These invariants represent quantities which are of considerable importance in the theory of electrons.[1]
Let[2]
w
1
=
w
x
1
−
w
2
,
w
2
=
w
y
1
−
w
2
,
w
3
=
w
z
1
−
w
2
,
w
4
=
1
1
−
w
2
,
{\displaystyle w_{1}={\frac {w_{x}}{\sqrt {1-w^{2}}}},\ w_{2}={\frac {w_{y}}{\sqrt {1-w^{2}}}},\ w_{3}={\frac {w_{z}}{\sqrt {1-w^{2}}}},\ w_{4}={\frac {1}{\sqrt {1-w^{2}}}},}
d
s
2
=
d
t
2
−
d
x
2
−
d
y
2
−
d
z
2
,
{\displaystyle ds^{2}=dt^{2}-dx^{2}-dy^{2}-dz^{2},}
so that
w
1
=
d
x
d
s
,
w
2
=
d
y
d
s
,
w
3
=
d
z
d
s
,
w
4
=
d
t
d
s
,
{\displaystyle w_{1}={\frac {dx}{ds}},\ w_{2}={\frac {dy}{ds}},\ w_{3}={\frac {dz}{ds}},\ w_{4}={\frac {dt}{ds}},}
Then
d
w
1
d
s
=
w
˙
x
1
−
w
2
+
w
x
(
w
w
˙
)
(
1
−
w
2
)
2
,
{\displaystyle {\frac {dw_{1}}{ds}}={\frac {{\dot {w}}_{x}}{1-w^{2}}}+{\frac {w_{x}(w{\dot {w}})}{\left(1-w^{2}\right)^{2}}},}
d
2
w
1
d
s
2
=
w
¨
x
(
1
−
w
2
)
1
2
+
3
w
˙
x
(
w
w
˙
)
(
1
−
w
2
)
1
2
+
w
x
(
1
−
w
2
)
1
2
{
w
w
¨
+
3
(
w
w
˙
)
2
1
−
w
2
+
w
˙
2
+
(
w
w
˙
)
2
1
−
w
2
}
.
{\displaystyle {\frac {d^{2}w_{1}}{ds^{2}}}={\frac {{\ddot {w}}_{x}}{\left(1-w^{2}\right)^{\frac {1}{2}}}}+{\frac {3{\dot {w}}_{x}(w{\dot {w}})}{\left(1-w^{2}\right)^{\frac {1}{2}}}}+{\frac {w_{x}}{\left(1-w^{2}\right)^{\frac {1}{2}}}}\left\{w{\ddot {w}}+{\frac {3(w{\dot {w}})^{2}}{1-w^{2}}}+{\dot {w}}^{2}+{\frac {(w{\dot {w}})^{2}}{1-w^{2}}}\right\}.}
Hence
(
d
w
1
d
s
)
2
+
(
d
w
2
d
s
)
2
+
(
d
w
3
d
s
)
2
−
(
d
w
4
d
s
)
2
=
w
˙
2
(
1
−
w
2
)
2
+
(
w
w
˙
)
2
(
1
−
w
2
)
3
+
w
2
(
w
w
˙
)
2
(
1
−
w
2
)
4
−
(
w
w
˙
)
2
(
1
−
w
2
)
4
=
w
˙
2
(
1
−
w
2
)
2
+
(
w
w
˙
)
2
(
1
−
w
2
)
3
{\displaystyle {\begin{array}{l}\left({\frac {dw_{1}}{ds}}\right)^{2}+\left({\frac {dw_{2}}{ds}}\right)^{2}+\left({\frac {dw_{3}}{ds}}\right)^{2}-\left({\frac {dw_{4}}{ds}}\right)^{2}\\\\\qquad ={\frac {{\dot {w}}^{2}}{\left(1-w^{2}\right)^{2}}}+{\frac {(w{\dot {w}})^{2}}{\left(1-w^{2}\right)^{3}}}+{\frac {w^{2}(w{\dot {w}})^{2}}{\left(1-w^{2}\right)^{4}}}-{\frac {(w{\dot {w}})^{2}}{\left(1-w^{2}\right)^{4}}}\\\\\qquad ={\frac {{\dot {w}}^{2}}{\left(1-w^{2}\right)^{2}}}+{\frac {(w{\dot {w}})^{2}}{\left(1-w^{2}\right)^{3}}}\end{array}}}
↑ Some of these are already known in the case of Lorentz's transformations. Cf. Planck, Ann. der Physik , Bd. 26, p. 1 (1908)Göttinger Nachrichten (1908)Born. Ann. d. Physik , Bd. 28 (1909).
↑ Cf. Minkowski, Gött. Nachr. (1908).
