*

formed at the point tends to a particular volume, then there exists at every point in the world a definite locus, viz. the boundary of the electron (averaged if necessary) which could potentially exist at that point. This locus must express some property of the structure of the field-entity; and it appears inevitable that the locus must be the locus of curvature of the world at that point. The locus will be judged to be a sphere of constant radius, not through any intrinsic shape or size (since shape and size are relative terms and cannot be intrinsic), but because our actual criterion of symmetry and scale at different points in the world rests ultimately on this locus. The whole principle of physical measurement rests on the convention that direction and position are not in themselves relevant to measurement, so that measurement conventionally assigns extension in such a way that the extensional properties of the unit of material structure are on the average independent of position and direction. The mathematical expression of our conclusion that the locus of curvature is a sphere of constant radius leads at once to Einstein's law of gravitation.

I think there can be little doubt that the law of gravitation arises in this way, through the fact that we have to study the structure of the field-entity with apparatus whose parts are themselves constituted in relation to the structure of the field-entity, so that the intrinsic properties of the field-entity disappear in the vicious circle. The law is not of the nature of a constraint (such as Least Action) imposed on the field-entity, but expresses the principle of measurement t. Admittedly the argument makes certain jumps, appealing to broad principles instead of following through a full theory of structure; and this is necessitated by our present ignorance of the details of material structure. That brings me back to the first contention in this paper; we cannot separate off the field-laws as a province of physics fully understood and

*This is, of course, not the locus of the centre of curvature, which lies in some fifth dimension which does not exist. In each direction a radius is drawn proportional to the radius of curvature of the corresponding section.

Cf. E. Cunningham, 'Relativity and the Electron Theory,' 2nd edition, p. 137. "The law of gravitation becomes the condition which singles out a measure-system." But whereas he appears to consider that our actual measure-system is determined so as to give the greatest possible simplicity to the laws of nature, I consider that it is determined by the structure of the measuring appliances which we have to use.

unaffected by our ignorance of the laws of matter. When for instance we ask why Einstein's equations of gravitation should be chosen rather than one of the more complex systems of covariant equations which are possible, we have to seek an answer, not in the structure of the field, but in the structure of the electron. It depends on whether the form of the electron corresponds to the locus of curvature or to some more complicated absolute locus in the world. Every thing points to the locus of curvature as the more likely; but until the precise mode of formation of the electron is decided, other solutions may remain as a perhaps far-fetched possibility.

APPENDIX.

=

Proof that the equations GM express the condition that the radius of curvature is the same at all points in all directions.

A four-dimensional manifold with general Riemannian geometry can be represented as a surface in Euclidean. space of ten dimensions. At any point the tangent and the normal give five dimensions, which osculate the surface to an order sufficient for the calculation of the curvature.

Let then the equation of this portion of the surface, referred to rectangular Euclidean coordinates (1, 2, 3, ,) along the lines of curvature and the normal, be

22 = k1x12 + k2 æq2 + k3x32+k4x4?

so that ki, ka, ka, k, are the reciprocals of the principal radii of curvature. We have, by Euclidean geometry,

— ds2 = dz2 + dæ12+dx22+dx32 +dx42.

Since four coordinates are sufficient to specify a point on. the surface, we eliminate, obtaining

-ds2= (1+k2x2)dx2+...+2k12xæ ̧ædæ1dæ2+...

− (1 + kμ2 xμ2), Iμv=-kuk,, (not summed).

Υμν

μ

Accordingly the first derivatives of the g's vanish at the

origin, and the g's themselves have Euclidean values there; but the second derivatives contain the curvatures.

To calculate Gu, at the origin, we have

since the remaining terms vanish with the first derivatives of the g's. Using values at the origin, this further reduces

Working out the terms in detail, we easily find

showing that the radius of curvature is isotropic and equal

to a constant.

The argument in the paper referred only to spacedimensions, so that our introduction of the time-dimension (or rather, imaginary time a) here may seem to be going beyond what has been justified. But the space-section of the four-dimensional continuum can be taken in any number of different directions, corresponding to the Lorentz-transformation, so that actually four dimensions are covered by the argument.

LXXXVIII. Does an Accelerated Electron necessarily
Radiate Energy on the Classical Theory?

To the Editors of the Philosophical Magazine.
GENTLEMEN,

N your March number Dr. Milner discusses the question of the radiation from an accelerated electron in connexion with a generalization of a solution of Maxwell's equations obtained by Schott. This solution gives the field of a Lorentz electron of large mass moving up from infinity and then back again along a line of force in a uniform electric field, and Dr. Milner adopts it to give the field of two electrons with equal and opposite charges moving first towards one another and then separating after coming simultaneously to rest, the motion again being along a line of force in a uniform electric field.

In the electromagnetic field thus presented by Milner, there is quite clearly no irreversible radiation of energy either away from or into the field, a fact which at first sight appears to contradict the generally accepted notion that an electron in accelerated motion is radiating energy. A closer examination will, however, soon prove that there is in fact no discrepancy.

The case for radiation may be presented in the following form-When the velocity of a moving electron is altered the necessary modification in the surrounding field is effected by means of a thin spherical shell of disturbance spreading out symmetrically from the instantaneous position of the electron, sweeping up the old field as it proceeds outwards and leaving behind the new. As this shell gets farther and farther from the electron its field approximates more nearly to the type associated with radiation, and the energy of this radiation, which never sinks below a finite positive limit, represents a proper loss by radiation of the intrinsic (localised velocity) energy of the electron.

It is quite clear from this presentation that there is no sense in the expression "radiation from the electron" unless the radiation shell created at any instant can in fact get away from the electron, that is unless the velocity and acceleration of the electron are small compared with the velocity of radiation. The limits have been calculated and amply cover all conceivable physical cases, so that there would appear to be full justification for the practical generalization of the notion of radiation from moving electrons.

In Dr. Milner's case, however, the electrons are moving with a rapidity which appreciably departs from the velocity of light only for a very short part of their total infinite path *, so that except perhaps at the turning point in each path, either the velocity or the acceleration would exceed the limits laid down for proper radiation; in other words, the radiation shell created at any instant cannot in general be expected to free itself from the electron, so that its field would always remain mixed up with the velocity field which properly determines the mass and energy of the electron. Near the turning point the ordinary conditions of small velocity and acceleration are realized, and here Milner verifies the usual formula for an instantaneous irreversible radiation.

It is evident, therefore, that the solution presented does not in reality contradict our previous notions of these matters except in so far, perhaps, as the conditions in it are such as are never presumed to be realized in actual practice.

The subsequent question as to the complete suitability of the retarded point potentials, and the consequent necessity for boundaries of discontinuity in the electromagnetic field, does not seem open to much doubt. Of course it involves regarding the whole subject from the point of view which concentrates mainly on the electrons and their motion. The alternative is to regard the infinite field as the fundamental entity, and then to deduce the motions of the various electrons involved in it as one aspect of the conditions in the field, but this is infinitely more difficult than the reverse process, if only for the reason that it involves in any problem. an a priori specification for a quadruply infinite number of space-time points instead of merely for a finite number of electrons. And ultimately there can be no discrepancy in the results for any given problem approached in the two ways; for two field solutions satisfying the electromagnetic equations and giving the same polar nuclei with the same motions must be identical.

The University, Manchester,
July 29th, 1921.

Yours faithfully,

G. H. LIVENS.

A point moving with the velocity of radiation and starting from infinity with one of the electrons, would only gain a finite distance in the infinite time required for the motion.