Tensors and Relativity: Chapter 6

Geodesic deviation

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We have shown that in a curved space [ for example on the surface of a balloon ] parallel lines when extended do not remain parallel. We will now formulate this mathematically in terms of the Riemann tensor.

Consider two geodesics with tangents tex2html_wrap_inline3398 and tex2html_wrap_inline3813 that begin parallel and near each other at points A and A' [ see Figure 6.4 ].

  figure2007
Figure 6.4: Geodesic deviation

Let the affine parameter on the geodesics be tex2html_wrap_inline3612 . We define a connecting vector tex2html_wrap_inline3821 which reaches from one geodesic to another, connecting points at equal intervals in tex2html_wrap_inline3612 .

To simplify things, let's adopt a local inertial frame at A, in which the coordinate tex2html_wrap_inline3827 points along the geodesics. Thus at A we have tex2html_wrap_inline3831 .

The geodesic equation at A is

equation3147

since all the connection coefficients vanish at A. The connection does not vanish at A', so the equation of the geodesic at A' is

equation3149

where again at A' we have arranged the coordinates so that tex2html_wrap_inline3831 . But, since A and A' are separated by the connecting vector tex2html_wrap_inline3821 we have

equation3153

the right hand side being evaluated at A, so

equation3155

Now tex2html_wrap_inline3853 so

equation3157

This then gives us an expression telling us how the components of tex2html_wrap_inline3821 change. Let us now compute the full second covariant derivative along tex2html_wrap_inline3398 , tex2html_wrap_inline3859 :

eqnarray3169

In a local inertial frame this is

eqnarray3183

where everything is again evaluated at A. Using the result for tex2html_wrap_inline3863 we get

eqnarray3191

since we have chosen tex2html_wrap_inline3831 .

The final expression is frame invariant, so we have in any basis

equation3197

So geodesics in flat space maintain their separation; those in curved space don't. This is called the equation of geodesic deviation  and it shows mathematically that the tidal forces of a gravitational field can be represented by the curvature of spacetime in which particles follow geodesics.


next up previous index
Next: The Bianchi identities; Ricci Up: Title page Previous: Properties of the Riemann