Appendix 1 â Formules van de Algemene Relativiteitstheorie
Hieronder geven we een samenvatting van een aantal eerder afgeleide formules uit de algemene relativiteitstheorie en de Schwarzschildâoplossing. Vervolgens leiden we alle formules af die relevant zijn voor berekeningen in verschillende hoofdstukken. In deze appendix passen we de Einsteinânotatie toe.
Algemene Relativiteitstheorie â Basisformules
Einsteins veldvergelijkingen:
\begin{equation}
R_{\mu\nu}
- \frac{1}{2} g_{\mu\nu} R
+ \lambda g_{\mu\nu}
=
\frac{8\pi G}{c^{4}}\, T_{\mu\nu}.
\end{equation}
Waarbij:
- \(R_{\mu\nu}\): de Ricciâtensor,
- \(g_{\mu\nu}\): de metrische tensor,
- \(R\): de Ricciâscalar,
- \(\lambda\): de kosmologische constante,
- \(T_{\mu\nu}\): de energieâimpulsâtensor.
Schwarzschildâmetriek (in sferische coördinaten)
\begin{equation}
ds^{2}
=
\left(1 - \frac{2GM}{r c^{2}}\right)c^{2} dt^{2}
-
\left(1 - \frac{2GM}{r c^{2}}\right)^{-1} dr^{2}
-
r^{2} d\theta^{2}
-
r^{2}\sin^{2}\theta\, d\phi^{2}.
\end{equation}
Waarbij:
- \(ds^{2}\): het ruimteâtijdinterval,
- \(G\): de gravitatieconstante,
- \(M\): de massa van het centrale object,
- \(r\): de radiale coördinaat,
- \(\theta\) en \(\phi\): de sferische coördinaten.
De metriekcoëfficiënten zijn dus niet afhankelijk van \(t\) en \(\phi\), maar alleen van \(r\) en \(\theta\).
Tijdvertraging voor een bolvormig object (Gravitational Time Dilation)
Voor een stilstaande waarnemer op afstand \(r\) van een bolvormige massa geldt:
\begin{equation}
\Delta \tau
=
\Delta t \,\sqrt{1 - \frac{2GM}{r c^{2}}}
\end{equation}
Waarbij:
- \(\Delta \tau\): de eigen tijd voor een waarnemer op afstand \(r\),
- \(\Delta t\): de tijd voor een verre waarnemer.
Baan van licht (nullâgeodeten)
Voor licht geldt \(ds^{2} = 0\). Daaruit volgt:
\begin{equation}
\left(1 - \frac{2GM}{r c^{2}}\right)c^{2} dt^{2}
=
\left(1 - \frac{2GM}{r c^{2}}\right)^{-1} dr^{2}
+ r^{2} d\theta^{2}
+ r^{2}\sin^{2}\theta\, d\phi^{2}.
\end{equation}
Krommingsradius van licht om een massa
De afwijking van een lichtstraal in de buurt van een massa wordt gegeven door:
\begin{equation}
\delta\phi = \frac{4GM}{r c^{2}}.
\end{equation}
Appendix 1.1 â Samenvatting en afleiding van verdere relevante formules
In deze sectie leiden we de relevante formules af voor de berekeningen in de hoofdstukken. Dit omvat:
- de metrische tensor in verschillende coördinatenstelsels,
- de geodetenvergelijkingen,
- de energieâimpulsâtensor in diverse configuraties.
Coördinatentransformaties
\begin{equation}
dx^{m} = \frac{\partial x^{m}}{\partial y^{r}}\, dy^{r}
\end{equation}
\begin{equation}
ds^{2} = \eta_{mn}\, d\xi^{m} d\xi^{n}
\end{equation}
\begin{equation}
ds^{2}
= g_{mn}(x)\, dx^{m} dx^{n}
= g_{pq}(y)\, dy^{p} dy^{q}
\end{equation}
\begin{equation}
g_{pq}(y)
=
g_{mn}(x)
\frac{\partial x^{m}}{\partial y^{p}}
\frac{\partial x^{n}}{\partial y^{q}}
\end{equation}
Transformatie van vectoren en tensoren
\begin{equation}
V'^{n}(y)
=
\frac{\partial y^{n}}{\partial x^{m}}\, V^{m}(x)
\end{equation}
\begin{equation}
W'_{p}(y)
=
\frac{\partial x^{q}}{\partial y^{p}}\, W_{q}(x)
\end{equation}
\begin{equation}
T_{mn}(x)
=
\frac{\partial V^{m}(x)}{\partial x^{n}}
\end{equation}
\begin{equation}
T_{mn}(y)
=
\frac{\partial x^{r}}{\partial y^{m}}
\frac{\partial x^{s}}{\partial y^{n}}
T_{rs}(x)
\end{equation}
\begin{equation}
T^{mn}(y)
=
\frac{\partial y^{m}}{\partial x^{r}}
\frac{\partial y^{n}}{\partial x^{s}}
T^{rs}(x)
\end{equation}
\begin{equation}
T^{rs}(x) = A_{x}^{r} B_{x}^{s}
\end{equation}
Indexverhoging en -verlaging
\begin{equation}
E_{\mu} = g_{\mu\nu} E^{\nu}
\end{equation}
\begin{equation}
E^{\mu}
= g^{\mu\nu} E_{\nu}
= g^{\mu\nu} g_{\nu\rho} E^{\rho}
= \delta^{\mu}_{\rho} E^{\rho}
= E^{\mu}
\end{equation}
Lijnsegment in klein gebied
Pythagoras:
\begin{equation}
ds^{2}
=
\delta_{mn}
\frac{\partial x^{m}}{\partial y^{n}}dy^{n}
\cdot
\frac{\partial x^{n}}{\partial y^{s}}
dy^{s}
\end{equation}
Transformeren naar ander frame:
\begin{equation}
ds^{2}
=
\delta_{mn}
\frac{\partial x^{m}}{\partial y^{r}}
\frac{\partial x^{n}}{\partial y^{s}}
dy^{r} dy^{s}
\end{equation}
Metrische tensor
\begin{equation}
g_{mn}
=
\delta_{mn}
\frac{\partial x^{m}}{\partial y^{r}}
\frac{\partial x^{n}}{\partial y^{s}}
\end{equation}
Einsteins veldvergelijkingen
\begin{equation}
R_{\mu\nu}
- \frac{1}{2} g_{\mu\nu} R
+ \lambda g_{\mu\nu}
=
\frac{8\pi G}{c^{4}}\, T_{\mu\nu}
\end{equation}
Geodetische vergelijking
\begin{equation}
\frac{d^{2} x^{\lambda}}{d\tau^{2}}
+
\Gamma^{\lambda}{}_{\mu\nu}
\frac{dx^{\mu}}{d\tau}
\frac{dx^{\nu}}{d\tau}
= 0
\end{equation}
\begin{equation}
\Gamma^{\lambda}{}_{\mu\nu}
\equiv
\frac{\partial x^{\lambda}}{\partial \xi^{\alpha}}
\frac{\partial^{2} \xi^{\alpha}}{\partial x^{\mu}\partial x^{\nu}}
\end{equation}
Tensortransformaties
\begin{equation}
T'_{\mu\nu}(y)
=
\frac{\partial x^{\alpha}}{\partial y^{\mu}}
\frac{\partial x^{\beta}}{\partial y^{\nu}}
T_{\alpha\beta}(x)
\end{equation}
\begin{equation}
T'^{\mu\nu}(y)
=
\frac{\partial y^{\mu}}{\partial x^{\alpha}}
\frac{\partial y^{\nu}}{\partial x^{\beta}}
T^{\alpha\beta}(x)
\end{equation}
\begin{equation}
T_{\mu}^{'\nu}(y)
=
\frac{\partial x^{\alpha}}{\partial y^{\mu}}
\frac{\partial y^{\nu}}{\partial x^{\beta}}
T_{\alpha}^{\beta}(x)
\end{equation}
\begin{equation}
g_{\mu\alpha} g^{\alpha\nu} = \delta_{\mu}^{\nu}
\end{equation}
Contractie
\begin{equation}
A^{\mu} = g^{\mu\nu} A_{\nu}
\end{equation}
\begin{equation}
A_{\mu} = g_{\mu\nu} A^{\nu}
\end{equation}
\begin{equation}
A \cdot B
=
g_{\mu\nu} A^{\mu} B^{\nu}
\equiv A_{\nu} B^{\nu}
\end{equation}
Ricciâtensor
\begin{equation}
R_{\mu\nu}
=
R^{\rho}_{\mu\rho\nu}
=
\Gamma^{\rho}_{\mu\nu,\rho}
-
\Gamma^{\rho}_{\mu\rho,\nu}
+
\Gamma^{\rho}_{\lambda\rho}\Gamma^{\lambda}_{\nu\mu}
-
\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\rho}
\end{equation}
\begin{equation}
G_{\mu\nu}
=
\Gamma^{\rho}_{\mu\nu,\rho}
-
\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\rho}
\quad\text{(alleen als } g = \det(g_{\mu\nu}) = -1\text{)}
\end{equation}
Christoffelâsymbolen
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu}
=
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\end{equation}
Ricciâscalar
\begin{equation}
R=R^\mu_\mu = g^{\mu\nu} R_{\mu\nu}
\end{equation}
\begin{equation}
R
=
g^{ab}
\left(
\Gamma^{c}{}_{ab,c}
-
\Gamma^{c}{}_{ac,b}
+
\Gamma^{d}{}_{ab}\Gamma^{c}{}_{dc}
-
\Gamma^{d}{}_{ac}\Gamma^{c}{}_{db}
\right)
\end{equation}
\begin{equation}
R
=
2 g^{ab}
\left(
\Gamma^{c}{}_{a[b,c]}
+
\Gamma^{d}{}_{a[b}\Gamma^{c}{}_{c]d}
\right)
\end{equation}
Appendix 1.2 â Schwarzschildâmetriek in sferische coördinaten
De Schwarzschildâmetriek luidt:
\begin{equation}
ds^{2}
=
\sigma^{2} c^{2} dt^{2}
-
\frac{dr^{2}}{\sigma^{2}}
-
r^{2} d\theta^{2}
-
r^{2}\sin^{2}\theta\, d\phi^{2},
\end{equation}
waarbij:
\begin{equation}
\sigma^{2} = 1 - \frac{R_{s}}{r},
\qquad
R_{s} = \frac{2GM}{c^{2}}.
\end{equation}
Identificatie van metriekcomponenten:
- \(g_{00} = g_{tt}\)
- \(g_{11} = g_{rr}\)
- \(g_{22} = g_{\theta\theta}\)
- \(g_{33} = g_{\phi\phi}\)
Schwarzschild in het vlak \(\theta = \frac{\pi}{2}\)
\begin{equation}
g_{00} = \sigma^{2},
\qquad
g^{00} = \frac{1}{\sigma^{2}},
\end{equation}
\begin{equation}
g_{11} = -\frac{1}{\sigma^{2}},
\qquad
g^{11} = -\sigma^{2},
\end{equation}
\begin{equation}
g_{22} = -r^{2},
\qquad
g^{22} = -\frac{1}{r^{2}},
\end{equation}
\begin{equation}
g_{33} = -r^{2}\sin^{2}\theta = -r^{2},
\qquad
g^{33} = -\frac{1}{r^{2}\sin^{2}\theta} = -\frac{1}{r^{2}}.
\end{equation}
Afgeleide van \(\sigma\):
\begin{equation}
\frac{d\sigma}{dr}
=
\frac{R_{s}}{2 r^{2} \sigma}.
\end{equation}
Eerste afgeleiden van de metriek
\begin{equation}
\frac{\partial g_{00}}{\partial r}
=
\frac{R_{s}}{r^{2}},
\qquad
\frac{\partial g_{11}}{\partial r}
=
\frac{R_{s}}{r^{2}\sigma^{4}},
\end{equation}
\begin{equation}
\frac{\partial g_{22}}{\partial r}
=
-2r,
\qquad
\frac{\partial g_{33}}{\partial r}
=
-2r\sin^{2}\theta = -2r,
\end{equation}
\begin{equation}
\frac{\partial g_{33}}{\partial \theta}
=
-2 r^{2}\sin\theta\cos\theta = 0.
\end{equation}
Tweede afgeleiden van de metriek
\begin{equation}
\frac{\partial^{2} g_{00}}{\partial r^{2}}
=
-\frac{2R_{s}}{r^{3}},
\qquad
\frac{\partial^{2} g_{11}}{\partial r^{2}}
=
-\frac{2R_{s}}{r^{3}\sigma^{6}},
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{22}}{\partial r^{2}}
=
-2,
\qquad
\frac{\partial^{2} g_{33}}{\partial r^{2}}
=
-2\sin^{2}\theta = -2,
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial \theta \partial r}
=
-4r\sin\theta\cos\theta = 0,
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial \theta^{2}}
=
2r^{2}(\sin^{2}\theta - \cos^{2}\theta)
= 2r^{2}.
\end{equation}
Christoffelâsymbolen voor Schwarzschild in polaire coördinaten
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu}
=
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\end{equation}
Nietânul componenten:
\begin{equation}
\Gamma^{0}{}_{10}
=
\Gamma^{0}{}_{01}
=
\frac{1}{2} g^{00}\frac{\partial g_{00}}{\partial r}
=
\frac{R_{s}}{2 r^{2} \sigma^{2}},
\end{equation}
\begin{equation}
\Gamma^{1}{}_{00}
=
\frac{1}{2} g^{11}\left(-\frac{\partial g_{00}}{\partial r}\right)
=
\sigma^{2}\frac{R_{s}}{2 r^{2}},
\end{equation}
\begin{equation}
\Gamma^{1}{}_{11}
=
\frac{1}{2} g^{11}\frac{\partial g_{11}}{\partial r}
=
-\frac{R_{s}}{2 r^{2} \sigma^{2}},
\end{equation}
\begin{equation}
\Gamma^{2}{}_{21}
=
\Gamma^{2}{}_{12}
=
\frac{1}{2} g^{22}\left(\frac{\partial g_{22}}{\partial r}\right)
=
\frac{1}{r},
\end{equation}
\begin{equation}
\Gamma^{3}{}_{31}
=
\Gamma^{3}{}_{13}
=
\frac{1}{2} g^{33}\left(\frac{\partial g_{33}}{\partial r}\right)
=
\frac{1}{r},
\end{equation}
\begin{equation}
\Gamma^{1}{}_{22}
=
\frac{1}{2} g^{11}\left(-\frac{\partial g_{22}}{\partial r}\right)
=
-r\sigma^{2},
\end{equation}
\begin{equation}
\Gamma^{1}{}_{33}
=
\frac{1}{2} g^{11}\left(-\frac{\partial g_{33}}{\partial r}\right)
=
-r\sigma^{2}\sin^{2}\theta,
\end{equation}
\begin{equation}
\Gamma^{3}{}_{32},
=
\frac{1}{2} g^{33}\left(-\frac{\partial g_{33}}{\partial \theta}\right)
=
\frac{\cos\theta}{\sin\theta},
\end{equation}
\begin{equation}
\Gamma^{2}{}_{33}
=
\frac{1}{2} g^{22}\left(-\frac{\partial g_{33}}{\partial \theta}\right)
=
-\sin\theta\cos\theta.
\end{equation}
Eerste afgeleiden van Christoffelâsymbolen
\begin{equation}
\frac{\partial \Gamma^{0}_{01}}{\partial r}
=
\frac{\partial \Gamma^{0}_{10}}{\partial r}
=
\frac{R_{s}(R_{s}-2r)}{2 r^{4} \sigma^{4}},
\qquad
\frac{\partial \Gamma^{1}{}_{00}}{\partial r}
=
\frac{R_{s}(3R_{s}-2r)}{2 r^{4}},
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{11}}{\partial r}
=
\frac{R_{s}(2r - R_{s})}{2 r^{4} \sigma^{4}},
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{22}}{\partial r}
=
-1,
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{33}}{\partial r}
=
-\sin^{2}\theta,
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{12}}{\partial r}
=
\frac{\partial \Gamma^{2}_{21}}{\partial r}
=
\frac{\partial \Gamma^{3}_{13}}{\partial r}
=
\frac{\partial \Gamma^{3}_{31}}{\partial r}
=
-\frac{1}{r^{2}},
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{33}}{\partial \theta}
=
-\cos^{2}\theta+\sin^{2}\theta=1,
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{3}_{23}}{\partial \theta}
=
\frac{\partial \Gamma^{3}_{32}}{\partial \theta}
=
-\frac{1}{\sin^{2}\theta} = -1,
\end{equation}
Eerste afgeleide van het Christoffelâsymbool (algemene vorm)
\begin{equation}
\begin{aligned}
\frac{\partial \Gamma^{\rho}_{\mu\nu}}{\partial x^{\delta}}
&=
\frac{1}{2}
\frac{\partial g^{\rho\alpha}}{\partial x^{\delta}}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\\
&\quad
+
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\delta}}
+
\frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\delta}}
-
\frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\delta}}
\right)
\end{aligned}
\end{equation}
Omdat:
\begin{equation}
\frac{\partial g^{\rho\alpha}}{\partial x^{\delta}}
=
\frac{\partial \frac{1}{g_{\rho\alpha}}}{\partial x^{\delta}}
=
\frac{-1}{g^2_{\rho\alpha}}\cdot\frac{\partial g_{\rho\alpha}}{\partial x^{\lambda}}
=
- (g^{\rho\alpha})^{2}
\frac{\partial g_{\rho\alpha}}{\partial x^{\delta}}
\end{equation}
krijgen we:
\begin{equation}
\begin{aligned}
\frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\delta}}
=
-\frac{1}{2}
(g^{\rho\alpha})^{2}
\frac{\partial g_{\rho\alpha}}{\partial x^{\delta}}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\\
+
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\delta}}
+
\frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\delta}}
-
\frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\delta}}
\right)
\end{aligned}
\end{equation}
Appendix 1.3 â Schwarzschildâmetriek in x, y, zâcoördinaten
Coördinatentransformatie
\begin{equation}
x_{0} = t_{\infty}, \qquad dx_{0} = dt_{\infty}
\end{equation}
\begin{equation}
x_{1} = \frac{r^{3}}{3}, \qquad dx_{1} = r^{2}\, dr, \qquad
\frac{dr}{dx_{1}} = \frac{1}{r^{2}}
\end{equation}
\begin{equation}
x_{2} = -\cos\theta, \qquad dx_{2} = \sin\theta\, d\theta = d\theta,
\qquad
\frac{d\theta}{dx_{2}} = \frac{1}{\sin\theta}
\end{equation}
\begin{equation}
x_{3} = \phi, \qquad dx_{3} = d\phi
\end{equation}
Schwarzschildâmetriek in xyzâcoördinaten
\begin{equation}
ds^{2}
=
\sigma^{2} c^{2} dt_{\infty}^{2}
-
\frac{dx_{1}^{2}}{r^{4}\sigma^{2}}
-
r^{2}\frac{dx_{2}^{2}}{\sin^{2}\theta}
-
r^{2}\sin^{2}\theta\, dx_{3}^{2},
\end{equation}
waarbij:
\begin{equation}
\sigma^{2} = 1 - \frac{R_{s}}{r},
\qquad
R_{s} = \frac{2GM}{c^{2}}.
\end{equation}
Aanname: equatorvlak \(\theta = \frac{\pi}{2}\)
\begin{equation}
\sin\theta = 1
\end{equation}
\begin{equation}
ds^{2}
=
\sigma^{2} c^{2} dt_{\infty}^{2}
-
\frac{dx_{1}^{2}}{r^{4}\sigma^{2}}
-
r^{2} dx_{2}^{2}
-
r^{2} dx_{3}^{2}.
\end{equation}
Metriekcomponenten in xyzâcoördinaten
\begin{equation}
g_{00} = \sigma^{2},
\qquad
g^{00} = \frac{1}{\sigma^{2}},
\end{equation}
\begin{equation}
g_{11} = -\frac{1}{r^{4}\sigma^{2}},
\qquad
g^{11} = -r^{4}\sigma^{2},
\end{equation}
\begin{equation}
g_{22} = -\frac{r^{2}}{\sin^{2}\theta},
\qquad
g^{22} = -\frac{\sin^{2}\theta}{r^{2}},
\end{equation}
\begin{equation}
g_{33} = -r^{2}\sin^{2}\theta=-r^2,
\qquad
g^{33} = -\frac{1}{r^{2}\sin^{2}\theta}=\frac{-1}{r^2}.
\end{equation}
Afhankelijkheden:
- \(g_{\mu\nu} = g_{\mu\nu}(r,\theta)\)
- \(\displaystyle \frac{dr}{dx_{1}} = \frac{1}{r^{2}}\)
- \(\displaystyle \frac{d\sigma}{dx_{1}} = \frac{R_{s}}{2 r^{4}\sigma}\)
- \(\displaystyle \frac{d\theta}{dx_{2}} = \frac{1}{\sin\theta}\)
Eerste afgeleiden van de metriek
\begin{equation}
\frac{\partial g_{00}}{\partial x_{1}}
=
\frac{\partial g_{00}}{\partial r}
\frac{dr}{dx_{1}}
=
2\sigma\frac{R_s}{2r^4\sigma}
=
\frac{R_{s}}{r^{4}}
\end{equation}
\begin{equation}
\frac{\partial g_{11}}{\partial x_{1}}
=
\frac{4r - 3R_{s}}{r^{8}\sigma^{4}}
\end{equation}
\begin{equation}
\frac{\partial g_{22}}{\partial x_{1}}
=
\frac{\partial g_{22}}{\partial r}
\frac{\partial r}{\partial x_{1}}
=
r^{-2}\left(\frac{-2r}{\sin^2\theta}\right)
=
\frac{-2}{r\sin^{2}\theta} =\frac{ -2}{r}
\end{equation}
\begin{equation}
\frac{\partial g_{33}}{\partial x_{1}}
=
r^{-2}\left(-2r\sin^2\theta\right)
=
\frac{-2r\sin^{2}\theta}{r} =\frac{ -2}{r}
\end{equation}
\begin{equation}
\frac{\partial g_{22}}{\partial x_{2}}
=
\frac{2r^2\cos\theta}{\sin^3\theta}.\frac{1}{\sin\theta}
=
\frac{2r^{2}\cos\theta}{ \sin^{4}\theta} = 0
\end{equation}
\begin{equation}
\frac{\partial g_{33}}{\partial x_{2}}
=
\frac{\partial g_{33}}{\partial \theta}
\frac{\partial \theta}{\partial x_{2}}
=
-2r^2\sin\theta\,\cos\theta\,\frac{1}{\sin\theta}
=
-2r^{2}\cos\theta = 0
\end{equation}
Tweede afgeleiden van de metriek
\begin{equation}
\frac{\partial^{2} g_{00}}{\partial x_{1}^{2}}
=
-\frac{4R_{s}}{r^{7}}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{11}}{\partial x_{1}^{2}}
=
-\frac{2(14r^{2} + 9R_{s}^{2} - 22rR_{s})}{r^{12}\sigma^{6}}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{22}}{\partial x_{1}^{2}}
=
\frac{2}{r^4\sin^2\theta}
=
\frac{2}{r^{4}}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{22}}{\partial x_{2}^{2}}
=
-2r^{2}\frac{1 + 3\cos^{2}\theta}{\sin^{6}\theta}
= -2r^{2}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{22}}{\partial x_{1}\partial x_{2}}
=
\frac{4\cos\theta}{r\sin^{4}\theta} = 0
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial x_{1}^{2}}
=
\frac2\sin^2\theta{}{r^4}
=
\frac{2}{r^{4}}
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial x_{1}\partial x_{2}}
=
\frac{-4\cos\theta}{r} = 0
\end{equation}
\begin{equation}
\frac{\partial^{2} g_{33}}{\partial x_{2}^{2}}
=
2r^2\,\sin\theta\,\frac{1}{\sin\theta}
=
2r^{2}
\end{equation}
Christoffelâsymbolen in xyzâcoördinaten
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu}
=
\frac{1}{2} g^{\rho\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\end{equation}
Nietânul componenten:
\begin{equation}
\Gamma^{0}{}_{10}
=
\Gamma^{0}{}_{01}
=
\frac{R_{s}}{2 r^{4}\sigma^{2}}
\end{equation}
\begin{equation}
\Gamma^{1}{}_{00}
=
\frac{R_{s}\sigma^{2}}{2}
\end{equation}
\begin{equation}
\Gamma^{1}{}_{11}
=
\frac{3R_{s} - 4r}{2 r^{4}\sigma^{2}}
\end{equation}
\begin{equation}
\Gamma^{1}{}_{22}
=
\frac{-r^{3}\sigma^{2}}{\sin^2\theta}
\end{equation}
\begin{equation}
\Gamma^{1}{}_{33}
=
-r^{3}\sigma^{2}\sin^{2}\theta
=-r^3\sigma^2
\end{equation}
\begin{equation}
\Gamma^{2}{}_{21}
=
\Gamma^{2}{}_{12}
=
\frac{1}{r^{3}}
\end{equation}
\begin{equation}
\Gamma^{2}{}_{33}
=
-\sin^2\theta\,\cos\theta
\end{equation}
\begin{equation}
\Gamma^2_{22}
=
\frac{-\cos\theta}{\sin^2\theta}
=0
\end{equation}
\begin{equation}
\Gamma^{3}{}_{31}
=
\Gamma^{3}{}_{13}
=
\frac{1}{2}
\frac{-1}{r^2\sin^2\theta}
\frac{-2\sin^2\theta}{r}
=
\frac{1}{r^{3}}
\end{equation}
\begin{equation}
\Gamma^{3}{}_{32}
=
\frac{1}{2}
\frac{-1}{r^2\sin^2\theta}
\left(-2r^2\cos\theta\right)
=
\frac{\cos\theta}{\sin^{2}\theta}
=0
\end{equation}
(in het equatorvlak \(\theta = \frac{\pi}{2}\)).
Eerste afgeleiden van Christoffelâsymbolen in xyzâcoördinaten
\begin{equation}
\frac{\partial \Gamma^{0}_{10}}{\partial x_{1}}
=
\frac{\partial \Gamma^{0}_{01}}{\partial x_{1}}
=
\frac{R_{s}(3R_{s} - 4r)}{2 r^{8}\sigma^{4}}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{00}}{\partial x_{1}}
=
\frac{R_{s}}{2 r^{4}}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{11}}{\partial x_{1}}
=
\frac{6}{r^{6}\sigma^{4}}
-
\frac{10R_{s} }{r^{7}\sigma^{4}}
+
\frac{4.5 R_{s}^{2} }{r^{8}\sigma^{4}}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{22}}{\partial x_{1}}
=
\frac{2R_s-3r}{r\sin^2\theta}
=
-3+\frac{2R_s}{r}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{33}}{\partial x_{1}}
=
\left(-3+\frac{2R_s}{r}\right)\sin^2\theta
=
-3+\frac{2R_s}{r}
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{21}}{\partial x_{1}}
=
\frac{\partial \Gamma^{3}_{13}}{\partial x_{1}}
=
-\frac{3}{r^{6}}
\end{equation}
\begin{equation}
\frac{\Gamma^2_{33}}{\partial x_{1}}=
\frac{\Gamma^2_{22}}{\partial x_{1}}=
\frac{\Gamma^3_{23}}{\partial x_{1}}=
\frac{\Gamma^3_{32}}{\partial x_{1}}
=0
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{22}}{\partial x_{2}}
=
\frac{2r^3\sigma^2\cos\theta}{\sin^4\theta}
=0
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{1}_{33}}{\partial x_{2}}
=-2r^3\sigma^2\cos\theta=0
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{33}}{\partial x_{2}}
=-3\cos^2\theta+1=1
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{2}_{22}}{\partial x_{1}}
=
\frac{1+\cos^2\theta}{\sin^{4}\theta}=1
\end{equation}
\begin{equation}
\frac{\partial \Gamma^{3}_{23}}{\partial x_{2}}
=
\frac{\partial \Gamma^{3}_{32}}{\partial x_{2}}
=
-\frac{1+\cos^2\theta}{\sin^4\theta}
=
-1
\end{equation}
Riemannâtensor
\begin{equation}
R^{i}_{jkl}
=
\Gamma^{i}_{jl,k}
-
\Gamma^{i}_{jk,l}
+
\Gamma^{u}_{jl}\Gamma^{i}{}_{uk}
-
\Gamma^{u}_{jk}\Gamma^{i}{}_{ul}
\end{equation}
Ricciâtensor
\begin{equation}
R_{\mu\nu}
=
R^{\rho}_{\mu\rho\nu}
=
\Gamma^{\rho}_{\mu\nu,\rho}
-
\Gamma^{\rho}_{\mu\rho,\nu}
+
\Gamma^{\lambda}_{\mu\nu}\Gamma^{\rho}_{\lambda\rho}
-
\Gamma^{\lambda}_{\mu\rho}\Gamma^{\rho}_{\lambda\nu}
\end{equation}
Of:
\begin{equation}
R_{\mu\nu}
=
R^{\rho}_{\mu\nu\rho}
=
-\Gamma^{\rho}_{\mu\nu,\rho}
+
\Gamma^{\rho}_{\mu\rho,\nu}
-
\Gamma^{\lambda}_{\mu\nu}\Gamma^{\rho}_{\lambda\rho}
+
\Gamma^{\lambda}_{\mu\rho}\Gamma^{\rho}_{\lambda\nu}.
\end{equation}
Opmerking over het teken van het Christoffelâsymbool
Uit de berekeningen volgt dat om alle Ricciâcomponenten in vacuĂŒm nul te krijgen, het Christoffelâsymbool moet beginnen met een positieve +1/2:
\begin{equation}
\Gamma^{\rho}_{\mu\nu}
=
+\frac{1}{2}
g^{\rho\alpha}
\left(
\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}
+
\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}
-
\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}
\right)
\end{equation}
Het teken beĂŻnvloedt alleen de afgeleideâtermen in de Ricciâtensor,
niet de producten van Christoffelâsymbolen.
Schwarzschildâsymmetrie van de Ricciâtensor
\begin{equation}
R_{\mu\nu}
=
\Gamma^0_{\mu\nu,0}
-
\Gamma^0_{0\mu,\nu}
+
\Gamma^{0}_{0\lambda}\Gamma^{\lambda}_{\mu\nu}
-
\Gamma^0_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu}
\end{equation}
\begin{equation}
+\Gamma^1_{\mu\nu,1}
-
\Gamma^1_{1\mu,\nu}
+
\Gamma^{1}_{1\lambda}\Gamma^{\lambda}_{\mu\nu}
-
\Gamma^1_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu}
\end{equation}
\begin{equation}
+\Gamma^2_{\mu\nu,2}
-
\Gamma^2_{2\mu,\nu}
+
\Gamma^{2}_{2\lambda}\Gamma^{\lambda}_{\mu\nu}
-
\Gamma^2_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu}
\end{equation}
\begin{equation}
+\Gamma^3_{\mu\nu,3}
-
\Gamma^3_{3\mu,\nu}
+
\Gamma^{3}_{3\lambda}\Gamma^{\lambda}_{\mu\nu}
-
\Gamma^3_{\nu\lambda}\Gamma^{\lambda}_{\mu\nu}
\end{equation}
In compacte vorm:
\begin{equation}
R_{\mu\nu}
=
\Gamma^{\rho}_{\mu\nu,\rho}
-
\Gamma^{\rho}_{\rho\mu,\nu}
+
\Gamma^{\rho}_{\rho\lambda}\Gamma^{\lambda}_{\nu\mu}
-
\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\rho\mu}
\end{equation}
Ricciâtensorcomponenten voor Schwarzschild
\begin{equation}
R_{00}
=
\Gamma_{00,1}^{1}
+
\Gamma_{11}^{1}\Gamma_{00}^{1}
+
\Gamma_{21}^{2}\Gamma_{00}^{1}
+
\Gamma_{31}^{3}\Gamma_{00}^{1}
-
\Gamma_{00}^{1}\Gamma_{10}^{0}
\end{equation}
\begin{equation}
=\frac{R_s^2}{2r^4}
-\frac{1}{2}\frac{4r-3R_s}{r^4\sigma^2}\frac{1}{2}R_s\sigma^2
-\frac{1}{2}R_s\sigma^2\frac{1}{2}\frac{R_s}{r^4\sigma^2}
-\frac{1}{2}\frac{R_s}{r^4\sigma^2}\frac{1}{2}R_s\sigma^2
\end{equation}
\begin{equation}
\begin{aligned}
R_{11}
&=
-\Gamma_{01,1}^{0}
-
\Gamma_{21,1}^{2}
-
\Gamma_{31,1}^{3}
+
\Gamma_{01}^{0}\Gamma_{11}^{1}
+
\Gamma_{21}^{2}\Gamma_{11}^{1}
\\
&\quad
+
\Gamma_{31}^{3}\Gamma_{11}^{1}
-
\Gamma_{10}^{0}\Gamma_{10}^{0}
-
\Gamma_{12}^{2}\Gamma_{21}^{2}
-
\Gamma_{13}^{3}\Gamma_{31}^{3}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{22}
&=
\Gamma_{22,1}^{1}
-
\Gamma_{32,2}^{3}
+
\Gamma_{01}^{0}\Gamma_{22}^{1}
+
\Gamma_{11}^{1}\Gamma_{22}^{1}
+
\Gamma_{21}^{2}\Gamma_{22}^{1}
\\
&\quad
+
\Gamma_{31}^{3}\Gamma_{22}^{1}
-
\Gamma_{22}^{1}\Gamma_{12}^{2}
-
\Gamma_{21}^{2}\Gamma_{22}^{1}
\end{aligned}
\end{equation}
\begin{equation}
R_{33}
=
\Gamma_{33,1}^{1}
+
\Gamma_{01}^{0}\Gamma_{33}^{1}
+
\Gamma_{11}^{1}\Gamma_{33}^{1}
+
\Gamma_{21}^{2}\Gamma_{33}^{1}
-
\Gamma_{33}^{1}\Gamma_{13}^{3}
\end{equation}
Ricciâtensorcomponenten voor Schwarzschild (\(\theta = \frac{\pi}{2}\))
Voor sferische coördinaten en de Schwarzschild-configuratie met \(đ=90^0\), zijn de volgende elementen van de Ricci-tensor relevant:
\begin{equation}
R_{00}
=
\Gamma_{00,1}^{1}
+
\Gamma_{00}^{1}\Gamma_{11}^{1}
+
\Gamma_{00}^{1}\Gamma_{12}^{2}
+
\Gamma_{00}^{1}\Gamma_{13}^{3}
-
\Gamma_{01}^{0}\Gamma_{00}^{1}
\end{equation}
\begin{equation}
\begin{aligned}
R_{11}
&=
-\Gamma_{10,1}^{0}
-
\Gamma_{12,1}^{2}
-
\Gamma_{13,1}^{3}
+
\Gamma_{11}^{1}\Gamma_{10}^{0}
+
\Gamma_{11}^{1}\Gamma_{12}^{2}
\\
&\quad
+
\Gamma_{11}^{1}\Gamma_{13}^{3}
-
\Gamma_{10}^{0}\Gamma_{01}^{0}
-
\Gamma_{12}^{2}\Gamma_{21}^{2}
-
\Gamma_{13}^{3}\Gamma_{31}^{3}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{22}
&=
\Gamma_{22,1}^{1}
-
\Gamma_{23,2}^{3}
+
\Gamma_{22}^{1}\Gamma_{10}^{0}
+
\Gamma_{22}^{1}\Gamma_{11}^{1}
+
\Gamma_{22}^{1}\Gamma_{13}^{3}
\\
&\quad
+\boxed{
\Gamma_{22}^{2}\Gamma_{32}^{3}}
-
\Gamma_{21}^{2}\Gamma_{22}^{1}
-
\Gamma_{23}^{3}\Gamma_{32}^{3}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{33}
&=
\Gamma_{33,1}^{1}
+
\Gamma_{33,2}^{2}
+
\Gamma_{33}^{1}\Gamma_{10}^{0}
+
\Gamma_{33}^{1}\Gamma_{11}^{1}
+
\Gamma_{33}^{1}\Gamma_{12}^{2}
\\
&\quad
+\boxed{
\Gamma_{33}^{2}\Gamma_{22}^{2}}
-
\Gamma_{31}^{3}\Gamma_{33}^{1}
-
\Gamma_{32}^{3}\Gamma_{33}^{2}
\end{aligned}
\end{equation}
\begin{equation}
R_{33} = \sin^{2}\theta\, R_{22}
\end{equation}
Wanneer \(\theta \neq \frac{\pi}{2}\) komen er dus extra termen voor:
\begin{equation}
R_{22} \to R_{22} + \Gamma_{22}^{2}\Gamma_{32}^{3},
\qquad
R_{33} \to R_{33} + \Gamma_{33}^{2}\Gamma_{22}^{2}.
\end{equation}