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Even if I have the details wrong, I have two questions:

1) Can you define a metric where geodesics are asymptotic?

2) Can you define the same metric with different coordinates to remove this behaviour?

The only thing I can think of is some kind of substitution, but I don't really know what to do and the textbook I am working through is not a lot of help. I even got the massive "Gravitation" book out of my library but if it did contain the solution, I didn't understand it!

Hopefully someone can give me a clue :)