  We present two modifications to the rotating Alcubierre metric [1], which was
shown to permit closed timelike curves (CTCs). We find that if the rotation
rate of the spacetime is made spatially dependent, in certain cases there exist
simple approximate timelike geodesics that are also CTCs, provided that the
velocity of the warp bubble varies slowly. The second modification is
essentially the original Alcubierre metric [ 2 ] with a periodic boundary,
resulting in a cylindrical spacetime which can also be related to the rotating
Alcubierre metric. Furthermore, this spacetime contains simple exact timelike
geodesics that are also CTCs. In both modifications, the CTC geodesics that we
have found allow for a simple model of a particle interacting with a CTC for a
finite proper time interval, entering and exiting in chronology-respecting
space. Given the simplicity of both of these exotic spacetimes, despite their
questionable physical realisability, we suggest that they may be useful in
studies of quantum fields near CTCs.
