Geodesic deviation and relative acceleration between the geodesics

Question

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?

Answer 1

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.

Answer 2

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.