# Geodesic deviation second covariant derivative

Suppose that $$V$$ is a vector field tangent to a family of geodesic curves. Furthermore suppose that there exists another vector field $$T$$ , where T is the tangent vector field that determines how far the geodesics are from each other. I know how to compute an expression such as $$\nabla_VV$$ however I do not know how to compute an expression such as $$\nabla_T\nabla_VV$$. How would I compute this second covariant derivative?

• You have $\nabla_V V = \mu V$ for some $\mathbb C$-valued function $\mu$ (which is zero if you take affine parametrization). So $\nabla_T \nabla_V V = \nabla_T (\mu \nabla V)$ ($=0$ with affine parametrization) Oct 19 '21 at 7:58
• I know that the expression equals 0 but how can I show that it equals 0?
– aygx
Oct 19 '21 at 12:19
• By definition, if the geodesic have an affine parameter (such as proper time in GR), $\nabla_V V = 0$. So its derivative (covariant or not) must also be zero. Oct 19 '21 at 21:36