Geodesic Deviation in 2D: Is There Directional Dependence?

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The discussion centers on whether the geodesic deviation equation in two dimensions is governed by a single scalar that is independent of the direction of the geodesics. Participants question if the equation d^2 ξ/ds^2 + Rξ = 0 implies no directional dependence in 2D. There is a suggestion to clarify if "2D" refers specifically to the coordinates t and x, and whether the metric can be explicitly worked out. The impact of geodesic velocities depending on time or space is also raised as a potential source of directional dependence. The conversation ultimately seeks to understand the implications of these factors on geodesic deviation in a two-dimensional context.
AcidRainLiTE
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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?
 
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AcidRainLiTE said:
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 ?
 

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