(_m_, -_ie_), and (_M_, -_iE_) are space-time vectors of the second
kind, (C, _i_p) is a space-time vector of the first kind.
This axiom I call the Principle of Relativity.
In fact these three axioms lead us from the previously mentioned
fundamental equations for bodies at rest to the equations for moving
bodies in an unambiguous way.
According to the second axiom, the magnitude of the velocity vector |
_u_ | is < 1 at any space-time point. In consequence, we can always
write, instead of the vector _u_, the following set of four allied
quantities
ω₁ = u_{_x_}/√(1 - _u²_),
ω₂ = u_{_y_}/√(1 - u²),
ω₃ = u_{_z_}/√(1 - u²),
ω₄ = _i_/√(1 - u²)
with the relation
(27) ω₁² + ω₂² + ω₃² + ω₄² = - |
From what has been said at the end of § 4, it is clear that in the case
of a Lorentz-transformation, this set behaves like a space-time vector
of the 1st kind.
Let us now fix our attention on a certain point (_x_, _y_, _z_) of
matter at a certain time (_t_). If at this space-time point _u_ = 0,
then we have at once for this point the equations (_A_), (_B_) (_V_) of
§ 7. If _u_ ≠ 0, then there exists according to 16), in case | _u_ | <
1, a special Lorentz-transformation, whose vector _v_ is equal to this
vector _u_ (_x_, _y_, _z_, _t_), and we pass on to a new system of
reference (_x′_ _y′_ _z′_ _t′_) in accordance with this transformation.
Therefore for the space-time point considered, there arises as in § 4,
the new values 28) ω′₁ = 0, ω′₂ = 0, ω′₃ = 0, ω′₄ = _i_, therefore the
new velocity vector ω′ = 0, the space-time point is as if transformed to
rest. Now according to the third axiom the system of equations for the
transformed point (_x′_ _y′_ _z′_ _t_) involves the newly introduced
magnitude (_u′_ ρ′, C′, _e′_, _m′_, _E′_, _M′_) and their differential
quotients with respect to (_x′_, _y′_, _z′_, _t′_) in the same manner as
the original equations for the point (_x_, _y_, _z_, _t_). But according
to the first axiom, when _u′_ = 0, these equations must be exactly
equivalent to
(1) the differential equations (_A′_), (_B′_), which are obtained from
the equations (_A_), (_B_) by simply dashing the symbols in (_A_) and
(_B_).
(2) and the equations
(V′) _e′_ = ε_E′_, _M’_ = μ_m′_, _C′_ = σ_E′_
where ε, μ, σ are the dielectric constant, magnetic permeability, and
conductivity for the system (_x′_ _y′_ _z′_ _t′_) _i.e._ in the
space-time point (_x_ _y_, _z_ _t_) of matter.
Now let us return, by means of the reciprocal Lorentz-transformation to
the original variables (_x_, _y_, _z_, _t_), and the magnitudes (_u_, ρ,
C, _e_, _m_, _E_, _M_) and the equations, which we then obtain from the
last mentioned, will be the fundamental equations sought by us for the
moving bodies.
Now from § 4, and § 6, it is to be seen that the equations _A_), as well
as the equations _B_) are covariant for a Lorentz-transformation, _i.e._
the equations, which we obtain backwards from _A′_) _B′_), must be
exactly of the same form as the equations _A_) and _B_), as we take them
for bodies at rest. We have therefore as the first result:—
The differential equations expressing the fundamental equations of
electrodynamics for moving bodies, when written in ρ and the vectors C,
_e_, _m_, E, M, are exactly of the same form as the equations for moving
bodies. The velocity of matter does not enter in these equations. In the
vectorial way of writing, we have