  The existence of time machines, understood as spacetime constructions
exhibiting physically realised closed timelike curves (CTCs), would raise
fundamental problems with causality and challenge our current understanding of
classical and quantum theories of gravity. In this paper, we investigate three
proposals for time machines which share some common features: cosmic strings in
relative motion, where the conical spacetime appears to allow CTCs; colliding
gravitational shock waves, which in Aichelburg-Sexl coordinates imply
discontinuous geodesics; and the superluminal propagation of light in
gravitational radiation metrics in a modified electrodynamics featuring
violations of the strong equivalence principle. While we show that ultimately
none of these constructions creates a working time machine, their study
illustrates the subtle levels at which causal self-consistency imposes itself,
and we consider what intuition can be drawn from these examples for future
theories.
