Page:Grundgleichungen (Minkowski).djvu/37

From Wikisource
Jump to: navigation, search
This page has been proofread, but needs to be validated.


For this matrix I shall use the shortened from lor.

Then if S is, as in (62), a space-time matrix of the II. kind, by lor S' will be understood the 1✕4 series matrix

\left|K_{1},\ K_{2},\ K_{3},\ K_{4}\right|

where

(64) K_{k}=\frac{\partial S_{1k}}{\partial x_{1}}+\frac{\partial S_{2k}}{\partial x_{2}}+\frac{\partial S_{3k}}{\partial x_{3}}+\frac{\partial S_{4k}}{\partial x_{4}}\qquad (k=1,2,3,4)

When by a Lorentz transformation \mathsf{A}, a new reference system x'_{1},\ x'_{2},\ x'_{3},\ x'_{4} is introduced, we can use the operator

lor'=\left|\frac{\partial}{\partial x'_{1}},\ \frac{\partial}{\partial x'_{2}},\ \frac{\partial}{\partial x'_{3}},\ \frac{\partial}{x'_{4}}\right|

Then S is transformed to S'=\bar{\mathsf{A}}S\mathsf{A}=\left|S'_{hk}\right|, so by lor' Sis meant the 1✕4 series matrix, whose element are

K'_{k}=\frac{\partial S'_{1k}}{\partial x'_{1}}+\frac{\partial S'_{2k}}{\partial x'_{2}}+\frac{\partial S'_{3k}}{\partial x'_{3}}+\frac{\partial S'_{4k}}{\partial x'_{4}}\qquad (k=1,2,3,4)

Now for the differentiation of any function of (x y z t) we have the rule

\frac{\partial}{\partial x'_{k}}=\frac{\partial}{\partial x_{1}}\frac{\partial x_{1}}{\partial x'_{k}}+\frac{\partial}{\partial x_{2}}\frac{\partial x_{2}}{\partial x'_{k}}+\frac{\partial}{\partial x_{3}}\frac{\partial x_{3}}{\partial x'_{k}}+\frac{\partial}{\partial x_{4}}\frac{\partial x_{4}}{\partial x'_{k}}

=\frac{\partial}{\partial x_{1}}\alpha_{1k}+\frac{\partial}{\partial x_{2}}\alpha_{2k}+\frac{\partial}{\partial x_{3}}\alpha_{3k}+\frac{\partial}{\partial x_{4}}\alpha_{4k},

so that, we have symbolically

lor'=lor\ (\mathsf{A}

Therefore it follows that

(65) lor'\ S'=lor(\mathsf{A}(\mathsf{A}^{-1}S\mathsf{A}))=(lor\ S)\mathsf{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