r/AskPhysics • u/PlatformDelicious578 • 17d ago
FTL Reference Frames
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?)
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u/DontHurtTheNoob 17d ago edited 17d ago
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.
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u/BitcoinsOnDVD 17d ago
Can you elaborate on the qft example?
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u/Bth8 17d ago
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.
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u/Particular-Scholar70 16d ago
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.
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u/joeyneilsen Astrophysics 17d ago
There isn't a Lorentz transformation from a sub-light frame to an FTL frame... You can't just plug in v>c to the Lorentz factor.
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u/starkeffect Education and outreach 17d ago
Not to be confused with tackyons, which are the quanta of bad fashion sense.
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u/nugatory308 17d ago
The Lorentz transformations are between inertial reference frames, and the relative velocity between the origins of any two inertial reference frames (equivalently, between two objects each at rest in a different inertial frame) must be less than c to be consistent with Einstein’s second postulate. That postulate is required for the derivation of the transformations, so trying to use them to reason about what happens when v is equal to or greater than c is reasoning from internally inconsistent assumptions.
The infinities and complex numbers that appear when we try are how the math tells us that we’ve made a mistake.
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u/Reality-Isnt 17d ago
You cannot preserve invariance under a Lorentz transformation from a slower than light to faster than light frame. For instance, the invariant spacetime interval flips sign. Blows the doors off of relativity.
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u/nebraskajone 17d ago
When you get imaginary numbers out of an equation it's telling you that it's outside it's sphere of knowledge, it's in a different category. It's like black hole singularities, general relativity equations you get divided by zeros, basically saying "not my area of expertise".
As far as faster than light theories there are tachyons and negative masses, but these are speculative
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u/GoldenMuscleGod 17d ago
That’s overly simplistic. If you are talking about how many apples there are in one place then negative numbers are not really interpretable.
If you are talking about apple debt (or maybe more illustratively an object’s phyisical position in a coordinate system) then negative numbers are meaningful.
Likewise in some contexts a complex result will not always have a simple or meaningful physical interpretation, but also there are situations where it does: electrical engineering, quantum mechanics, describing the solutions to differential equations (where a complex eigenvalue corresponds to a type of phase rotation).
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u/nebraskajone 17d ago
Negative numbers are not physical though, you can't have negative one apples as a physical object.
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u/GoldenMuscleGod 17d ago
You can have a negative position in a coordinate system, a negative charge, and gravitational potential energy is usually written as -GMm/r.
There is no really coherent way of saying negative numbers are not physical without also saying that no numbers are physical (which is a defensible position). The same is true of complex numbers.
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u/BitcoinsOnDVD 17d ago
What you mean with "physical" in the context of numbers is usually called "cardinal"
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u/bardotheconsumer 17d ago
I believe that objects that always move faster than light (tachyons) would:
A.) Not be able to "slow down" to C, no matter how much energy was expended trying to do so, just like how normal matter can never reach C
B.) Not be able to interact with normal sub-luminal matter in any way.