ON GRAVITY AND ETHERIAL TENSION

In the arithmetical examples of Chapter [IX] we reckon merely the force between two bodies; but the Newtonian tension mentioned in Chapter [VIII] does not signify that force, but rather a certain condition or state of the medium, to variations in which, from place to place, the force is due. This Newtonian tension is a much greater quantity than the force to which it gives rise; and, moreover, it exists at every point of space, instead of being integrated all through an attracted body.

It rises to a maximum value near the surface of any spherical mass; and if the radius be R and the gravitational intensity is g, the tension at the surface is T₀ = gR. At any distance r, further away, the tension is T = gR²/r.

This follows at once thus:—

Stating the law of gravitation as F = γmm´ / r², the meaning here adopted for etherial tension at the surface of the earth is

T = ∫_R^∞ γE / r² dr = γE / R;

so that the ordinary intensity of gravity is

g = −dT / dR = γE / R² = 4 / 3πργR.

Accordingly, near the surface of a planet the tension is T₀ = gR, or for different planets is proportional to ρR².

The velocity of free fall from infinity to such a planet is √(2T₀); the velocity of free fall from circumference to centre, assuming uniform distribution of density, is √(T₀); and from infinity to centre it is √(3T₀).

Expanding all this into words:—

The etherial tension near the earth's surface, required to explain gravity by its rate of variation, is of the order 6 × 10¹¹ c.g.s. units. The tension near the sun is 2500 times as great (p. [103]). With different spheres in general, it is proportional to the density and to the superficial area. Hence, near a bullet one inch in diameter, it is of the order 10^-6; and near an atom or an electron about 10^-21 c.g.s.

If ever the tension rose to equal the constitutional elasticity or intrinsic kinetic energy of the ether,—which we have seen is 10³³ dynes per square centimetre (or ergs per c.c.) or 10²² tons weight per square millimetre,—it seems likely that something would give way. But no known mass of matter is able to cause anything like such a tension.

A smaller aggregate of matter would be able to generate the velocity of light in bodies falling towards it from a great distance; and it may be doubted whether any mass so great as to be able to do even that can exist in one lump.

In order to set up a tension equal to what is here suspected of being a critical, or presumably disruptive, stress in the ether [10³³ c.g.s.], a globe of the density of the earth would have to have a radius of eight light years. In order to generate a velocity of free fall under gravity equal to the velocity of light, a globe of the earth's density would have to be equal in radius to the distance of the earth from the sun, or say 26,000 times the earth's radius. If the density were less, the superficial area would have to be increased in proportion, so as to keep ρ R² constant.

The whole visible universe within a parallax of ¹/₁₀₀₀ second of arc, estimated by Lord Kelvin as the equivalent of 10⁹ suns, would be quite incompetent to raise etherial tension to the critical point 10³³ c.g.s. unless it were concentrated to an absurd degree; but it could generate the velocity of light with a density comparable to that of water, if mass were constant.

If the average density of the above visible universe (which may be taken as 1.6 × 10^-23 grammes per c.c.) continued without limit, a disruptive tension of the ether would be reached when the radius was comparable to 10¹³ light years; and the velocity of light would be generated by it when the radius was 10⁷ light years. But heterogeneity would enable these values to be reached more easily.

Gravitation is thus supposed to be the result of a mechanical tension inherently, and perhaps instantaneously, set up throughout space whenever the etherial structure called an electric charge comes into existence; the tension being directly proportional to the square of the charge and inversely as its linear dimensions. Cohesion is quite different, and is due to a residual electrical attraction between groups of neutral molecules across molecular distances: a variant or modification of chemical affinity.

APPENDIX 2

CALCULATIONS IN CONNEXION WITH
ETHER DENSITY

Just as the rigidity of the ether is of a purely electric character, and is not felt mechanically—since mechanically it is perfectly fluid,—so its density is likewise of an electromagnetic character, and again is not felt mechanically, because it cannot be moved by mechanical means. It is by far the most stationary body in existence; though it is endowed with high intrinsic energy of local movement, analogous to turbulence, conferring on it gyrostatic properties.

Optically, its rigidity and density are both felt, since optical disturbances are essentially electromotive. Matter loads the ether optically, in accordance with the recognised fraction μ²−1 / μ²; and this loading, being part and parcel of the matter, of course travels with it. It is the only part amenable to mechanical force.

The mechanical density of matter is a very small portion of the etherial density; whereas the optical or electrical density of matter—being really that of ether affected by the intrinsic or constitutional electricity of matter—is not so small. The relative optical virtual density of the ether inside matter is measured by μ²; but it may be really a defect of elasticity, at least in non-magnetic materials.

Electrical and optical effects depend upon e. Mechanical or inertia effects depend upon e². Electric charges can load the ether optically, quite appreciably; but as regards mechanical loading, the densest matter known is trivial and gossamer-like compared with the unmodified ether in the same space.

Massiveness of the Ether deduced from Electrical
Principles.

Each electron, moving like a sphere through a fluid, has a certain mass associated with it; dependent on its size, and, at very high speeds, on its velocity also.

If we treat the electron merely as a sphere moving through a perfect liquid, its behaviour is exactly as if its mass were increased by half that of the fluid displaced and the surrounding fluid were annihilated.

Ether being incompressible, the density of fluid inside and outside an electron must be the same. So, dealing with it in this simplest fashion, the resultant inertia is half as great again as that of the volume of fluid corresponding to the electron: that is to say the effective mass is 2πρα³, where ρ is the uniform density. If an electron is of some other shape than a sphere, then the numerical part is modified, but remains of the same order of magnitude, so long as there are no sharp edges.

If, however, we consider the moving electron as generating circular lines of magnetic induction, by reason of some rotational property of the ether, and if we attribute all the magnetic inertia to the magnetic whirl thus caused round its path,—provisionally treating this whirl as an actual circulation of fluid excited by the locomotion,—then we shall proceed thus:—

Let a spherical electron e of radius a be flying at moderate speed u, so that the magnetic field at any point, rθ, outside, is

H = eu sinθ / r²,

and the energy per unit volume everywhere is μH²/8π.

But a magnetic field has been thought of by many mathematicians as a circulation of fluid along the lines of magnetic induction—which are always closed curves—at some unknown velocity w.

So consider the energy per unit volume anywhere: it can be represented by the equivalent expressions

½ρw² = μH² / 8π = μ / 8π · e²u² sin²θ / r²;

wherefore

w / u = √(μ / 4πρ) · e sinθ / r².

The velocity of the hypothetical circulation must be a maximum at the equator of the sphere, where r=a and θ=90; so, calling this w₀,

w₀ / u = √(μ / 4πρ) e / a²,

and

w / wₒ = a² sinθ / r²

wherefore the major part of the circulation is limited to a region not far removed from the surface of the electron.

The energy of this motion is

½ρ ∫₀^π ∫_a^∞ w² · 2π r sin θ · rdθ · dr,

whence, substituting the above value of w, the energy comes out equal to 4/3πρa³w₀².

Comparing this with a mass moving with speed u,

m = (8 / 3)πρa³(w₀ / u)².

This agrees with the simple hydrodynamic estimate of effective inertia if w₀ = ½ √3·u; that is to say, if the whirl in contact with the equator of the sphere is of the same order of magnitude as the velocity of the sphere.

Now for the real relation between w₀ and u we must make a hypothesis. If the two are considered equal, the effectively disturbed mass comes out as twice that of the bulk of the electron. If w₀ is smaller than u, then the mass of the effectively disturbed fluid is less even than the bulk of an electron; and in that case the estimate of the fluid-density ρ must be exaggerated in order to supply the required energy. It is difficult to suppose the equatorial circulation w₀ greater than u, since it is generated by it; and it is most reasonable to treat them both as of the same order of magnitude. So, taking them as equal,

e = a² √(4πρ / μ)

and m = twice the spherical mass.

Hence all the estimates of the effective inertia of an electron are of the same order of magnitude, being all comparable with that of a mass of ether equal to the electron in bulk. But the linear dimension of an electron is 10^-13 centimetre diameter, and its mass is of the order 10^-27 gram. Consequently the density of its material must be of the order 10¹² grams per cubic centimetre.

This, truly, is enormous, but any reduction in the estimate of the circulation-speed, below that of an electron, would only go to increase it. And, since electrons move sometimes at a speed not far below that of light, we cannot be accused of under-estimating the probable velocity of magnetic spin by treating it as of the same order of magnitude, at the bounding surface of the electron, as its own speed: a relation suggested, though not enforced, by gyrostatic analogies.

Some Consequences of this Great Density.

The amplitude of a wave of light, in a place where it is most intense, namely near the sun where its energy amounts to 2 ergs per c.c., comes out only about 10^-17 of the wave-length. The maximum tangential stress called out by such strain is of the order 10¹¹ atmospheres.

The hypothetical luminous circulation-velocity, conferring momentum on a wave-front, in accordance with Poynting's investigation, comes out 10^-22 cm. per sec. These calculations are given in the concluding chapter of the new edition of Modern Views of Electricity.

The supposed magnetic etherial drift, along the axis of a solenoid or other magnetic field, if it exist, is comparable to ·003 centim. per sec., or 4 inches an hour, for a field of intensity 12,000 c.g.s.

But it is not to be supposed that this hypothetical velocity is slow everywhere. Close to an electron the speed of magnetic drift is comparable to the locomotion-velocity of the electron itself, and may therefore rise to something near the speed of light; say 1/30th of that speed: but in spite of that, at a distance of only 1 millimetre away, it is reduced to practical stagnation, being less than a millimicron per century.

In any solenoid, the ampere-turns per linear inch furnish a measure of the speed of the supposed magnetic circulation along the axis—no matter what the material of the core may be—in millimicrons per sec.

[1 micron = 10^-6 metre; 1 millimicron is 10^-9 metre = 10^-7 centimetre, or a millionth of a millimetre.]

To get up an etherial speed of 1 centimetre per second—such as might be detected experimentally by refined optical appliances, through its effect in accelerating or retarding the speed of light sent along the lines of magnetic force,—would need a solenoid of great length, round every centimetre of which 1000 amperes circulated 3000 times. That is to say, a long field of four million c.g.s. units of intensity.

In other words, any streaming along magnetic lines of force, such as could account for the energy of a magnetic field, must be comparable, in centimetres per second, to one four-millionth of the number of c.g.s. units of intensity in the magnetic field.

APPENDIX 3

FRESNEL'S LAW A SPECIAL CASE OF A
UNIVERSAL POTENTIAL FUNCTION

The modern view of Fresnel's Law may be worded thus:—

Inside a region occupied by matter, in addition to the universal ether of space, are certain modified or electrified specks, which build up the material atoms. These charged particles, when they move, have specific inertia, due to the magnetic field surrounding each of them. And by reason of this property, and as a consequence of their discontinuity, they virtually increase the optical density of the ether of space, acting in analogy with weights distributed along a flexible cord. Thus they reduce the velocity of light in the ratio of the refractive index μ:1, and therefore may be taken as increasing the virtual density of the ether in the ratio 1:μ².

That is to say, their loading makes the ether behave to optical waves as if—being a homogeneous medium without these discontinuous loads—it had a density μ² times that which it has in space outside matter. Calling the density outside 1, the extra density inside must be μ²−1, so as to make up the total to μ².

The μ²−1 portion is that which we call "matter," and this portion is readily susceptible to locomotion, being subject to—that is, accelerated by—mechanical force. The free portion of normal density 1 is absolutely stationary as regards locomotion, whether it be inside or outside a region occupied by ordinary matter, for it is not amenable to either mechanical or electric forces. They are transmitted by it, but never terminate upon it; except, indeed, at the peculiar structure called a wave-front, which simulates some of the properties of matter.

(If free or unmodified ether can ever be moved at all, it must be by means of a magnetic field; along the lines of which it has, in several theories, been supposed to circulate. Even this, however, is not real locomotion.)

Fizeau tested that straightforward consequence of this theory which is known as Fresnel's Law, and ascertained by experiment that a beam of light was accelerated or retarded by a stream of water, according as it travelled with or against the stream. And he found the magnitude of the effect precisely in accordance with the ratio of the locomotive portion of the ether to the whole,—the fraction (μ²−1)/μ² of the speed of the water being added to or subtracted from the velocity of light, when a beam was sent down or up the stream.

But even if another mode of expression be adopted, the result to be anticipated from this experiment would be the same.

For instead of saying that a modified portion of the ether is moving with the full velocity of the body while the rest is stationary, it is permissible for some purposes to treat the whole internal ether as moving with a fraction of the velocity of the body.

On this method of statement the ether outside a moving body is still absolutely stationary, but, as the body advances, ether may be thought of as continually condensing in front, and, as it were, evaporating behind; while, inside, it is streaming through the body in its condensed condition at a pace such that what is equivalent to the normal quantity of ether in space may remain absolutely stationary. To this end its speed backwards relative to the body must be u/μ² and accordingly its speed forward in space must be u(1 − 1/μ²).

For consider a slab of matter moving flatways with velocity u; let its internal etherial density be μ², and let the external ether of density 1 be stationary. Let the forward speed of the internal ether through space be xu, so that a beam of light therein would be hurried forward with this velocity. Then consider two imaginary parallel planes moving with the slab, one in advance of it and the other inside it, and express the fact that the amount of ether between those two planes must continue constant. The amount streaming relatively backwards through the first plane as it moves will be measured by u times the external density, while the amount similarly streaming backwards through the second plane will be (u − xu) times the internal density. But this latter amount must equal the former amount. In other words,

u × 1 must equal (u − xu) × μ².

Consequently x comes out x = (μ² − 1) / μ²; which is Fresnel's incontrovertible law for the convective effect of moving transparent matter on light inside it.

The whole subject, however, may be treated more generally, and for every direction of the ray, on the lines of Chapter [X], thus:—

Inside a transparent body light travels at a speed V/μ; and the ether, which outside drifts at velocity v, making an angle θ with the ray, inside may be drifting with velocity v´ and angle θ´.

Hence the equation to a ray inside such matter is

T´ = ∫ ds / ((V/μ) cos ε´ + v´ cos θ´) = min.,

where sin ε´ / sin θ´ = v´ / (V/μ) = α´.

This may be written

T´ = ∫ cos ε´ ds / V/μ (1 − α´²) − ∫ v´ cos θ´ ds / V²/μ² (1 − α´²);

the second term alone involves the first power of the motion, and assuming that μ²v´ cos θ´ = dφ´/ds, and treating α´ as a quantity too small for its possible variations to need attention, the expression becomes

T´ = μT cos ε´ / (1 − α´²) − (φ´B − φ´A) / V²(1 − α´²),

T being the time of travel through the same space when empty. Now, if the time of journey and course of ray, however they be affected by the dense body, are not to be more affected by reason of etherial drift through it than if it were so much empty space, it is necessary that the difference of potential between two points A and B should be the same whether the space between is filled with dense matter or not (or, say, whether the ray-path is taken through or outside a portion of dense medium). In other words (calling φ the outside and φ´ the inside potential function), in order to secure that T´ shall not differ from μT by anything depending on the first power of motion, it is necessary that φ´B−φ´A shall equal φB−φA: i.e. that the potential inside and outside matter shall be the same up to a constant, or that μ²v´ cos θ´ = v cos θ; which for the case of drift along a ray is precisely Fresnel's hypothesis.

Another way of putting the matter is to say that to the first power of drift velocity

T´ = μ T − ∫ (μ² v´ cos θ´ − v cos θ) ds / V²,

and that the second or disturbing term must vanish.

Hence Fresnel's hypothesis as to the behaviour of ether inside matter is equivalent to the assumption that a potential function, ∫ μ² v cos θ ds , exists throughout all transparent space, so far as motion of ether alone is concerned.

Given that condition, no first-order interference effect due to drift can be obtained from stationary matter by sending rays round any kind of closed contour; nor can the path of a ray be altered by etherial drift through any stationary matter. Hence filling a telescope tube with water cannot modify the observed amount of stellar aberration.

The equation to a ray in transparent matter moving with velocity u in a direction φ, and subject to an independent ether drift of speed v in direction θ, is

∫ ds / (V/μ cos ε + v/μ² cos θ + u[1 − (1/μ²)] cos φ) = const.

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