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Timelike Geodesics

In order to simplify the computation we take the case of the radial geodesics, this means that and k=0 the flat universe. This gives the system:

In order to find the timelike vector we use the geodesic equations and the fact that for timelike vectors.

This gives us

Where c is the constant equal to (from the geodesic equation).

Now we compute the covariant derivative of the tangent timelike vector of the geodesics.

One can ask for what c we have . For say c>0 one must have

For a flat space . It is not possible for this to be true if . In other words for an universe in expansion .

If we now take c to be negative then there exist a case of convergence for a flat space with .



Peggy Varniere
Fri Jul 24 11:57:38 EDT 1998