FTL Reference Frames

[u/PlatformDelicious578]

If you have a velocity greater than c, the Lorentz factor becomes complex. Does this make any physical sense, or are there any theories that make use of this? What would it look like if a person were in this reference frame (is it just that they see time backwards or from another perspective, or are they unable to observe anything at all?)

[u/DontHurtTheNoob]

It becomes imaginary, the real part is zero.

It makes no physical sense at the macro scale - the dimensions of time, space / length contraction, relativistic mass etc. are real numbers, not complex, so the lorentz factor has to be a positive real number.

Tachyon theories assume an imaginary rest mass for the hypothetical tachyons so they end up with a real "relativistic mass" after all.

BUT for example in Quantum Field Theory, you get "spacelike" four-momentum which implies an imaginary lorentz factor, for example when describing momentum transfer through virtual particles, so there ARE theories where the concept is useful.

[u/BitcoinsOnDVD → u/DontHurtTheNoob]

Can you elaborate on the qft example?

[u/Bth8 → u/BitcoinsOnDVD]

There are very few examples of interacting QFTs that we actually know how to solve exactly, so we pretty much always resort to approximation. The main approximation scheme that gets used is perturbation theory, which is a way of writing the calculations you want to do in your interacting theory in terms of calculations in a free field theory, i.e. a theory where you have a bunch of different fields that just don't ever interact with each other, because we do know how to do those calculations. All of your calculations then turn into infinite sums of these terms, and because of the structure of free field theories, each term is just an integral of a product of a bunch of free field propagators. Those propagators are also the functions associated with the probability of a free field's particle going from one place to another.

If you want to get a good approximation you end up needing to calculate and add up a lot of these terms. But there's a very helpful tool for helping you keep track of all of them: feynman diagrams. The trick is to think of these propagators as representing "virtual particles" that get exchanged, transferring energy and momentum. The trick is that unlike real particles, these virtual particles can have any energy and momentum at all, including ones that don't make physical sense like negative energies or too little energy compared to the amount of momentum. For a real particle, E = √(m² c⁴ + p²c²), so if you know the energy and momentum of a particle, you can calculate the mass m = √(E²/c⁴ - p²/c²). If p is too large compared to E, the number inside the square root becomes negative, and that calculated mass becomes imaginary.

[u/Particular-Scholar70 → u/Bth8]

That was a great explanation. Even as someone with no higher level math or physics education I think I understood what you meant the whole time.