As for the relation E), which introduces the conductivity σ we have -ωS
= -(ω₁_s₁_ + ω₂_s₂_ + ω₃_s₃_ + ω₄_s₄_) = (- | _u_ | C_{_u_} + ρ)/√(1 -
_u²_) = ρ′.
This expression gives us the rest-density of electricity (see §8 and
§4).
Then 61) = _s_ + (ω_ṡ_)ω represents a space-time vector of the 1st kind,
which since ωω = -1, is normal to ω, and which I may call the
rest-current. Let us now conceive of the first three component of this
vector as the (_x_-_y_-_z_) co-ordinates of the space-vector, then the
component in the direction of _u_ is
C_{_u_} - (| _u_ | ρ′)/√(1 - _u²_)
= (_c__{_u_} - | _u_ |ρ)/√(1 - _u²_)
= J_{_u_}/(1 - _u²_)
and the component in a perpendicular direction is C_{_u_} = J_{_ū_}.
This space-vector is connected with the space-vector J = C - ρ_u_, which
we denoted in §8 as the conduction-current.
Now by comparing with Φ = -ωF, the relation (E) can be brought into the
form
{E} _s_ + (ω_ṡ_)ω = - σωF,
This formula contains four equations, of which the fourth follows from
the first three, since this is a space-time vector which is
perpendicular to ω.
Lastly, we shall transform the differential equations (A) and (B) into a
typical form.
§12. The Differential Operator Lor.
A 4 × 4 series matrix 62) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | S_{_kh_} |
| S₂₁ S₂₂ S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ |
with the condition that in case of a Lorentz transformation it is to be
replaced by ĀSA, may be called a space-time matrix of the II kind. We
have examples of this in:—
1) the alternating matrix _f_, which corresponds to the space-time
vector of the II kind,—
2) the product _f_F of two such matrices, for by a transformation A, it
is replaced by (A⁻¹_f_A·A⁻¹FA) = A⁻¹_f_FA,
3) further when (ω₁, ω₂, ω₃, ω₄) and (Ω₁, Ω₂, Ω₃, Ω₄) are two space-time
vectors of the 1st kind, the 4 × 4 matrix with the element S_{_hk_} =
ω_{_h_}Ω_{_k_},
lastly in a multiple L of the unit matrix of 4 × 4 series in which all
the elements in the principal diagonal are equal to L, and the rest are
zero.
We shall have to do constantly with functions of the space-time point
(_x_, _y_, _z_, _it_), and we may with advantage
employ the 1 × 4 series matrix, formed of differential symbols,—
| ∂/∂_x_, ∂/∂_y_, ∂/∂_z_, ∂/_i_∂_t_,|
or (63) | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ |