Page:The principle of relativity (1920).djvu/258

 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 = | s_{1}, s_{2}, s_{3} s_{4} |

and   s = | s_{1} | | s_{2} | | s_{3} | | s_{4} |

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

Div S = Lim [integral] P_{n}dS/ΔΣ. ΔΣ = 0

Now if for ΔΣ we choose the four-dimensional parallelopiped with sides (dx_{1}, dx_{2}, dx_{3}, dx_{4}), we have then

Div S = [part]s_{1}/[part]x_{1} + [part]s_{2}/[part]x_{2} + [part]s_{3}/[part]x_{3} + [part]s_{4}/[part]x_{4}. = 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