Page:Grundgleichungen (Minkowski).djvu/7

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as regards permutation with the indices (1,2,3,4).

§ 3. The Theorem of Relativity of Lorentz.

It is well-known that by writing the equations I) to IV) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis. through an amount \varphi, keeping \mathfrak{e,m,w} fixed in space, and introduce new variables x'_{1},\ x'_{2},\ x'_{3},\ x'_{4} instead of x_{1},\ x_{2},\ x_{3},\ x_{4}, where

x'_{1}=x_{1}\cos\varphi+x_{2}\sin\varphi,\ x'_{2}=-x_{1}\sin\varphi+x_{2}\cos\varphi,\ x'_{3}=x_{3},\ x'_{4}=x_{4},

and introduce magnitudes

\varrho'_{1},\ \varrho'_{2},\ \varrho'_{3},\ \varrho'_{4},

where

\varrho'_{1}=\varrho_{1}\cos\varphi+\varrho_{2}\sin\varphi,\ \varrho'_{2}=-\varrho_{1}\sin\varphi+\varrho_{2}\cos\varphi,\ \varrho'_{3}=\varrho_{3},\ \varrho'_{4}=\varrho_{4},

and f'_{12},\dots f'_{34}, where

f'_{23}=f_{23}\cos\varphi+f_{31}\sin\varphi,\ f'_{31}=-f_{23}\sin\varphi+f_{31}\cos\varphi,\ f'_{12}=f_{12},

f'_{14}=f_{14}\cos\varphi+f_{24}\sin\varphi,\ f'_{24}=-f_{14}\sin\varphi+f_{24}\cos\varphi,\ f'_{34}=f_{34},

f'_{kh} = - f'_{hk}\qquad (h, k = 1, 2, 3, 4),

then out of the equations (A) would follow a corresponding system of dashed equations (A') composed of the newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equations (A) and (B) concerning the suffixes (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all.

I will denote by i\psi a purely imaginary magnitude, and consider the substitution

(1) \begin{array}{ccc}
 & x'_{1}=x_{1},\ x'_{2}=x_{2},\\
x'_{3}=x_{3}\cos\ i\psi+x_{4}\sin\ i\psi, &  & x'_{4}=-x_{3}\sin\ i\psi+x_{4}\cos\ i\psi\end{array}

Putting

(2) -i\ tg\ i\psi=\frac{e-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q,\ \psi=\frac{1}{2}\log\ nat\ \frac{1+q}{1-q}