ρ₀ (φ₁, φ₂, φ₃) = ρ (_d_ + 1/_c_ [_v_·_h_]),
_i.e._, ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the electron.
Compare Lorentz, Theory of Electrons, page 14.
The fourth component φ₄ when multiplied by ρ₀ represents _i_-times the
rate at which work is done by the moving electron, for ρ₀ φ₄ = _i_ρ
[_v__{_x_}_d__{_x_} + _v__{_y_}_d__{_y_} + _v__{_z_}_d__{_z_}] =
_v__{_x_} ρ₀φ₁ + _v__{_y_} ρ₀φ₂ + _v__{_z_} ρ₀φ₃. -√(-1) times the power
possessed by the electron therefore represents the fourth component, or
the time component of the force-four-vector. This component was first
introduced by Poincare in 1906.
The four-vector ψ = _i_ωF^* has a similar relation to the force acting
on a moving magnetic pole.
[M. N. S.]
Note 17.
Operator “Lor” (§ 12, p. 41).
The operation | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ | which plays in
four-dimensional mechanics a rôle similar to that of the operator
(_i_∂/∂_x_, + _j_∂/∂_y_, + _k_∂/∂_z_ = ▽) in three-dimensional geometry
has been called by Minkowski ‘Lorentz-Operation’ or shortly ‘lor’ in
honour of H. A. Lorentz, the discoverer of the theorem of relativity.
Later writers have sometimes used the symbol □ to denote this operation.
In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38)
Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation),
Grad (gradient) as four-dimensional extensions of the corresponding
three-dimensional operations in place of the general symbol lor. The
physical significance of these operations will become clear when along
with Minkowski’s method of treatment we also study the geometrical
method of Sommerfeld. Minkowski begins here with the case of lor S,
where S is a six-vector (space-time vector of the 2nd kind).
This being a complicated case, we take the simpler case of lor _s_,
where _s_ is a four-vector = | _s₁_, _s₂_, _s₃_, _s₄_ |
and _s_ = | _s₁_ |
| _s₂_ |
| _s₃_ |
| _s₄_ |
The following geometrical method is taken from Sommerfeld.
Scalar Divergence—Let ΔΣ denote a small four-dimensional volume of any
shape in the neighbourhood of the space-time point Q, _d_S denote the
three-dimensional bounding surface of ΔΣ, _n_ be the outer normal to
_d_S. Let S be any four-vector, P_{_n_} its normal component. Then
Div S = Lim ∫ P_{_n_}_d_S/ΔΣ.
ΔΣ = 0
Now if for ΔΣ we choose the four-dimensional parallelopiped with sides
(_dx₁_, _dx₂_, _dx₃_, _dx₄_), we have then
Div S = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_ = lor
S.
If _f_ denotes a space-time vector of the second kind, lor _f_ is
equivalent to a space-time vector of the first kind. The geometrical
significance can be thus brought out. We have seen that the operator
‘lor’ behaves in every respect like a four-vector. The vector-product of
a four-vector and a six-vector is again a four-vector. Therefore it is
easy to see that lor S will be a four-vector. Let us find the component
of this four-vector in any direction _s_. Let S denote the three-space
which passes through the point Q (_x₁_, _x₂_, _x₃_, _x₄_) and is
perpendicular to _s_, ΔS a very small part of it in the region of Q,
_d_σ is an element of its two-dimensional surface. Let the perpendicular
to this surface lying in the space be denoted by _n_, and let _f__{_s_
_n_} denote the component of _f_ in the plane of (_sn_) which is
evidently conjugate to the plane _d_σ. Then the _s_-component of the
vector divergence of _f_ because the operator lor multiplies _f_
vectorially.
= Div _f__{_s_} = Lim (∫ _f__{_s_ _n_}_d_σ)/ΔS.
Δ_s_ = 0
Where the integration in _d_σ is to be extended over the whole surface.
If now _s_ is selected as the _x_-direction, Δ_s_ is then a
three-dimensional parallelopiped with the sides _dy_, _dz_, _dl_, then
we have
$$ Div f_{x} = \frac{1}{dy dz dl} {dz. dl. \frac{\partial
f_{xy}}{\partial y} dy + dl dy \frac{\partial f_{xy}}{\partial z} dz +
dy dz \frac{\partial f_{xy}}{\partial l} dl} = \frac{\partial
f_{xy}}{\partial y} + \frac{\partial f_{xy}}{\partial z} +
\frac{\partial f_{xy}}{\partial l} $$
and generally
Div _f__{_j_} = ∂_f__{_j_ _x_}/∂_x_ + ∂_f__{_j_ _y_}/∂_y_ +
∂_f__{_j_ _z_}/∂_z_ + ∂_f__{_j_ _l_}/∂_l_ (where _f__{_j_, _j_} =
0).
Hence the four-components of the four-vector lor S or Div. _f_ is a
four-vector with the components given on page 42.
According to the formulae of space geometry, D_{_x_} denotes a
parallelopiped laid in the (_y_-_z_-_l_) space, formed out of the
vectors (P_{_y_} P_{_z_} P_{_l_}), (U_{_y_}^* U_{_z_}^* U_{_l_}^*)
(V_{_y_}^* V_{_z_}^* V_{_l_}^*).
D_{_x_} is therefore the projection on the _y-z-l_ space of the
parallelopiped formed out of these three four-vectors (P, U^*, V^*), and
could as well be denoted by Dyzl. We see directly that the four-vector
of the kind represented by (D_{_x_}, D_{_y_}, D_{_z_}, D_{_l_}) is
perpendicular to the parallelopiped formed by (P U^* V^*).