Runaway solutions of geodesic equations Given a metric tensor $g_{\mu\nu}$ it is possible to calculate the geodesic equations from:
$$\dfrac{d^2x^{\mu}}{ds^2}=\Gamma^\mu_{\nu \eta}\dfrac{dx^\nu}{ds}\dfrac{dx^\eta}{ds}$$
where the $\Gamma^\mu_{\nu \eta}$ are the Christoffel symobols. How is it possible to know if there are runaway solutions in the geodesic equations? If the Ricci scalar is zero, this mean there are not this kind solutions?
 A: A geodesic tells us the motion of a test mass, i.e., a mass that is very small. Runaway solutions are a phenomenon that occurs when you try to describe a pointlike particle whose mass is big enough to make radiation reaction nonnegligible. Radiation reaction effects go like the square of the mass, which is negligible for a test mass.
This whole framework doesn't even make sense conceptually when you try to apply it to the geodesic equation. We don't have any criterion that would define whether a geodesic was or was not a runaway solution. It's not like the situation in electromagnetism, where we can define inertial motion as the motion of an uncharged particle, and runaway solutions for charged particles are those that deviate exponentially from inertial motion.
A: Maybe what you are asking about is whether geodesics emanating from near-by points in parallel directions deviate from each other, moving farther and farther apart the more they extend. A typical example of this phenomenon are the geodesics of hyperbolic space, while the geodesics of a spherical or Euclidean space do not have this property. I believe the time-like geodesics of anti-deSitter space are the general relativistic analogue of the hyperbolic space, as it has space-like slices which are in fact embedded copies of hyperbolic space. In contrast, deSitter space is the analogue of the sphere (again, space-like slices that are embdeded three-spheres) and Minkowski space is the Euclidean analogue.
Then, I think the basic stuff is to look at the extended system of a geodesic together with a Jacobi field: $\big(\, x(\tau), \,\, J(\tau)\,\big)$. Here $\tau$ is the proper time, in the case of time-like geodesics in general relativity, or it is arc-length parametrization in the case of a Riemannian manifold.
\begin{align}
&\nabla_{\frac{dx}{d\tau}}\, \frac{dx}{d\tau} = 0 \\
& \nabla_{\frac{dx}{d\tau}}\left(\nabla_{\frac{dx}{d\tau}}\, J \right) + \text{R}\left(\,J,\, \,\frac{dx}{d\tau} \,\right)\,  \frac{dx}{d\tau} = 0
\end{align}
Roughly speaking, Jacobi fields give you the first derivative of the distance between two near-by geodesics. I am not sure, but I seriously doubt that the scalar curvature of a manifold can provide sufficient criterion about the tendency of geodesics to  diverge from each other. Some classical results in the field tell you that if the manifold has negative sectional curvature, bounded away from zero, near-by geodesics tend to separate, running away from each other. So it is the sectional curvature that encodes this information better than the scalar curvature. And in general relativity it is possibly the sectional curvature of the space-like slices that has to be negative.
