Appendix 2 — Afleiding van de Afgeleide van de Christoffel‑Symbolen in Algemene Vorm
Er wordt aangetoond hoe het Christoffel‑symbool uitsluitend afhangt van de elementen van de metrische tensor en diens afgeleiden. Dit is bijzonder handig bij implementatie in spreadsheets of computerprogramma’s.
Christoffel‑symbool
\[ \Gamma^{\rho}{}_{\mu\nu} = \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) \]
Afgeleide van het Christoffel‑symbool
\[ \frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\gamma}} = \frac{1}{2} \frac{\partial g^{\rho\alpha}}{\partial x^{\gamma}} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\gamma}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\gamma}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\gamma}} \right) \]
Afgeleide van de inverse metriek
\[ \frac{\partial g^{\rho\alpha}}{\partial x^{\gamma}} = - g^{\rho\lambda} g^{\alpha\sigma} \frac{\partial g_{\lambda\sigma}}{\partial x^{\gamma}} \]
Dus: \[ \frac{\partial g^{\rho\alpha}}{\partial x^{\gamma}} = - (g^{\rho\alpha})^{2} \frac{\partial g_{\rho\alpha}}{\partial x^{\gamma}} \]
Volledige afgeleide van het Christoffel‑symbool
\[ \frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\gamma}} = -\frac{1}{2} (g^{\rho\alpha})^{2} \frac{\partial g_{\rho\alpha}}{\partial x^{\gamma}} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\gamma}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\gamma}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\gamma}} \right) \]
Compacte vorm
\[ \frac{\partial \Gamma^{\rho}{}_{\mu\nu}}{\partial x^{\gamma}} = - g^{\rho\alpha} \frac{\partial g_{\rho\alpha}}{\partial x^{\gamma}} \Gamma^{\rho}{}_{\mu\nu} + \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial^{2} g_{\nu\alpha}}{\partial x^{\mu}\partial x^{\gamma}} + \frac{\partial^{2} g_{\mu\alpha}}{\partial x^{\nu}\partial x^{\gamma}} - \frac{\partial^{2} g_{\mu\nu}}{\partial x^{\alpha}\partial x^{\gamma}} \right) \]