I think there needs to be a little bit of clarification here. In GR, the concept of closed time-like curves is not investigated so people can built a Time Cop machine or for Einstein to go back and kill Hitler, ala 'Red Alert', but because CTC can play a critical role in things like the big bang, the overall properties of the universe and the centres of black holes. For instance, 'Can the universe create itself'. Through the use of CTC, you can construct consistent, self generating universes. Similar things happen when you start talking about paths through the middle of rotating black holes, due to weird properties of the metric near the centre of a ring singularity. Considerable amounts of space are devoted to CTC research and discussions in books by Wald, Gibbons and particularly Hawking and Penrose. They are the 'extremal' things in GR, the ones with all the weird and wonderful properties, which (as usual) are also mind bendingly complicated to analyse. So I don't think it's correct to call them 'fringe science', because they play a very deep role in relativity. It's just that the depth at which they appear is far beyond what 99.999% of people who even do physics degrees will ever touch upon. You don't need to know about CTC to build the GPS network, but if you're addressing things like 'What would a black hole in CERN appear like?' then CTCs are something you need to really be familiar with.