Appendix 1 Formules van de Algemene Relativiteitstheorie | De Algemene Relativiteitstheorie van Einstein

Appendix 1 — Formules van de Algemene Relativiteitstheorie

Hieronder geven we een samenvatting van een aantal eerder afgeleide formules uit de algemene relativiteitstheorie en de Schwarzschild‑oplossing. Vervolgens leiden we alle formules af die relevant zijn voor berekeningen in verschillende hoofdstukken. In deze appendix passen we de Einstein‑notatie toe.

Algemene Relativiteitstheorie — Basisformules

Einsteins veldvergelijkingen: \[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}}\, T_{\mu\nu}. \]

Waarbij:

\(R_{\mu\nu}\): de Ricci‑tensor,
\(g_{\mu\nu}\): de metrische tensor,
\(R\): de Ricci‑scalar,
\(\lambda\): de kosmologische constante,
\(T_{\mu\nu}\): de energie‑impuls‑tensor.

Schwarzschild‑metriek (in sferische coördinaten)

\[ ds^{2} = \left(1 - \frac{2GM}{r c^{2}}\right)c^{2} dt^{2} - \left(1 - \frac{2GM}{r c^{2}}\right)^{-1} dr^{2} - r^{2} d\theta^{2} - r^{2}\sin^{2}\theta\, d\phi^{2}. \]

Waarbij:

\(ds^{2}\): het ruimte‑tijdinterval,
\(G\): 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: \[ d\tau = \sqrt{1 - \frac{2GM}{r c^{2}}}\; dt, \] waarbij \(d\tau\) de lokale (eigen)tijd is en \(dt\) de coördinaattijd op grote afstand.

\[ \Delta \tau = \Delta t \,\sqrt{1 - \frac{2GM}{r c^{2}}} \]

Waarbij:

\(\Delta \tau\): de eigen tijd voor een waarnemer op afstand \(r\),
\(\Delta t\): de tijd voor een verre waarnemer.

Baan van licht (null‑geodeten)

Voor licht geldt \(ds^{2} = 0\). Daaruit volgt: \[ \left(1 - \frac{2GM}{r c^{2}}\right)c^{2} dt^{2} = \left(1 - \frac{2GM}{r c^{2}}\right)^{-1} dr^{2} + r^{2} d\theta^{2} + r^{2}\sin^{2}\theta\, d\phi^{2}. \]

Krommingsradius van licht om een massa

De afwijking van een lichtstraal in de buurt van een massa wordt gegeven door: \[ \delta\phi = \frac{4GM}{r c^{2}}. \]

Appendix 1.1 — Samenvatting en afleiding van verdere relevante formules

In deze sectie zullen we de relevante formules voor de specifieke berekeningen in de hoofdstukken afleiden. Dit omvat de afleiding van:

de metrische tensor in verschillende coördinatenstelsels,
de geodetenvergelijkingen,
de energie‑impuls‑tensor in diverse configuraties.

\[ \Delta \tau = \Delta t \,\sqrt{1 - \frac{2GM}{r c^{2}}} \]

Waarbij:

\(\Delta \tau\): de eigen tijd voor een waarnemer op afstand \(r\),
\(\Delta t\): de tijd voor een verre waarnemer.

Baan van licht (null‑geodeten)

Krommingsradius van licht om een massa

De afwijking van een lichtstraal in de buurt van een massa wordt gegeven door: \[ \delta\phi = \frac{4GM}{r c^{2}}. \]

Appendix 1.1 — 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 energie‑impuls‑tensor in diverse configuraties.

Coördinatentransformaties

\[ dx^{m} = \frac{\partial x^{m}}{\partial y^{r}}\, dy^{r} \]

\[ ds^{2} = \eta_{mn}\, d\xi^{m} d\xi^{n} \]

\[ ds^{2} = g_{mn}(x)\, dx^{m} dx^{n} = g_{pq}(y)\, dy^{p} dy^{q} \]

\[ g_{pq}(y) = g_{mn}(x) \frac{\partial x^{m}}{\partial y^{p}} \frac{\partial x^{n}}{\partial y^{q}} \]

Transformatie van vectoren en tensoren

\[ V'^{n}(y) = \frac{\partial y^{n}}{\partial x^{m}}\, V^{m}(x) \]

\[ W'_{p}(y) = \frac{\partial x^{q}}{\partial y^{p}}\, W_{q}(x) \]

\[ T_{mn}(x) = \frac{\partial V_{m}(x)}{\partial x^{n}} \]

\[ T_{mn}(y) = \frac{\partial x^{r}}{\partial y^{m}} \frac{\partial x^{s}}{\partial y^{n}} T_{rs}(x) \]

\[ T_{mn}(y) = \frac{\partial y^{m}}{\partial x^{r}} \frac{\partial y^{n}}{\partial x^{s}} T_{rs}(x) \]

\[ T_{rs}(x) = A^{x}_{r} B^{x}_{s} \]

Indexverhoging en -verlaging

\[ E_{\mu} = g_{\mu\nu} E^{\nu} \]

\[ E^{\mu} = g^{\mu\nu} E_{\nu} = g^{\mu\nu} g_{\nu\rho} E^{\rho} = \delta^{\mu}{}_{\rho} E^{\rho} = E^{\mu} \]

Lijnsegment in klein gebied

Pythagoras: \[ ds^{2} = \delta_{mn} \frac{\partial x^{m}}{\partial y^{r}} \frac{\partial x^{n}}{\partial y^{s}} dy^{r} dy^{s} \]

Transformeren naar ander frame: \[ ds^{2} = \delta_{mn} \frac{\partial x^{m}}{\partial y^{r}} \frac{\partial x^{n}}{\partial y^{s}} dy^{r} dy^{s} \]

Metrische tensor

\[ g_{mn} = \delta_{mn} \frac{\partial x^{m}}{\partial y^{r}} \frac{\partial x^{n}}{\partial y^{s}} \]

Einsteins veldvergelijkingen

\[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}}\, T_{\mu\nu} \]

Geodetische vergelijking

\[ \frac{d^{2} x^{\lambda}}{d\tau^{2}} + \Gamma^{\lambda}{}_{\mu\nu} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} = 0 \]

\[ \Gamma^{\lambda}{}_{\mu\nu} \equiv \frac{\partial x^{\lambda}}{\partial \xi^{\alpha}} \frac{\partial^{2} \xi^{\alpha}}{\partial x^{\mu}\partial x^{\nu}} \]

Tensortransformaties

\[ T'_{\mu\nu}(y) = \frac{\partial x^{\alpha}}{\partial y^{\mu}} \frac{\partial x^{\beta}}{\partial y^{\nu}} T_{\alpha\beta}(x) \]

\[ T'^{\mu\nu}(y) = \frac{\partial y^{\mu}}{\partial x^{\alpha}} \frac{\partial y^{\nu}}{\partial x^{\beta}} T^{\alpha\beta}(x) \]

\[ T_{\mu}{}^{\nu}(y) = \frac{\partial x^{\alpha}}{\partial y^{\mu}} \frac{\partial y^{\nu}}{\partial x^{\beta}} T_{\alpha}{}^{\beta}(x) \]

\[ g_{\mu\alpha} g^{\alpha\nu} = \delta_{\mu}{}^{\nu} \]

Contractie

\[ A_{\mu} = g_{\mu\nu} A^{\nu} \]

\[ A \cdot B = g_{\mu\nu} A^{\mu} B^{\nu} \equiv A_{\nu} B^{\nu} \]

Ricci‑tensor

\[ R_{\mu\nu} = R^{\rho}{}_{\mu\rho\nu} = \Gamma^{\rho}{}_{\mu\nu,\rho} - \Gamma^{\rho}{}_{\mu\rho,\nu} + \Gamma^{\rho}{}_{\lambda\rho}\Gamma^{\lambda}{}_{\nu\mu} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\rho} \]

\[ G_{\mu\nu} = \Gamma^{\rho}{}_{\mu\nu,\rho} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\rho} \quad\text{(alleen als } g = \det(g_{\mu\nu}) = -1\text{)} \]

Christoffel‑symbolen

\[ \Gamma^{\rho}{}_{\mu\nu} = \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) \]

Ricci‑scalar

\[ R = g^{\mu\nu} R_{\mu\nu} \]

\[ R = g^{ab} \left( \Gamma^{c}{}_{ab,c} - \Gamma^{c}{}_{ac,b} + \Gamma^{d}{}_{ab}\Gamma^{c}{}_{dc} - \Gamma^{d}{}_{ac}\Gamma^{c}{}_{db} \right) \]

\[ R = 2 g^{ab} \left( \Gamma^{c}{}_{a[b,c]} + \Gamma^{d}{}_{a[b}\Gamma^{c}{}_{c]d} \right) \]

Appendix 1.2 — Schwarzschild‑metriek in polaire coördinaten

De Schwarzschild‑metriek luidt: \[ ds^{2} = \sigma^{2} c^{2} dt^{2} - \frac{dr^{2}}{\sigma^{2}} - r^{2} d\theta^{2} - r^{2}\sin^{2}\theta\, d\phi^{2}, \] waarbij: \[ \sigma^{2} = 1 - \frac{R_{s}}{r}, \qquad R_{s} = \frac{2GM}{c^{2}}. \]

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 \(\theta = \frac{\pi}{2}\)

\[ g_{00} = \sigma^{2}, \qquad g^{00} = \frac{1}{\sigma^{2}}, \] \[ g_{11} = -\frac{1}{\sigma^{2}}, \qquad g^{11} = -\sigma^{2}, \] \[ g_{22} = -r^{2}, \qquad g^{22} = -\frac{1}{r^{2}}, \] \[ g_{33} = -r^{2}\sin^{2}\theta = -r^{2}, \qquad g^{33} = -\frac{1}{r^{2}\sin^{2}\theta} = -\frac{1}{r^{2}}. \]

Afgeleide van \(\sigma\): \[ \frac{d\sigma}{dr} = \frac{R_{s}}{2 r^{2} \sigma}. \]

Eerste afgeleiden van de metriek

\[ \frac{\partial g_{00}}{\partial r} = \frac{R_{s}}{r^{2}}, \qquad \frac{\partial g_{11}}{\partial r} = \frac{R_{s}}{r^{2}\sigma^{4}}, \] \[ \frac{\partial g_{22}}{\partial r} = -2r, \qquad \frac{\partial g_{33}}{\partial r} = -2r\sin^{2}\theta = -2r, \] \[ \frac{\partial g_{33}}{\partial \theta} = -2 r^{2}\sin\theta\cos\theta = 0. \]

Tweede afgeleiden van de metriek

\[ \frac{\partial^{2} g_{00}}{\partial r^{2}} = -\frac{2R_{s}}{r^{3}}, \qquad \frac{\partial^{2} g_{11}}{\partial r^{2}} = -\frac{2R_{s}}{r^{3}\sigma^{6}}, \] \[ \frac{\partial^{2} g_{22}}{\partial r^{2}} = -2, \qquad \frac{\partial^{2} g_{33}}{\partial r^{2}} = -2\sin^{2}\theta = -2, \] \[ \frac{\partial^{2} g_{33}}{\partial \theta \partial r} = -4r\sin\theta\cos\theta = 0, \] \[ \frac{\partial^{2} g_{33}}{\partial \theta^{2}} = 2r^{2}(\sin^{2}\theta - \cos^{2}\theta) = 2r^{2}. \]

Christoffel‑symbolen voor Schwarzschild in polaire coördinaten

Niet‑nul componenten:

\[ \Gamma^{0}{}_{10} = \Gamma^{0}{}_{01} = \frac{1}{2} g^{00}\frac{\partial g_{00}}{\partial r} = \frac{R_{s}}{2 r^{2} \sigma^{2}}, \] \[ \Gamma^{1}{}_{00} = \frac{1}{2} g^{11}\left(-\frac{\partial g_{00}}{\partial r}\right) = \sigma^{2}\frac{R_{s}}{2 r^{2}}, \] \[ \Gamma^{1}{}_{11} = \frac{1}{2} g^{11}\frac{\partial g_{11}}{\partial r} = -\frac{R_{s}}{2 r^{2} \sigma^{2}}, \] \[ \Gamma^{2}{}_{21} = \Gamma^{2}{}_{12} = \frac{1}{r}, \qquad \Gamma^{3}{}_{31} = \Gamma^{3}{}_{13} = \frac{1}{r}, \] \[ \Gamma^{1}{}_{22} = -r\sigma^{2}, \qquad \Gamma^{1}{}_{33} = -r\sigma^{2}\sin^{2}\theta, \] \[ \Gamma^{3}{}_{32} = \frac{\cos\theta}{\sin\theta}, \qquad \Gamma^{2}{}_{33} = -\sin\theta\cos\theta. \]

Eerste afgeleiden van Christoffel‑symbolen

\[ \frac{\partial \Gamma^{0}{}_{10}}{\partial r} = \frac{R_{s}(R_{s}-2r)}{2 r^{4} \sigma^{4}}, \qquad \frac{\partial \Gamma^{1}{}_{00}}{\partial r} = \frac{R_{s}(3R_{s}-2r)}{2 r^{4}}, \] \[ \frac{\partial \Gamma^{1}{}_{11}}{\partial r} = \frac{R_{s}(2r - R_{s})}{2 r^{4} \sigma^{4}}, \] \[ \frac{\partial \Gamma^{2}{}_{21}}{\partial r} = -\frac{1}{r^{2}}, \qquad \frac{\partial \Gamma^{3}{}_{31}}{\partial r} = -\sin^{2}\theta, \] \[ \frac{\partial \Gamma^{3}{}_{32}}{\partial \theta} = -\cos^{2}\theta + \sin^{2}\theta = 1, \] \[ \frac{\partial \Gamma^{2}{}_{33}}{\partial \theta} = -\frac{1}{\sin^{2}\theta} = -1. \]

Eerste afgeleide van het Christoffel‑symbool (algemene vorm)

\[ \frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\delta}} = \frac{1}{2} \frac{\partial g^{\rho\alpha}}{\partial x^{\delta}} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\delta}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\delta}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\delta}} \right) \]

Omdat: \[ \frac{\partial g^{\rho\alpha}}{\partial x^{\delta}} = - g^{\rho\lambda} g^{\alpha\sigma} \frac{\partial g_{\lambda\sigma}}{\partial x^{\delta}}, \] krijgen we: \[ \frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\delta}} = -\frac{1}{2} (g^{\rho\alpha})^{2} \frac{\partial g_{\rho\alpha}}{\partial x^{\delta}} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\delta}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\delta}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\delta}} \right). \]

Appendix 1.3 — Schwarzschild‑metriek in x, y, z‑coördinaten

Coördinatentransformatie

\[ x^{0} = t_{\infty}, \qquad dx^{0} = dt_{\infty} \] \[ x^{1} = r^{3/3} = r, \qquad dx^{1} = r^{2}\, dr, \qquad \frac{dr}{dx^{1}} = \frac{1}{r^{2}} \] \[ x^{2} = -\cos\theta, \qquad dx^{2} = \sin\theta\, d\theta = d\theta, \qquad \frac{d\theta}{dx^{2}} = \frac{1}{\sin\theta} \] \[ x^{3} = \phi, \qquad dx^{3} = d\phi \]

Schwarzschild‑metriek in xyz‑coördinaten

\[ ds^{2} = \sigma^{2} c^{2} dt_{\infty}^{2} - \frac{dx_{1}^{2}}{r^{4}\sigma^{2}} - r^{2}\frac{dx_{2}^{2}}{\sin^{2}\theta} - r^{2}\sin^{2}\theta\, dx_{3}^{2}, \] waarbij: \[ \sigma^{2} = 1 - \frac{R_{s}}{r}, \qquad R_{s} = \frac{2GM}{c^{2}}. \]

Aanname: equatorvlak \(\theta = \frac{\pi}{2}\)

\[ \sin\theta = 1 \] \[ ds^{2} = \sigma^{2} c^{2} dt_{\infty}^{2} - \frac{dx_{1}^{2}}{r^{4}\sigma^{2}} - r^{2} dx_{2}^{2} - r^{2} dx_{3}^{2}. \]

Metriekcomponenten in xyz‑coördinaten

\[ g_{00} = \sigma^{2}, \qquad g^{00} = \frac{1}{\sigma^{2}}, \] \[ g_{11} = -\frac{1}{r^{4}\sigma^{2}}, \qquad g^{11} = -r^{4}\sigma^{2}, \] \[ g_{22} = -r^{2}\sin^{2}\theta, \qquad g^{22} = -\frac{1}{r^{2}\sin^{2}\theta}, \] \[ g_{33} = -r^{2}\sin^{2}\theta, \qquad g^{33} = -\frac{1}{r^{2}\sin^{2}\theta}. \]

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

\[ \frac{\partial g_{00}}{\partial x^{1}} = \frac{\partial g_{00}}{\partial r} \frac{dr}{dx^{1}} = \frac{R_{s}}{r^{4}} \] \[ \frac{\partial g_{11}}{\partial x^{1}} = \frac{4r - 3R_{s}}{r^{8}\sigma^{4}} \] \[ \frac{\partial g_{22}}{\partial x^{1}} = -2r\sin^{2}\theta = -2r \] \[ \frac{\partial g_{33}}{\partial x^{1}} = -2r\sin^{2}\theta = -2r \] \[ \frac{\partial g_{22}}{\partial x^{2}} = 2r^{2}\cos\theta\, \sin^{-4}\theta = 0 \] \[ \frac{\partial g_{33}}{\partial x^{2}} = -2r^{2}\cos\theta = 0 \]

Tweede afgeleiden van de metriek

\[ \frac{\partial^{2} g_{00}}{\partial x_{1}^{2}} = -\frac{4R_{s}}{r^{7}} \] \[ \frac{\partial^{2} g_{11}}{\partial x_{1}^{2}} = -\frac{2(14r^{2} + 9R_{s}^{2} - 22rR_{s})}{r^{12}\sigma^{6}} \] \[ \frac{\partial^{2} g_{22}}{\partial x_{1}^{2}} = 2r^{4} \] \[ \frac{\partial^{2} g_{22}}{\partial x_{2}^{2}} = -2r^{2}\frac{1 + 3\cos^{2}\theta}{\sin^{6}\theta} = -2r^{2} \] \[ \frac{\partial^{2} g_{22}}{\partial x_{1}\partial x_{2}} = 4r\cos\theta\, \sin^{-4}\theta = 0 \] \[ \frac{\partial^{2} g_{33}}{\partial x_{1}^{2}} = 2r^{4} \] \[ \frac{\partial^{2} g_{33}}{\partial x_{1}\partial x_{2}} = -4r\cos\theta = 0 \] \[ \frac{\partial^{2} g_{33}}{\partial x_{2}^{2}} = 2r^{2} \]

Christoffel‑symbolen in xyz‑coördinaten

Niet‑nul componenten:

\[ \Gamma^{0}{}_{10} = \Gamma^{0}{}_{01} = \frac{R_{s}}{2 r^{4}\sigma^{2}} \] \[ \Gamma^{1}{}_{00} = \frac{R_{s}}{2\sigma^{2}} \] \[ \Gamma^{1}{}_{11} = \frac{3R_{s} - 4r}{2 r^{4}\sigma^{2}} \] \[ \Gamma^{2}{}_{21} = \Gamma^{2}{}_{12} = \frac{1}{r^{3}} \] \[ \Gamma^{3}{}_{31} = \Gamma^{3}{}_{13} = \frac{1}{r^{3}} \] \[ \Gamma^{1}{}_{22} = -r^{3}\sigma^{2} \] \[ \Gamma^{1}{}_{33} = -r^{3}\sigma^{2}\sin^{2}\theta \] \[ \Gamma^{3}{}_{32} = \frac{\cos\theta}{\sin^{2}\theta} \] \[ \Gamma^{2}{}_{33} = -\cos\theta\, \sin\theta \]

Eerste afgeleiden van Christoffel‑symbolen

\[ \frac{\partial \Gamma^{0}{}_{10}}{\partial x^{1}} = \frac{R_{s}(3R_{s} - 4r)}{2 r^{8}\sigma^{4}} \] \[ \frac{\partial \Gamma^{0}{}_{01}}{\partial x^{1}} = \frac{R_{s}(3R_{s} - 4r)}{2 r^{8}\sigma^{4}} \] \[ \frac{\partial \Gamma^{1}{}_{00}}{\partial x^{1}} = \frac{R_{s}}{2 r^{4}} \] \[ \frac{\partial \Gamma^{1}{}_{11}}{\partial x^{1}} = \frac{6r^{6}}{\sigma^{4}} - \frac{10R_{s} r^{7}}{\sigma^{4}} + \frac{4.5 R_{s}^{2} r^{8}}{\sigma^{4}} \] \[ \frac{\partial \Gamma^{2}{}_{21}}{\partial x^{1}} = -\frac{3}{r^{6}} \] \[ \frac{\partial \Gamma^{3}{}_{31}}{\partial x^{1}} = -\frac{3}{r^{6}} \] \[ \frac{\partial \Gamma^{3}{}_{32}}{\partial x^{2}} = 1 \] \[ \frac{\partial \Gamma^{2}{}_{33}}{\partial x^{2}} = 1 \] \[ \frac{\partial \Gamma^{3}{}_{23}}{\partial x^{2}} = -1 \]

Christoffel‑symbolen (aanvullende componenten)

\[ \Gamma^{2}{}_{33} = \Gamma^{3}{}_{23} = \frac{1}{2} g^{33} \frac{\partial g_{33}}{\partial x^{2}} = \frac{1}{2} \left(-\frac{1}{r^{2}\sin^{2}\theta}\right) \left(-2 r^{2}\cos\theta\right) = \cos\theta\, \sin^{-2}\theta = 0 \] (in het equatorvlak \(\theta = \frac{\pi}{2}\)).

Eerste afgeleiden van Christoffel‑symbolen in xyz‑coördinaten

\[ \frac{\partial \Gamma^{0}{}_{10}}{\partial x^{1}} = \frac{\partial \Gamma^{1}{}_{00}}{\partial x^{1}} = \frac{R_{s}(3R_{s}-4r)}{2 r^{8}\sigma^{4}} \] \[ \frac{\partial \Gamma^{0}{}_{01}}{\partial x^{1}} = \frac{R_{s}}{2 r^{4}} \] \[ \frac{\partial \Gamma^{1}{}_{11}}{\partial x^{1}} = \frac{6 r^{6}}{\sigma^{4}} - \frac{10 R_{s} r^{7}}{\sigma^{4}} + \frac{4.5 R_{s}^{2} r^{8}}{\sigma^{4}} \] \[ \frac{\partial \Gamma^{2}{}_{21}}{\partial x^{1}} = \frac{2R_{s} - 3r}{r\sin^{2}\theta} = -3 + \frac{2R_{s}}{r} \] \[ \frac{\partial \Gamma^{3}{}_{31}}{\partial x^{1}} = -3 + \frac{2R_{s}}{r} \] \[ \frac{\partial \Gamma^{1}{}_{22}}{\partial x^{1}} = \frac{\partial \Gamma^{2}{}_{12}}{\partial x^{1}} = \frac{\partial \Gamma^{1}{}_{33}}{\partial x^{1}} = \frac{\partial \Gamma^{3}{}_{13}}{\partial x^{1}} = -\frac{3}{r^{6}} \] \[ \frac{\partial \Gamma^{3}{}_{32}}{\partial x^{2}} = -3\cos^{2}\theta + 1 = 1 \] \[ \frac{\partial \Gamma^{2}{}_{22}}{\partial x^{2}} = 1 + \frac{\cos^{2}\theta}{\sin^{4}\theta} = 1 \] \[ \frac{\partial \Gamma^{3}{}_{23}}{\partial x^{2}} = -1 - \frac{\cos^{2}\theta}{\sin^{4}\theta} = -1 \]

Riemann‑tensor

\[ R^{j}{}_{kli} = \Gamma^{j}{}_{l,k i} - \Gamma^{j}{}_{k,l i} + \Gamma^{j}{}_{l u}\Gamma^{u}{}_{k i} - \Gamma^{j}{}_{k u}\Gamma^{u}{}_{l i} \]

Ricci‑tensor

\[ R_{\mu\nu} = R^{\rho}{}_{\mu\rho\nu} = \Gamma^{\rho}{}_{\mu\nu,\rho\rho} - \Gamma^{\rho}{}_{\mu\rho,\nu\rho} + \Gamma^{\rho}{}_{\mu\nu\lambda}\Gamma^{\lambda}{}_{\rho\rho} - \Gamma^{\rho}{}_{\mu\rho\lambda}\Gamma^{\lambda}{}_{\nu\rho} \]

In compacte vorm: \[ R_{\mu\nu} = \Gamma^{\rho}{}_{\mu\nu,\rho\rho} - \Gamma^{\rho}{}_{\mu\rho,\nu\rho} + \Gamma^{\rho}{}_{\lambda\rho}\Gamma^{\lambda}{}_{\nu\mu} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\rho}. \]

Opmerking over het teken van het Christoffel‑symbool

Uit de berekeningen volgt dat om alle Ricci‑componenten in vacuüm nul te krijgen, het Christoffel‑symbool moet beginnen met: \[ \Gamma^{\mu}{}_{\nu\rho} = +\frac{1}{2} g^{\mu\alpha} \left( \partial_{\nu} g_{\rho\alpha} + \partial_{\rho} g_{\nu\alpha} - \partial_{\alpha} g_{\nu\rho} \right) \] Het teken beïnvloedt alleen de afgeleide‑termen in de Ricci‑tensor, niet de producten van Christoffel‑symbolen.

Schwarzschild‑symmetrie van de Ricci‑tensor

\[ R_{\mu\nu} = \Gamma_{\mu\nu,00} - \Gamma_{0\mu,\nu 0} + \Gamma^{0}{}_{\lambda 0}\Gamma^{\lambda}{}_{\nu\mu} - \Gamma^{\nu}{}_{\lambda 0}\Gamma^{0}{}_{\mu\lambda} + \Gamma_{\mu\nu,11} - \Gamma_{1\mu,\nu 1} + \Gamma^{1}{}_{\lambda 1}\Gamma^{\lambda}{}_{\nu\mu} - \Gamma^{\nu}{}_{\lambda 1}\Gamma^{1}{}_{\mu\lambda} + \cdots \]

In compacte vorm: \[ R_{\mu\nu} = \Gamma_{\mu\nu,\rho\rho} - \Gamma_{\rho\mu,\nu\rho} + \Gamma^{\rho}{}_{\lambda\rho}\Gamma^{\lambda}{}_{\nu\mu} - \Gamma^{\nu}{}_{\lambda\rho}\Gamma^{\rho}{}_{\mu\lambda} \]

Ricci‑tensorcomponenten voor Schwarzschild (\(\theta = \frac{\pi}{2}\))

\[ R_{00} = \Gamma_{00,11} + \Gamma_{001}\Gamma_{111} + \Gamma_{001}\Gamma_{122} + \Gamma_{001}\Gamma_{133} - \Gamma_{010}\Gamma_{001} \]

\[ R_{11} = -\Gamma_{10,10} - \Gamma_{12,12} - \Gamma_{13,13} + \Gamma_{111}\Gamma_{100} + \Gamma_{111}\Gamma_{122} + \Gamma_{111}\Gamma_{133} - \Gamma_{100}\Gamma_{010} - \Gamma_{122}\Gamma_{212} - \Gamma_{133}\Gamma_{313} \]

\[ R_{22} = \Gamma_{22,11} - \Gamma_{23,23} + \Gamma_{221}\Gamma_{100} + \Gamma_{221}\Gamma_{111} + \Gamma_{221}\Gamma_{133} + \Gamma_{222}\Gamma_{323} - \Gamma_{212}\Gamma_{221} - \Gamma_{233}\Gamma_{323} \]

\[ R_{33} = \Gamma_{33,11} + \Gamma_{33,22} + \Gamma_{331}\Gamma_{100} + \Gamma_{331}\Gamma_{111} + \Gamma_{331}\Gamma_{122} + \Gamma_{332}\Gamma_{222} - \Gamma_{313}\Gamma_{331} - \Gamma_{323}\Gamma_{332} \]

\[ R_{33} = \sin^{2}\theta\, R_{22} \]

Wanneer \(\theta \neq \frac{\pi}{2}\) komen extra termen voor: \[ R_{22} \to R_{22} + \Gamma_{222}\Gamma_{323}, \qquad R_{33} \to R_{33} + \Gamma_{332}\Gamma_{222}. \]