Christoffel symbols

The Christoffel symbols are related to the metric tensor by

${\mathrm{\Gamma }}_{\mu \nu }^{\lambda }=\frac{1}{2}{g}^{\lambda \rho }(g_{\mu \rho }{,}_{\nu }+g_{\rho \nu }{,}_{\mu }-g_{\mu \nu }{,}_{\rho })$

where the comma is a partial derivative. For example

$g_{\mu \nu }{,}_{\rho }=\frac{\partial g_{\mu \nu }}{\partial x^{\rho }}$

The Christoffel symbols are part of a covariant derivative opperation, represented by a semicolin or capitalized D ,mapping tensor elements to tensor elements. For example

$T^{\lambda }{;}_{\rho }=\frac{D{T}^{\lambda }}{\partial x^{\rho }}=\frac{\partial T^{\lambda }}{\partial x^{\rho }}+{\mathrm{\Gamma }}_{\mu \rho }^{\lambda }{T}^{\mu }$

Also for example

$T_{\lambda }{;}_{\rho }=\frac{D{T}_{\lambda }}{\partial x^{\rho }}=\frac{\partial T_{\lambda }}{\partial x^{\rho }}-{\mathrm{\Gamma }}_{\lambda \rho }^{\mu }{T}_{\mu }$

And differentiating with respect to an invariant example

$\frac{D{T}^{\lambda }}{d\tau }=\frac{d{T}^{\lambda }}{d\tau }+{\mathrm{\Gamma }}_{\mu \nu }^{\lambda }{T}^{\mu }\frac{\partial x^{\nu }}{d\tau }$

References

• Relativity