Page:Grundgleichungen (Minkowski).djvu/21

In fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way.

According to the second axiom, the magnitude of the velocity vector $\left|{\mathfrak {w}}\right|$ is < 1 at any space-time point. In consequence, we can always write, instead of the vector ${\mathfrak {w}}$, the following set of four allied quantities

$w_{1}={\frac {{\mathfrak {w}}_{x}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{2}={\frac {{\mathfrak {w}}_{y}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{3}={\frac {{\mathfrak {w}}_{z}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{4}={\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}$,

with the relation

(27)	$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=-1$

From what has been said at the end of § 4, it is clear that in the case of a Lorentz-transformation, this set behaves like a space-time vector of the 1st kind, and we want to call it space-time vector velocity.

Let us now fix our attention on a certain point x, y, z of matter at a certain time t. If at this space-time point ${\mathfrak {w}}=0$, then we have at once for this point the equations (A), (B) (V) of § 7. If ${\mathfrak {w}}\neq 0$, then there exists according to 16), in case $\left|{\mathfrak {w}}\right|<1$, a special Lorentz-transformation, whose vector ${\mathfrak {v}}$ is equal to this vector ${\mathfrak {w}}(x,\ y,\ z,\ t)$ and we pass on to a new system of reference x', y', z', t' in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values

(28)	$w'_{0}=0,\ w'_{2}=0,\ w'_{3}=0,\ w'_{4}=i$,

therefore the new velocity vector ${\mathfrak {w}}'=0$, the space-time point is as if transformed to rest. Now according to the third axiom the system of equations for the transformed point x, y, z, t involves the newly introduced magnitude ${\mathfrak {w}}',\varrho ',{\mathfrak {s',e',m',E',M'}}$ and their differential quotients with respect to x', y', z, t' in the same manner as the original equations for the point x, y, z, t. But according to the first axiom, when ${\mathfrak {w}}'=0$ these equations must be exactly equivalent to

(1) the differential equations (A'), (B'), which are obtained from the equations (A), (B) by simply dashing the symbols in (A) and (B).

(2) and the equations