Page:The principle of relativity (1920).djvu/116

 Therefore it follows that

lor´S´ = lor (A A^{-1} SA) = (lor S)A.

i.e., lor S behaves like a space-time vector of the first kind.

If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements


 * [part]L/[part]x_{1}    [part]L/[part]x_{2}     [part]L/[part]x_{3}     [part]L/[part]x_{4} |

If s is a space-time vector of the 1st kind, then

lor 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}.

In case of a Lorentz transformation A, we have

lor´[=s]´ = lor A. Ās = lor s.

i.e., lor s is an invariant in a Lorentz-transformation.

In all these operations the operator lor plays the part of a space-time vector of the first kind.

If [function] represents a space-time vector of the second kind,—lor [function] denotes a space-time vector of the first kind with the components

[part][function]_{12}/[part]x_{2} + [part][function]_{13}/[part]x_{3} + [part][function]_{14}/[part]x_{4}, [part][function]_{21}/[part]x_{1} + [part][function]_{23}/[part]x_{3} + [part][function]_{24}/[part]x_{4}, [part][function]_{31}/[part]x_{1} + [part][function]_{32}/[part]x_{2} + [part][function]_{34}/[part]x_{4}, [part][function]_{41}/[part]x_{1} + [part][function]_{42}/[part]x_{2} + [part][function]_{43}/[part]x_{3}