Is it because the lightcones are not tipped all the way over in which the time axis points downwards?
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Prior to the Tipler cylinder being created all the light cones point upwards. So if you attempted to follow time like path taking you backwards in time (that is backwards in the the time coordinate of the observer far from the cylinder) when you reached the time the cylinder was created you'd have to stop. At this point all light cones point upwards so your timelike path would move in the direction of increasing coordinate time again.
John Rennie
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