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What would theoretically happen to an observed object's length (let it be a 5 meter line) moving at some speed greater than that of light in a straight path?

Qmechanic
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user2761204
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  • Have a look at http://physics.stackexchange.com/q/80511/. Feed in a velocity greater than $c$ and see what happens. – John Rennie Oct 13 '13 at 15:00

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A basic logical requirement for the Lorentz transformation is that it needs to be a one-to-one function. Otherwise, for example, different observers could disagree about whether two particles had collided. The Lorentz transformation is one-to-one for $v<c$, but at $v=c$ it isn't anymore. To make sense out of Lorentz transformations with $v>c$, one would therefore have to come up with a real-valued Lorentz transformation for $v>c$ and "glue" it onto the $v<c$ version. Recami (1986) shows how this can be done in 1+1 dimensions, but Gorini (1971) proves that it's impossible in 3+1 dimensions. (Basically you can only do this in $n+n$ dimensions, because FTL observers would interpret themselves as being in the STL sector, which requires a symmetry between spacelike and timelike vectors.)

In practical terms, if you tried to accelerate a physical object such as a meter-stick to $v>c$, you wouldn't be able to do it, since no continuous process of acceleration can provide a boost from a frame $\text{K}$ to a frame $\text{K}'$ in motion relative to $\text{K}$ at $v>c$. (This is a kinematic fact, and it holds regardless of dynamical considerations such as relativistic inertia.)

References

V. Gorini, "Linear Kinematical Groups," Commun. Math. Phys. 21 no. 2 (1971), pp. 150-163. Open access via project Euclid.

E. Recami, "Special Relativity Extended to (antimatter and) Superluminal motions: A Review" (also "Classical tachyons and possible applications"), Riv. Nuovo Cimento 9 no. 6 (1986) pp. 1-178. Available from Recami's web page.

Emilio Pisanty
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mathematically when a particle moves faster than the speed of light then lorentz factor will be an imaginary number that can be written as a real number multiplied imaginary unit, $$\gamma=-i\Omega$$ where the $\Omega$ denotes Estakhr Omega Factor (check the link above for more information)

and if faster than the speed of Light becomes technically possible then it will be possible to build FTL (Faster Than Light) Particle accelerator and hadron collider. FTL-LHC

FTL
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