Geodesic Deviation in 2D: Is There Directional Dependence?

[AcidRainLiTE]

In 2 dimensions, is the geodesic deviation equation governed by a single scalar, independent of the direction of the geodesics? That is, if ξ is the separation of two nearby geodesics, do we have $$d^2 \xi/ds^2 + R\xi = 0$$ where R is a scalar that is completely independent of the direction of the geodesics?

If so, how can we see that there can be no directional dependence in 2d?

 

[Mentz114]

In 2 dimensions, is the geodesic deviation equation governed by a single scalar, independent of the direction of the geodesics? That is, if ξ is the separation of two nearby geodesics, do we have $$d^2 \xi/ds^2 + R\xi = 0$$ where R is a scalar that is completely independent of the direction of the geodesics?

If so, how can we see that there can be no directional dependence in 2d?

By 2D do you mean $t,x$ ? So if you have a metric $ds^2=-g_{00} dt^2 + g_{11}dx^2$, can you work it out explicitly ? If the velocities of the geodesics depend on $t$ or $x$, is that directional dependence ?