# Aether and Matter/Chapter 10

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Aether and Matter, Cambridge University Press, 1900, Online and at Commons (djvu)
Chapters X-XI (pp. 161-193) contain Larmor's version of the Lorentz transformation and the derivation of time dilation and length contraction.
102. We shall now consider the material system as consisting of free aether pervaded by a system of electrons which are to be treated individually, some of them free or isolated, but the great majority of them grouped into material molecules: and we shall attempt to compare the relative motions of these electrons when they form, or belong to, a material system devoid of translatory motion through the aether, with what it would be when a translatory velocity is superposed, say for shortness a velocity v parallel to the axis of x. The medium in which the activity occurs is for our present purpose the free aether itself, whose dynamical equations have been definitely ascertained in quite independent ways from consideration of both the optical side and the electrodynamic side of its activity: so that there will be nothing hypothetical in our analysis on that score. An electron e will occur in this analysis as a singular point in the aether, on approaching which the elastic strain constituting the aethereal displacement (f, g, h) increases indefinitely, according to the type
$-e/4\pi \ .\ (d/dx,\ d/dy,\ d/dz)r^{-1}:$
it is in fact analogous to what is called a simple pole in the two-dimensional representation that is employed in the theory of a function of a complex variable. It is assumed that this singularity represents a definite structure, forming a nucleus of strain in the aether, which is capable of transference across that medium independently of motion of the aether itself: the ​portion of the surrounding aethereal strain, of which the displacement-vector (f, g, h) is the expression, which is associated with the electron and is carried along with the electron in its motion, being as above $-e/4\pi \ .\ (d/dx,\ d/dy,\ d/dz)r^{-1}$. It is to be noticed that the energy of this part of the displacement is closely concentrated around the nucleus of the electron, and not widely diffused as might at first sight appear. The aethereal displacement satisfies the stream-condition
$df/dx+dg/dy+dh/dz=0,\$
except where there are electrons in the effective element of volume: these are analogous to the so-called sources and sinks in the abstract theory of liquid flow, so that when electrons are present the integral of the normal component of the aethereal displacement over the boundary of any region, instead of being null, is equal to the quantity Σe of electrons existing in the region. The other vector which is associated with the aether, namely the magnetic induction (a, b, c), also possesses the stream property; but singular points in its distribution, of the nature of simple poles, do not exist. The motion of an electron involves however a singularity in (a, b, c), of a rotational type, with its nucleus at the moving electron; and the time-average of this singularity for a very rapid minute steady orbital motion of an electron is analytically equivalent, at distances considerable compared with the dimensions of the orbit, to a magnetic doublet analogous to a source and ​associated equal sink. Finally, the various parts of the aether are supposed to be sensibly at rest, so that for example the time-rate of change of the strain of any element of the aether is represented by differentiation with respect to the time without any additional terms to represent the change due to the element of aether being carried on in the meantime to a new position; in this respect the equations of the aether are much simpler than those of the dynamics of fluid motion, being in fact linear. The aether is stagnant on this theory, while the molecules constituting the Earth and all other material bodies flit through it without producing any finite flow in it; hence the law of the astronomical aberration of light is rigorously maintained, and the Doppler change of wave-length of radiation from a moving source holds good; but it will appear that all purely terrestrial optical phenomena are unaffected by the Earth's motion.
103. Subject to this general explanation, the analytical equations which express the dynamics of the field of free aether, existing between and around the nuclei of the electrons, are
* $4\pi {\frac {d}{dt}}(f,\ g,\ h)=\operatorname {curl} (a,\ b,\ c)$
* $-{\frac {d}{dt}}(a,\ b,\ c)=4\pi c^{2}\operatorname {curl} (f,\ g,\ h),$
in which the symbol $\operatorname {curl} (a,\ b,\ c)$ represents, after Maxwell, the vector
$\left({\frac {dc}{dy}}-{\frac {db}{dz}},\ {\frac {da}{dz}}-{\frac {dc}{dx}},\ {\frac {db}{dx}}-{\frac {da}{dy}}\right),$
and in which c is the single physical constant of the aether, being the velocity of propagation of elastic disturbances through it. These are the analytical equations derived by Maxwell in his mathematical development of Faraday's views as to an electric medium: and they are the same as the equations arrived at by MacCullagh a quarter of a century earlier in his formulation of the dynamics of optical media. It may fairly be claimed that the theoretical investigations of Maxwell, in combination with the experimental verifications of Hertz and his successors in that field, have imparted to this analytical formulation of the dynamical relations of free aether an exactness ​and precision which is not surpassed in any other department of physics, even in the theory of gravitation.
Where a more speculative element enters is in the construction of a kinematic scheme of representation of the aether-strain, such as will allow of the unification of the various assumptions here enumerated. It is desirable for the sake of further insight, and even necessary for various applications, to have concrete notions of the physical nature of the vectors (f, g, h) and (a, b, c) which specify aethereal disturbances, in the form of representations such as will implicitly and intuitively involve the analytical relations between them, and will also involve the conditions and restrictions to which each is subject, including therein the permanence and characteristic properties of an electron and its free mobility through the aether.
104. But for the mere analytical development of the aether-scheme as above formulated, a concrete physical representation of the constitution of the aether is not required: the abstract relations and conditions above given form a sufficient basis. In point of fact these analytical relations are theoretically of an ideal simplicity for this purpose: for they give explicitly the time-rates of change of the vectors of the problem at each instant, so that from a knowledge of the state of the system at any time t the state at the time t + δt can be immediately expressed, and so by successive steps, or by the use of Taylor's differential expansion-theorem, its state at any further time can theoretically be derived. The point that requires careful attention is as to whether the solution of these equations in terms of a given initial state of the system determines the motions of the electrons or strain-nuclei through the medium, as well as the changes of strain in the medium itself: and it will appear on consideration that under suitable hypotheses this is so. For the given initial state will involve given motions of the electrons, that is the initial value of (a, b, c) will involve rotational singularities at the electrons around their directions of motion, just such as in the element of time δt will shift the electrons themselves into their new positions: and so on step by step continually. This however ​presupposes that the nucleus of the electron is quite labile as regards displacement through the aether, in other words that its movement is not influenced by any inertia or forces except such as are the expression of its relation to the aether: we in fact assume the completeness of the aethereal scheme of relations as above given. Any difficulty that may be felt on account of the infinite values of the vectors at the nucleus itself may be removed, in the manner customary in analytical discussions on attractions, by considering the nucleus to consist of a volume distribution of electricity of finite but very great density, distributed through a very small space instead of being absolutely concentrated in a point: then the quantities will not become infinite. Of the detailed structure of electrons nothing is assumed: so long as the actual dimensions of their nuclei are extremely small in comparison with the distances between them, it will suffice for the theory to consider them as points, just as for example in the general gravitational theory of the Solar System it suffices to consider the planets as attracting points. This method is incomplete only as regards those portions of the energy and other quantities that are associated with the mutual actions of the parts of the electron itself, and are thus molecularly constitutive.
105. It is to be observed that on the view here being developed, in which atoms of matter are constituted of aggregations of electrons, the only actions between atoms are what may be described as electric forces. The electric character of the forces of chemical affinity was an accepted part of the chemical views of Davy, Berzelius, and Faraday; and more recent discussions, while clearing away crude conceptions, have invariably tended to the strengthening of that hypothesis. The mode in which the ordinary forces of cohesion could be included in such a view is still quite undeveloped. Difficulties of this kind have however not been felt to be fundamental in the vortex-atom illustration of the constitution of matter, which has exercised much fascination over high authorities on molecular physics: yet in the concrete realization of Maxwell's theory of the aether above referred to, the atom of matter ​possesses all the dynamical properties of a vortex ring in a frictionless fluid, so that everything that can be done in the domain of vortex-ring illustration is implicitly attached to the present scheme. The fact that virtually nothing has been achieved in the department of forces of cohesion is not a valid objection to the development of a theory of the present kind. For the aim of theoretical physics is not a complete and summary conquest of the modus operandi of natural phenomena: that would be hopelessly unattainable if only for the reason that the mental apparatus with which we conduct the search is itself in one of its aspects a part of the scheme of Nature which it attempts to unravel. But the very fact that this is so is evidence of a correlation between the process of thought and the processes of external phenomena, and is an incitement to push on further and bring out into still clearer and more direct view their inter-connexions. When we have mentally reduced to their simple elements the correlations of a large domain of physical phenomena, an objection does not lie because we do not know the way to push the same principles to the explanation of other phenomena to which they should presumably apply, but which are mainly beyond the reach of our direct examination.
The natural conclusion would rather be that a scheme, which has been successful in the simple and large-scale physical phenomena that we can explore in detail, must also have its place, with proper modifications or additions on account of the difference of scale, in the more minute features of the material world as to which direct knowledge in detail is not available. And in any case, whatever view may be held as to the necessity of the whole complex of chemical reaction being explicable in detail by an efficient physical scheme, a limit is imposed when vital activity is approached: any complete analysis of the conditions of the latter, when merely superficial sequences of phenomena are excluded, must remain outside the limits of our reasoning faculties. The object of scientific explanation is in fact to coordinate mentally, but not to exhaust, the interlaced maze of natural phenomena: a theory which gives an adequate correlation of a portion of this field maintains its ​place until it is proved to be in definite contradiction, not removable by suitable modification, with another portion of it.
106. We now recall the equations of the free aether, with a view to changing from axes (x, y, z) at rest in the aether to axes (x', y', z') moving with translatory velocity v parallel to the axis of x; so as thereby to be in a position to examine how phenomena are altered when the observer and his apparatus are in uniform motion through the stationary aether. These equations are
${\begin{array}{ccc}4\pi {\frac {df}{dt}}={\frac {dc}{dy}}-{\frac {db}{dz}}&&-(4\pi c^{2})^{-1}{\frac {da}{dt}}={\frac {dh}{dy}}-{\frac {dg}{dz}}\\\\4\pi {\frac {dg}{dt}}={\frac {da}{dz}}-{\frac {dc}{dx}}&&-(4\pi c^{2})^{-1}{\frac {db}{dt}}={\frac {df}{dz}}-{\frac {dh}{dx}}\\\\4\pi {\frac {dh}{dt}}={\frac {db}{dx}}-{\frac {da}{dy}}&&-(4\pi c^{2})^{-1}{\frac {dh}{dt}}={\frac {dg}{dx}}-{\frac {df}{dy}}.\end{array}}$
When they are referred to the axes (x', y', z') in uniform motion, so that $(x',\ y',\ z')=(x-vt,\ y,\ z),\ t'=t$, then $d/dx,\ d/dy,\ d/dz$ become $d/dx',\ d/dy',\ d/dz'$, but d/dt becomes $d/dt'-vd/dx'$: thus
${\begin{array}{ccc}4\pi {\frac {df}{dt'}}={\frac {dc'}{dy'}}-{\frac {db'}{dz'}}&&-(4\pi c^{2})^{-1}{\frac {da}{dt'}}={\frac {dh'}{dy'}}-{\frac {dg'}{dz'}}\\\\4\pi {\frac {dg}{dt'}}={\frac {da'}{dz'}}-{\frac {dc'}{dx'}}&&-(4\pi c^{2})^{-1}{\frac {db}{dt'}}={\frac {df'}{dz'}}-{\frac {dh'}{dx'}}\\\\4\pi {\frac {dh}{dt'}}={\frac {db'}{dx'}}-{\frac {da'}{dy'}}&&-(4\pi c^{2})^{-1}{\frac {dh}{dt'}}={\frac {dg'}{dx'}}-{\frac {df'}{dy'}}.\end{array}}$
where
* $(a',\ b',\ c')=(a,\ b+4\pi vh,\ c-4\pi vg)$
* $(f',\ g',\ h')=\left(f,\ g-{\frac {v}{4\pi c^{2}}}c,\ h+{\frac {v}{4\pi c^{2}}}b\right).$
We can complete the elimination of (f, g, h) and (a, b, c) so ​that only the vectors denoted by accented symbols shall remain, by substituting from these latter formulae: thus
$g=g'+{\frac {v}{4\pi c^{2}}}(c'+4\pi vg),$
so that
$\epsilon ^{-1}g=g'+{\frac {v}{4\pi c^{2}}}c',$
where ε is equal to $\left(1-v^{2}/c^{2}\right)^{-1}$, and exceeds unity;
and
$b=b'-4\pi v\left(h'-{\frac {v}{4\pi c^{2}}}b\right)$
so that
$\epsilon ^{-1}b=b'-4\pi vh';$
giving the general relations
* $\epsilon ^{-1}(a,\ b,\ c)=\left(\epsilon ^{-1}a',\ b'-4\pi vh',\ c'+4\pi vg'\right)$
* $\epsilon ^{-1}(f,\ g,\ h)=\left(\epsilon ^{-1}f',\ g'+{\frac {v}{4\pi c^{2}}}c',\ h'-{\frac {v}{4\pi c^{2}}}b'\right).$
Hence
* $4\pi {\frac {df'}{dt'}}={\frac {dc'}{dy'}}-{\frac {db'}{dz'}}$
* $4\pi \epsilon {\frac {dg'}{dt'}}={\frac {da'}{dz'}}-\left({\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}\right)c'$
* $4\pi \epsilon {\frac {dh'}{dt'}}=\left({\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}\right)b'-{\frac {da'}{dy'}}$
* $-\left(4\pi c^{2}\right)^{-1}{\frac {da'}{dt'}}={\frac {dh'}{dy'}}-{\frac {dg'}{dz'}}$
* $-\left(4\pi c^{2}\right)^{-1}\epsilon {\frac {db'}{dt'}}={\frac {df'}{dz'}}-\left({\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}\right)h'$
* $-\left(4\pi c^{2}\right)^{-1}\epsilon {\frac {dc'}{dt'}}=\left({\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}\right)g'-{\frac {df'}{dy'}}.$
Now change the time-variable from t' to t", equal to $t'-{\frac {v}{c^{2}}}\epsilon x'$; this will involve that ${\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}$ is replaced by ${\frac {d}{dx'}}$, while the other differential operators remain unmodified; thus the scheme of equations reverts to the same type as when it was referred to axes at rest, except as regards the factors ε on the left-hand sides.
​107. It is to be observed that this factor ε only differs from unity by $(v/c)^{2}$, which is of the second order of small quantities; hence we have the following correspondence when that order is neglected. Consider any aethereal system, and let the sequence of its spontaneous changes referred to axes (x', y', z') moving uniformly through the aether with velocity (v, 0, 0) be represented by values of the vectors (f, g, h) and (a, b, c) expressed as functions of x', y', z' and t', the latter being the time measured in the ordinary manner: then there exists a correlated aethereal system whose sequence of spontaneous changes referred to axes (x', y', z') at rest are such that its electric and magnetic vectors (f', g' , h') and (a', b', c') are functions of the variables x', y', z' and a time-variable t", equal to $t'-{\frac {v}{c^{2}}}x'$, which are the same as represent the quantities
$\left(f,\ g-{\frac {v}{4\pi c^{2}}}c,\ h+{\frac {v}{4\pi c^{2}}}b\right)$
and
$(a,\ b+4\pi vh,\ c-4\pi vg)$
belonging to the related moving system when expressed as functions of the variables x', y', z' and t'.
Conversely, taking any aethereal system at rest in the aether, let the sequence of its changes be represented by (f', g', h') and (a', b', c') expressed as functions of the coordinates (x, y, z) and of the time t'. In these functions change t' into $t-{\frac {v}{c^{2}}}x$: then the resulting expressions are the values of
$\left(f,\ g-{\frac {v}{4\pi c^{2}}}c,\ h+{\frac {v}{4\pi c^{2}}}b\right),$
and
$(a,\ b+4\pi vh,\ c-4\pi vg),$
for a system in uniform motion through the aether, referred to axes (x, y, z) moving along with it, and to the time t. In comparing the states of the two systems, we have to the first order
| ${\frac {df}{dx}}$ | equal to | ${\frac {df'}{dx}}-{\frac {v}{c'}}{\frac {df'}{dt}}$ |
| ${\frac {d}{dy}}\left(g-{\frac {v}{4\pi c^{2}}}c\right)$ | „ | ${\frac {dg'}{dy}}$ |
| ${\frac {d}{dz}}\left(h+{\frac {v}{4\pi c^{2}}}b\right)$ | „ | ${\frac {dh'}{dz}};$ |
${\frac {dc'}{dy}}-{\frac {db'}{dz}}=4\pi {\frac {df'}{dt'}},$
or, what is the same,
${\frac {dc'}{dy}}-{\frac {db'}{dz}}=4\pi \left({\frac {df'}{dt}}-v{\frac {df'}{dx}}\right),$
we have, to the first order,
${\frac {df}{dx}}+{\frac {dg}{dy}}+{\frac {dh}{dz}}={\frac {df'}{dx}}+{\frac {dg'}{dy}}+{\frac {dh'}{dz}}.$
Thus the electrons in the two systems here compared, being situated at the singular points at which the concentration of the electric displacement ceases to vanish, occupy corresponding positions. Again, these electrons are of equal strengths: for, very near an electron, fixed or moving, the values of (f, g, h) and (a, b, c) are practically those due to it, the part due to the remainder of the system being negligible in comparison: also in this correspondence the relation between (f, g, h) and the accented variables is, by § 106
$\epsilon ^{-1}(f,\ g,\ h)=\left(\epsilon ^{-1}f',\ g'+{\frac {v}{4\pi c^{2}}}c',\ h'-{\frac {v}{4\pi c^{2}}}b'\right);$
hence, since for the single electron at rest (a', b', c') is null, we have, very close to the correlative electron in the moving system, (f, g, h) equal to (f', εg', εh'), where ε, being $\left(1-v^{2}/c^{2}\right)^{-1}$, differs from unity by the second order of small quantities. Thus neglecting the second order, (f, g, h) is equal to (f', g', h') for corresponding points very close to electrons; and, as the amount of electricity inside any boundary is equal to the integral of the normal component of the aethereal displacement taken over the boundary, it follows by taking a very contracted boundary that the strengths of the corresponding electrons in the two systems are the same, to this order of approximation.
108. It is to be observed that the above analytical transformation of the equations applies to any isotropic dielectric medium as well as to free aether: we have only to alter c into ​the velocity of radiation in that medium, and all will be as above. The transformation will thus be different for different media. But we are arrested if we attempt to proceed to compare a moving material system, treated as continuous, with the same system at rest; for the motion of the polarized dielectric matter has altered the mathematical type of the electric current. It is thus of no avail to try to effect in this way a direct general transformation of equations of a material medium in which dielectric and conductive coefficients occur.
109. The correspondence here established between a system referred to fixed axes and a system referred to moving axes will assume a very simple aspect when the former system is a steady one, so that the variables are independent of the time. Then the distribution of electrons in the second system will be at each instant precisely the same as that in the first, while the second system accompanies its axes of reference in their uniform motion through the aether. In other words, given any system of electrified bodies at rest, in equilibrium under their mutual electric influences and imposed constraints, there will be a precisely identical system in equilibrium under the same constraints, and in uniform translatory motion through the aether. That is, uniform translatory motion through the aether does not produce any alteration in electric distributions as far as the first order of the ratio of the velocity of the system to the velocity of radiation is concerned. Various cases of this general proposition will be verified subsequently in connexion with special investigations.
Moreover this result is independent of any theory as to the nature of the forces between material molecules: the structure of the matter being assumed unaltered to the first order by motion through the aether, so too must be all electric distributions. What has been proved comes to this, that if any configuration of ionic charges is the natural one in a material system at rest, the maintenance of the same configuration as regards the system in uniform motion will not require the aid of any new forces. The electron taken by itself must be on any conceivable theory a simple singularity of the aether whose ​movements when it is free, and interactions with other electrons if it can be constrained by matter, are traceable through the differential equations of the surrounding free aether alone: and a correlation has been established between these equations for the two cases above compared. It is however to be observed (cf. § 99) that though the fixed and the moving system of electrons of this correlation are at corresponding instants identical, yet the electric and magnetic displacements belonging to them differ by terms of the first order.
* ↑ Namely as the distance r from it diminishes indefinitely, the magnetic induction tends to the form $evr^{-2}\sin {\theta }$, at right angles to the plane of the angle θ between r and the velocity v of the electron: this arises as the disturbance of the medium involved in annulling the electron in its original position and restoring it in the new position to which it has moved. The relations will appear more clearly when visualized by the kinematic representation of Appendix E; or when we pass to the limit in the formulae of Chapter ix relating to the field of a moving charged body of finite dimensions. The specification in the text, as a simple pole, only applies for an electron moving with velocity v, when terms of the order $(v/c)^{2}$ are neglected: otherwise the aethereal field close around it is not isotropic and an amended specification derivable from the formulae of Chapter ix must be substituted. In the second-order discussion of Chapter xi this more exact form is implicitly involved, the strength of the electron being determined (§ 111) by the concentration of the aethereal displacement around it. The singularity in the magnetic field which is involved in the motion of the electron, not of course an intrinsic one, has no concentration.
* ↑ See Appendix E.
* ↑ Cf. footnote, p. 162.
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