Geodesics

The Geodesic equation is $V^{\mu }{\nabla }_{\mu }{V}^{\nu }=0$ We can write this in terms ofChristoffel-symbols as $V^{\mu }{\nabla }_{\mu }{V}^{\nu }=V^{\mu }({\partial }_{\mu }{V}^{\nu }+{\Gamma }_{\nu \lambda }^{\mu }{V}^{\lambda })=\frac{d^2{x}^{\mu }}{d{\lambda }^2}+{\Gamma }_{\nu \lambda }^{\mu }\frac{d{x}^{\nu }}{d\lambda }\frac{d{x}^{\lambda }}{d\lambda }=0$ When we choose the metric as theMinkowski metric, theChristoffel-symbols vanish and hence the geodesic equation recovers straight lines. A consequence is that the length of the tangent vector $V^{\lambda }{\partial }_{\lambda }(g_{\mu \nu }{V}^{\mu }{V}^{\nu })=V^{\lambda }{\nabla }_{\lambda }(g_{\mu \nu }{V}^{\mu }{V}^{\nu })=2V^{\lambda }{V}^{\mu }{g}_{\mu \nu }{\nabla }_{\lambda }{V}^{\nu }=0$ By choice of the parameter, we can set the length to be ±1,0 (Proper length (+1) or Proper time (-1)). Alternatively, we can define geodesics as an extrema of proper length or proper time. This method of finding the Euler-Lagrangian equations resulting from the proper length can be an efficient way to calculate theChristoffel-symbols in curved geometry. **Steps:

• Write the EL equation and choose the parameter as proper length. Set the length of the tangent vector to 1.
• Obtain the equations and compare with the Geodesic equation.
• Write theChristoffel-symbols .