No, there's no (experimentally supported) theory that would work at all with an object going faster than light. It just isn't in the geometry - asking for a hyperbolic rotation that stretches your motion's vector past c is like trying to turn an apple around in your hand in a way that makes it twice as big.

What they're doing here is based on the fact that you can make light move at under c if you force it to travel through a dielectric material. The c in E=mc^2 is the speed of light in a vacuum: not necessarily through a carefully designed atomic obstacle course.