∴ _v_ = _v₀_ + _u_,
making Fresnelian convection co-efficient simply unity.
Equations (1·21) and (2·21) may be obtained more simply from physical
considerations.
According to Heaviside and Hertz, the real seat of both electric and
magnetic polarisation is the moving medium itself. Now at a point which
is fixed with respect to the ether, the rate of change of electric
polarisation is δD/δ_t_.
Consider a slab of matter moving with velocity _u__{_x_} along the
_x_-axis, then even in a stationary field of electrostatic polarisation,
that is, for a field in which δD/δ_t_ = 0, there will be some change in
the polarisation of the body due to its motion, given by
_u__{_x_}(δD/δ_x_). Hence we must add this term to a purely temporal
rate of change δD/δ_t_. Doing this we immediately arrive at equations
(1·21) and (2·21) for the special case considered there.
Thus the Hertz-Heaviside form of field equations gives _unity_ as the
value for the Fresnelian convection co-efficient. It has been shown in
the historical introduction how this is entirely at variance with the
observed optical facts. As a matter of fact, Larmor has shown (Aether
and Matter) that 1 - 1/μ² is not only sufficient but is also necessary,
in order to explain experiments of the Arago prism type.
A short summary of the electromagnetic experiments bearing on this
question, has already been given in the introduction.
According to Hertz and Heaviside the total polarisation is situated in
the medium itself and is completely carried away by it. Thus the
electromagnetic effect outside a moving medium should be proportional to
K, the specific inductive capacity.
_Rowland_ showed in 1876 that when a charged condenser is rapidly
rotated (the dielectric remaining stationary), the magnetic effect
outside is proportional to K, the Sp. Ind. Cap.
_Röntgen_ (Annalen der Physik 1888, 1890) found that if the dielectric
is rotated while the condenser remains stationary, the effect is
proportional to K - 1.
_Eichenwald_ (Annalen der Physik 1903, 1904) rotated together both
condenser and dielectric and found that the magnetic effect was
proportional to the potential difference and to the angular velocity,
but was completely independent of K. This is of course quite consistent
with Rowland and Röntgen.
_Blondlot_ (Comptes Rendus, 1901) passed a current of air in a steady
magnetic field H_{_y_}, (H = H_{_z_} = 0). If this current of air moves
with velocity _u__{_x_} along the _x_-axis, an electromotive force would
be set up along the _z_-axis, due to the relative motion of matter and
magnetic tubes of induction. A pair of plates at _z_ = ±_a_, will be
charged up with density ρ = D_{_z_} = KE = K. _u__{_s_} H_{_y_}/c. But
Blondlot failed to detect any such effect.
_H. A. Wilson_ (Phil. Trans. Royal Soc. 1904) repeated the experiment
with a cylindrical condenser made of ebony, rotating in a magnetic field
parallel to its own axis. He observed a change proportional to K — 1 and
not to K.
Thus the above set of electro-magnetic experiments contradict the
Hertz-Heaviside equations, and these must be abandoned.
[P. C. M.]
Note 2.
Lorentz Transformation.
Lorentz. Versuch einer theorie der elektrischen und optischen
Erscheinungen im bewegten Körpern.
(Leiden—1895).
Lorentz. Theory of Electrons (English edition), pages 197-200, 230, also
notes 73, 86, pages 318, 328.
Lorentz wanted to explain the Michelson-Morley null-effect. In order to
do so, it was obviously necessary to explain the Fitzgerald contraction.
Lorentz worked on the hypothesis that an electron itself undergoes
contraction when moving. He introduced new variables for the moving
system defined by the following set of equations.
_x¹_ = β(_x_ - _ut_), _y¹_ = _y_, _z¹_ = _z_, _t¹_ = β(_t_ -
(_u_/_c²_)·_x_)
and for velocities, used
_v__{_x_}¹ = β²_v__{_x_} + _u_, _v__{_y_}¹ = β_v__{_y_}, _v__{_z_}¹
= β_v__{_z_} and ρ¹ = ρ/β.
With the help of the above set of equations, which is known as the
Lorentz transformation, he succeeded in showing how the Fitzgerald
contraction results as a consequence of “fortuitous compensation of
opposing effects.”
It should be observed that the Lorentz transformation is not identical
with the Einstein transformation. The Einsteinian addition of velocities
is quite different as also the expression for the “relative” density of
electricity.
It is true that the Maxwell-Lorentz field equations remain _practically_
unchanged by the Lorentz transformation, but they _are_ changed to some
slight extent. One marked advantage of the Einstein transformation
consists in the fact that the field equations of a moving system
preserve _exactly_ the same form as those of a stationary system.
It should also be noted that the Fresnelian convection coefficient comes
out in the theory of relativity as a direct consequence of Einstein’s
addition of velocities and is quite independent of any electrical theory
of matter.
[P. C. M.]
Note 3.
See Lorentz, Theory of Electrons (English edition), § 181, page 213.
H. Poincare, Sur la dynamique ‘electron, Rendiconti del circolo
matematico di Palermo 21 (1906).
[P. C. M.]
Note 4.
Relativity Theorem and Relativity-Principle.
Lorentz showed that the Maxwell-Lorentz system of electromagnetic
field-equations remained practically unchanged by the Lorentz
transformation. Thus the electromagnetic laws of Maxwell and Lorentz
_can be definitely proved_ “to be independent of the manner in which
they are referred to two coordinate systems which have a uniform
translatory motion relative to each other.” (See “Electrodynamics of
Moving Bodies,” page 5.) Thus so far as the electromagnetic laws are
concerned, the principle of relativity _can be proved to be true_.
But it is not known whether this principle will remain true in the case
of other physical laws. We can always proceed on the assumption that it
does remain true. Thus it is always possible to construct physical laws
in such a way that they retain their form when referred to moving
coordinates. The ultimate ground for formulating physical laws in this
way is merely a subjective conviction that the principle of relativity
is universally true. There is no _a priori_ logical necessity that it
should be so. Hence the Principle of Relativity (so far as it is applied
to phenomena other than electromagnetic) must be regarded as a
_postulate_, which we have assumed to be true, but for which we cannot
adduce any definite proof, until after the generalisation is made and
its consequences tested in the light of actual experience.
[P. C. M.]
Note 5.
See “Electrodynamics of Moving Bodies,” p. 5-8.
Note 6.
Field Equations in Minkowski’s Form.