The Principle of Relativity

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 = iρ [v_xd_x + v_yd_y + v_zd_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 = iωF^* 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 |

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, dS denote the three-dimensional bounding surface of ΔΣ, n be the outer normal to dS. Let S be any four-vector, P_n its normal component. Then

Now if for ΔΣ we choose the four-dimensional parallelopiped with sides (dx₁, dx₂, dx₃, dx₄), we have then

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, dσ 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 dσ. Then the s-component of the vector divergence of f because the operator lor multiplies f vectorially.

Where the integration in dσ is to be extended over the whole surface.

If now s is selected as the x-direction, Δs is then a three-dimensional parallelopiped with the sides dy, dz, dl, then we have

and generally

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^*).

Generally we have

∴ The vector of the third type represented by (Pf) is given by the geometrical sum of the two four-vectors of the first type PD and P^*D^*.

[M. N. S.]