Some of the popular ideas utilize frame dragging, hypothesized to occur in the vicinity of Kerr black holes. The idea is that you can accelerate close to light speed, then cross into the ergosphere, and with the frame dragging effect your velocity relative to your starting point is predicted to be higher than light speed.
Another idea using high energy lasers is perhaps more ripe for experimentation. Apparently light exhibits the frame dragging effect more potently than was at first believed. The idea here is that a tunnel of laser beams will (rather like a coil/solenoid) amplify the effect enough to cause a photon passing through the tunnel (at light speed) to be dragged into a greater (FTL) velocity.
These ideas are primarily time machines.
You travel (a little) faster than light into the past - and, unless you use truly impractical (and I mean IMPRACTICAL) amounts of energy (or find a huge rotating black hole around which you..rotate, going nowhere in space), your travel through space is only a little better than near-light-speed travel.
Fortunately, john titor doesn't seem to realise this

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