|MadSci Network: Physics|
Speeds greater than light imply imaginary energies, not imaginary time. (Although I like the concept of imaginary time, and I'll have to think about what it might mean! You do need to keep in mind that time is not a variable, but rather a coordinate. I think having a complex coordinate would be equivalent to adding a new dimension to the universe. New dimensions of time have been speculated upon, but it's not a popular idea.)
For many decades, the fact that a faster-than-light object would have imaginary energy was enough to "disprove" it in many people's minds. After all, even though imaginary numbers are necessary in quantum mechanics, any observable quantity must be real (there's a theorem that proves this in quantum mechanics). When imaginary numbers are used in engineering, it is simply a tool to make the math easier, and the real part of the result is always used at the end. Because energies are observable, they can't be complex.
But in 1967 a loophole was discovered. What if the mass of the particle was also imaginary? (Mass is not directly observable, so that's okay by quantum.) In that case, it turned out that the energy of the faster-than-light particle would now be real! In fact, this imaginary-mass particle couldn't go less than lightspeed -- it must always go faster than light. As you may have guessed, this type of particle was named a tachyon. Tachyons have never been detected, but no one has proven they can't exist. So maybe faster-than-light particles are possible after all...
Here is a speculative article on tachyons. If you're interested in why observable quantities can't be complex, I'm afraid I'll have to direct you to a quantum mechanics text.
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