_w₂_ _s₃_ - _w₃_ _s₂_,
_w₃_ _s₁_ - _w₁_ _s₃_,
_w₁_ _s₂_ - _w₂_ _s₁_,
_w₁_ _s₄_ - _w₄_ _s₁_,
_w₂_ _s₄_ - _w₄_ _s₂_,
_w₃_ _s₄_ - _w₄_ _s₃_,
behaves in case of a Lorentz-transformation as a space-time vector of
the II kind. The vector of the second kind with the components (44) are
denoted by [_w_, _s_]. We see easily that Det^{½} [_w_, _s_] = 0. The
dual vector of [_w_, _s_] shall be written as [_w_, _s_].
If _ẇ_ is a space-time vector of the 1st kind, _f_ of the second
kind, _w_ _f_ signifies a 1 × 4 series matrix. In case of a
Lorentz-transformation A, _w_ is changed into _w′_ = _w_A, _f_ into
_f′_ = A⁻¹ _f_ A,—therefore _w′_ _f′_ becomes = (_w_A A⁻¹ _f_ A) =
_w_ _f_ A _i.e._ _w_ _f_ is transformed as a space-time vector of
the 1st kind.[23] We can verify, when _w_ is a space-time vector of
the 1st kind, _f_ of the 2nd kind, the important identity
(45) [_w_, _w__f_] + [_w_, _w__f_*]* = (_w_] _ẇ_)_f_.
The sum of the two space time vectors of the second kind on the left
side is to be understood in the sense of the addition of two alternating
matrices.
For example, for ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_,
ω_f_ = | _i__f_₄₁, _i__f_₄₂, _i__f_₄₃, 0 |;
ω_f_* = | _i__f_₃₂, _i__f_₁₃, _i__f_₂₁, 0 |
[ω · ω_f_] = 0, 0, 0, _f_₄₁, _f_₄₂, _f_₄₃;
[ω · ω_f_*]* = 0, 0, 0, _f_₃₂, _f_₁₃, _f_₂₁.
The fact that in this special case, the relation is satisfied, suffices
to establish the theorem (45) generally, for this relation has a
covariant character in case of a Lorentz transformation, and is
homogeneous in (ω₁, ω₂, ω₃, ω₄).
After these preparatory works let us engage ourselves with the equations
(C,) (D,) (E) by means which the constants ε μ, σ will be introduced.
Instead of the space vector _u_, the velocity of matter, we shall
introduce the space-time vector of the first kind ω with the components.
ω₁ = _u__{_x_}/√(1 - _u²_),
ω₂ = _u__{_y_}/√(1 - _u²_),
ω₃ = _u__{_z_}/√(1 - _u²_),
ω₄ = _i_/√(1 - _u²_).
(40) where ω₁² + ω₂² + ω₃² + ω₄² = -1 and -_i_ω₄ > 0.
By F and _f_ shall be understood the space time vectors of the second
kind M - _i_E, _m_ - _ie_.