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 and that begin parallel and near each other at points A and A' [ see Figure 6.4 ].
Figure 6.4: Geodesic deviation
Let the affine parameter on the geodesics be . We define a connecting vector which reaches from one geodesic to another, connecting points at equal intervals in .
To simplify things, let's adopt a local inertial frame at A, in which the coordinate points along the geodesics. Thus at A we have .
The geodesic equation at A is
since all the connection coefficients vanish at A. The connection does not vanish at A', so the equation of the geodesic at A' is
where again at A' we have arranged the coordinates so that . But, since A and A' are separated by the connecting vector we have
the right hand side being evaluated at A, so
This then gives us an expression telling us how the components of change. Let us now compute the full second covariant derivative along , :
In a local inertial frame this is
where everything is again evaluated at A. Using the result for we get
since we have chosen .
The final expression is frame invariant, so we have in any basis
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.