas a ray of light is 

 deflected. It would be difficult to reconcile this with 

 the principle of equivalence, which seems to require 

 that the trajectory of a material particle should 

 approach that of a light-pulse as the velocity 

 approaches that of light. 



The differential equation of the orbit of a material 

 particle moving with anv speed is [Report, Phvsical 

 Society, p. 51, equation (31-2)] 



d'u/de' + u^m/h'+^mu', (u=x/r), 



where the constant h = r^dO/ds. It is from this exact 

 equation that the motion of perihelion of Mercury is 

 obtained. For motion with the speed of light ds = o, 

 so that h is infinite, and the equation becomes 



The solution is 



neglecting m^R*- 



In Cartesian co-ordinates this becomes 

 r=R-^' -^' + 2^ 



The asymptotes are found by taking y very large 

 compared with x, giving 



Hence the angle between them is 4W/R, agreeing 

 with the result for the deflection of light rays. 



I have verified by the usual methods the othe.*- 

 principal result given by Mr. Page, that for radial 

 motion the force (relative to the co-ordinates used) is 

 a repulsion if the speed exceeds i/'s/^ times the 

 velocity of light. 



With regard to the question whether the svstem 

 of an atom on the sun can be identical with that of 

 an atom on the earth, inasmuch as the warping of 

 space-time is different in the two places, it is clear 

 that the identity cannot be exact ; but this loophole 

 for escape from the predicted shift of the Fraunhofer 

 lines does not seem to be very promising. If the 

 "intervals" of vibration of the two atoms are not 

 the same, the difference must depend on some in- 

 variant of space-time which differs at the two places. 

 T do not think that any invariant of order tn/r exists. 

 The simplest invariant which does not vanish is 



it is rather laborious to work out the actual value of 

 this (since it consists of 65,536 terms), but it appears 

 to be of order m'/r^. The* Fraunhofer displacement 

 depends on terms of the much greater order of magni- 

 tude mjr. A. S. Eddingtox. 

 Observatory, Cambridge. 



NO. 2628, VOL. 105] 



Gravitational Shift of Speetral Linet. 



The assumption that the equations of motion in a 

 gravitational field can be deduced from a condition of 



the form ?J'ds = o is in itself little more than a very 

 natural way of expressing the principle of least 

 action. Ihe greatness of Einstein's theory really lies 

 in the suggestion, made apparently on purely a priori 

 grounds, that a certain set of six relations between the 

 coefficients in the formula for ds', which are true 

 when no heavy body is near, still hold near one. 

 These are found to make the coefficients determinate, 

 whereas previously they were quite arbitrary, and the 

 observed motions of the planets, including the advance 

 of the perihelion of Mercury, are at once deduced. 



The displacement of star images during an eclipse 

 is based on the further very plausible assumption 

 that a light-wave moves like a material particle of 

 zero mass starting from an infinite distance with the 

 velocity of light there. Now that this displacement 

 has become a result of observation, the data are just 

 enough to make it possible to reverse the argument 

 and deduce the fundamental assumption of the theory 

 from observation, as I have done in a forth- 

 coming paper in the Monthly Notices of the Royal 

 Astronomical Society. Neither in Einstein's discussion 

 nor in mine is any identification of ds with an invari- 

 I able line element in four-dimensional space-time 

 I relevant to the theory; and as the application of the 

 j theory is purely physical, I think it undesirable that 

 ] any such abstract idea should be made to appear as 

 I part of it. Physically, the invariance of ds means 

 I simply that the "motion of a particle can be described 

 I in terms of any set of co-ordinates we like to choose. 

 ; In discussing these phenomena all positions and 

 times are referred to an observer at the centre of the 

 sun, and it is not necessary to determine the relations 

 between his measures and ours, for the uncertainty in 

 these would not affect the observed quantities 

 appreciably. The problem of the shift of spectral 

 lines, however, depends essentially on such a com- 

 parison. About part of the theory of it there can be 

 no reasonable doubt, namely, the assumption that 

 the vibration on the earth appears to any observer 

 to have the same period as the vibration on the sun 

 that causes it. What is doubtful is whether the atom 

 on the sun vibrates in the same time as a similar atom 

 on the earth. Einstein assumes that it does not, but 

 that the increase in ds in a period is the same for 

 both, and deduces the shift of the spectral lines. 



There is nothing very bizarre about this ; it only 

 means that when we move about we must refer our 

 observations to time standards in the place where 

 these were originally used, and not expect that they 

 will serve the same functions if we carry them about 

 with us. An analogy from colour will illustrate this. 

 Suppose we have a standard of redness in the form 

 of a particular red body. We judge the redness of 

 other bodies by comparison with this. Now suppose 

 we go to a place where the prevailing illumination is 

 green, but where our standard of redness is still visible 

 through a window. We then say that none of the 

 things in the room look red, but our judgments as to 

 what outside bodies look red are the same as before. 

 Our standard is now brought into the room. Are we 

 going to .say that it looks red still? If we do, we 

 shall have to say that the red external bodies that 

 have not been rrioved have been changed in colour 

 bv the motion of our standard, which is at least in- 

 convenient, and which most people would call absurd. 

 Therefore we sav that our colour standard has been 

 altered bv its displacement, and choose another 

 standard from among the visible external bodies. 

 Similarlv, if an observer on the earth went to the 



