Geodesic Deviation

Geodesics in flat space maintain their separation; those in curved space don’t. Consider a one-parameter family of geodesics with tangent V and geodesic deviation vector U, pointing from one geodesic to another. Note: $V^{\mu }{\nabla }_{\mu }{U}^{\nu }-U^{\mu }{\nabla }_{\mu }{V}^{\nu }=0$ Then the relative acceleration is A^\mu=V^\mu\nabla_\mu(V^\lambda\nabla_\lambda U^\nu)=V^\mu\nabla_\mu(U^\lambda\nabla_\lambda V^\nu)=V^\mu\nabla_\mu(U^\lambda\nabla_\lambda V^\nu)-U^\mu\nabla_\mu(V^\lambda\nabla_\lambda V^\nu)$$$$\implies A^\mu = V^\mu U^\lambda R^\nu_{\rho\mu\lambda}V^\rho This equation is called the geodesic deviation equation. In flat space,Riemann-Tensor is zero ⇒ A = 0 ⇒ parallel lines stay parallel only in flat space.