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, dS denote the three-dimensional bounding surface of ΔΣ, n be the outer normal to dS. Let S be any four-vector, Pn its normal component. Then
Div S = Lim ∫ PndS/ΔΣ.