aus Wikisource, der freien Quellensammlung
man durch direkte Ausführung der Variation die Differentialgleichungen der geodätischen Linie bildet und aus diesen die Komponenten abliest. Die Differentialgleichungen der geodätischen Linie für das Linienelement (9) ergeben sich durch die Variation unmittelbar in der Form:
=
f
1
d
2
x
1
d
s
2
+
1
2
∂
f
4
∂
x
1
(
d
x
4
d
s
)
2
+
1
2
∂
f
1
∂
x
1
(
d
x
1
d
s
)
2
−
1
2
∂
f
2
∂
x
1
[
1
1
−
x
2
2
(
d
x
2
d
s
)
2
+
(
1
−
x
2
2
)
(
d
x
3
d
s
)
2
]
{\displaystyle =f_{1}{\frac {d^{2}x_{1}}{ds^{2}}}+{\frac {1}{2}}{\frac {\partial f_{4}}{\partial x_{1}}}\left({\frac {dx_{4}}{ds}}\right)^{2}+{\frac {1}{2}}{\frac {\partial f_{1}}{\partial x_{1}}}\left({\frac {dx_{1}}{ds}}\right)^{2}-{\frac {1}{2}}{\frac {\partial f_{2}}{\partial x_{1}}}\left[{\frac {1}{1-x_{2}^{2}}}\left({\frac {dx_{2}}{ds}}\right)^{2}+\left(1-x_{2}^{2}\right)\left({\frac {dx_{3}}{ds}}\right)^{2}\right]\,}
=
f
2
1
−
x
2
2
d
2
x
2
d
s
2
+
∂
f
2
∂
x
1
1
1
−
x
2
2
d
x
1
d
s
d
x
2
d
s
+
f
2
x
2
(
1
−
x
2
2
)
2
(
d
x
2
d
s
)
2
+
f
2
x
2
(
d
x
3
d
s
)
2
{\displaystyle ={\frac {f_{2}}{1-x_{2}^{2}}}{\frac {d^{2}x_{2}}{ds^{2}}}+{\frac {\partial f_{2}}{\partial x_{1}}}{\frac {1}{1-x_{2}^{2}}}{\frac {dx_{1}}{ds}}{\frac {dx_{2}}{ds}}+{\frac {f_{2}\ x_{2}}{\left(1-x_{2}^{2}\right)^{2}}}\left({\frac {dx_{2}}{ds}}\right)^{2}+f_{2}\ x_{2}\left({\frac {dx_{3}}{ds}}\right)^{2}\,}
=
f
2
(
1
−
x
2
2
)
d
2
x
3
d
s
2
+
∂
f
2
∂
x
1
(
1
−
x
2
2
)
d
x
1
d
s
d
x
3
d
s
−
2
f
2
x
2
d
x
2
d
s
d
x
3
d
s
{\displaystyle =f_{2}\left(1-x_{2}^{2}\right){\frac {d^{2}x_{3}}{ds^{2}}}+{\frac {\partial f_{2}}{\partial x_{1}}}\left(1-x_{2}^{2}\right){\frac {dx_{1}}{ds}}{\frac {dx_{3}}{ds}}-2f_{2}\ x_{2}{\frac {dx_{2}}{ds}}{\frac {dx_{3}}{ds}}\,}
=
f
4
d
2
x
4
d
s
2
+
∂
f
4
∂
x
1
d
x
1
d
s
d
x
4
d
s
.
{\displaystyle =f_{4}{\frac {d^{2}x_{4}}{ds^{2}}}+{\frac {\partial f_{4}}{\partial x_{1}}}{\frac {dx_{1}}{ds}}{\frac {dx_{4}}{ds}}.\,}
Der Vergleich mit (2) gibt die Komponenten des Gravitationsfeldes
Γ
11
1
=
−
1
2
1
f
1
∂
f
1
∂
x
1
,
Γ
22
1
=
+
1
2
1
f
1
∂
f
2
∂
x
1
1
1
−
x
2
2
,
{\displaystyle \Gamma _{11}^{1}=-{\frac {1}{2}}{\frac {1}{f_{1}}}{\frac {\partial f_{1}}{\partial x_{1}}},\quad \Gamma _{22}^{1}=+{\frac {1}{2}}{\frac {1}{f_{1}}}{\frac {\partial f_{2}}{\partial x_{1}}}{\frac {1}{1-x_{2}^{2}}},\,}
Γ
33
1
=
+
1
2
1
f
1
∂
f
2
∂
x
1
(
1
−
x
2
2
)
,
{\displaystyle \Gamma _{33}^{1}=+{\frac {1}{2}}{\frac {1}{f_{1}}}{\frac {\partial f_{2}}{\partial x_{1}}}\left(1-x_{2}^{2}\right),\,}
Γ
44
1
=
−
1
2
1
f
1
∂
f
4
∂
x
1
,
{\displaystyle \Gamma _{44}^{1}=-{\frac {1}{2}}{\frac {1}{f_{1}}}{\frac {\partial f_{4}}{\partial x_{1}}},\,}
Γ
21
2
=
−
1
2
1
f
2
∂
f
2
∂
x
1
,
Γ
22
2
=
−
x
2
1
−
x
2
2
,
Γ
33
2
=
−
x
2
(
1
−
x
2
2
)
,
{\displaystyle \Gamma _{21}^{2}=-{\frac {1}{2}}{\frac {1}{f_{2}}}{\frac {\partial f_{2}}{\partial x_{1}}},\quad \Gamma _{22}^{2}=-{\frac {x_{2}}{1-x_{2}^{2}}},\quad \Gamma _{33}^{2}=-x_{2}\left(1-x_{2}^{2}\right),\,}
Γ
31
3
=
−
1
2
1
f
2
∂
f
2
∂
x
1
,
Γ
32
3
=
+
x
2
1
−
x
2
2
,
{\displaystyle \Gamma _{31}^{3}=-{\frac {1}{2}}{\frac {1}{f_{2}}}{\frac {\partial f_{2}}{\partial x_{1}}},\quad \Gamma _{32}^{3}=+{\frac {x_{2}}{1-x_{2}^{2}}},\,}
Γ
41
4
=
−
1
2
1
f
4
∂
f
4
∂
x
1
{\displaystyle \Gamma _{41}^{4}=-{\frac {1}{2}}{\frac {1}{f_{4}}}{\frac {\partial f_{4}}{\partial x_{1}}}\,}
(die übrigen null).
Bei der Rotationssymmetrie um den Nullpunkt genügt es, die Feldgleichungen nur für den Äquator
(
x
2
=
0
)
{\displaystyle \left(x_{2}=0\right)\,}
1
−
x
2
2
{\displaystyle 1-x_{2}^{2}\,}
a)
∂
∂
x
1
(
1
f
1
∂
f
1
∂
x
1
)
=
1
2
(
1
f
1
∂
f
1
∂
x
1
)
2
+
(
1
f
2
∂
f
2
∂
x
4
)
2
+
1
2
(
1
f
4
∂
f
4
∂
x
1
)
2
,
{\displaystyle {\frac {\partial }{\partial x_{1}}}\left({\frac {1}{f_{1}}}{\frac {\partial f_{1}}{\partial x_{1}}}\right)={\frac {1}{2}}\left({\frac {1}{f_{1}}}{\frac {\partial f_{1}}{\partial x_{1}}}\right)^{2}+\left({\frac {1}{f_{2}}}{\frac {\partial f_{2}}{\partial x_{4}}}\right)^{2}+{\frac {1}{2}}\left({\frac {1}{f_{4}}}{\frac {\partial f_{4}}{\partial x_{1}}}\right)^{2},\,}
b)
∂
∂
x
1
(
1
f
1
∂
f
2
∂
x
1
)
=
2
+
1
f
1
f
2
(
∂
f
2
∂
x
1
)
2
,
{\displaystyle {\frac {\partial }{\partial x_{1}}}\left({\frac {1}{f_{1}}}{\frac {\partial f_{2}}{\partial x_{1}}}\right)=2+{\frac {1}{f_{1}f_{2}}}\left({\frac {\partial f_{2}}{\partial x_{1}}}\right)^{2},\,}
c)
∂
∂
x
1
(
1
f
1
∂
f
4
∂
x
1
)
=
1
f
1
f
4
(
∂
f
4
∂
x
1
)
2
.
{\displaystyle {\frac {\partial }{\partial x_{1}}}\left({\frac {1}{f_{1}}}{\frac {\partial f_{4}}{\partial x_{1}}}\right)={\frac {1}{f_{1}f_{4}}}\left({\frac {\partial f_{4}}{\partial x_{1}}}\right)^{2}.\,}
![{\displaystyle =f_{1}{\frac {d^{2}x_{1}}{ds^{2}}}+{\frac {1}{2}}{\frac {\partial f_{4}}{\partial x_{1}}}\left({\frac {dx_{4}}{ds}}\right)^{2}+{\frac {1}{2}}{\frac {\partial f_{1}}{\partial x_{1}}}\left({\frac {dx_{1}}{ds}}\right)^{2}-{\frac {1}{2}}{\frac {\partial f_{2}}{\partial x_{1}}}\left[{\frac {1}{1-x_{2}^{2}}}\left({\frac {dx_{2}}{ds}}\right)^{2}+\left(1-x_{2}^{2}\right)\left({\frac {dx_{3}}{ds}}\right)^{2}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15a7fb1c86cbbb0f27d343336ed968f7eda8f683)
