# Geodesic deviation and relative acceleration between the geodesics

Consider a one parameter family of curves with curve parameter $$\lambda$$. We denote the deviation vector $$W^\mu=\frac{\partial x^\mu}{\partial \epsilon}$$ where $$\epsilon$$ can be thought of as the transverse coordinate. In this scenario the relative acceleration of the geodesics $$A^\mu=T^\omega\nabla_\omega T^\nu \nabla_\nu W^\mu$$ which results $$A^\mu=R^\mu_{\delta \omega \nu} T^\omega T^\nu W^\delta$$ Where $$T^\mu$$ are the tangent vectors of the curves. So, the relative acceleration is determined by the curvature tensor or if there has a gravitational field, the relative acceleration will not be zero. But can we prove from here the effect of the gravitational field on the geodesics are attractive?

The expression is general, so it is not possible to prove that the field is attractive. A counter-example is to postulate a repulsive Newton gravitation: $$\frac{d^2\mathbf X}{dt^2} = \nabla \phi$$ where $$\phi = -\frac{GM}{r}$$ Following the steps of this answer, the geodesic equation is:$$\frac{d^2 X^j}{d\tau^2} - \frac{\partial \phi}{\partial x^j} \left(\frac{dt}{d\tau}\right)\left(\frac{dt}{d\tau}\right) = 0$$

Where $$\Gamma^j_{00} = -\frac{\partial \phi}{\partial x^j}$$ We have well defined connections, a geodesic equation, the curvature tensor doesn't vanish, all from a repulsive gravitation.

The eqs. you wrote are called geodesics deviation eqs.

Any differential geometry book derives them.

This is purely a mathematical result. You can plug in the Riemann tensor for a sphere, a hyperboloid or any another 2D surface and visualize how the geodesics diverge/converge and compare with your intuition/3D graph.

But...if you plug in the Riemann tensor for any physically realistic solution of the Einstein field equations (EFE) - using physically realistic stress-energy tensor $$T_{ab}$$ - you will always get converging geodesics.

Because this is the key point in general relativity: the stress-energy tensor determines the curvature.

We make different possible assumptions regarding $$T_{ab}$$ called Energy conditions. The most realistic/experimentally-tested ones, show that geodesics tend to converge toward massive bodies. However this is better studied using the Raychaudhuri equation.