[Physics] Does time dilation mean that faster than light travel is backwards time travel?

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Ok. So my question is, I've always heard it that Faster Than Light travel is supposedly backwards time travel.

However, the time dilation formula is
$$T=\frac{T_0}{\sqrt{1-v^2/c^2}}$$
And while it is true that speeds greater than $c$ turn the denominator negative, doesn't the whole thing get rendered a complex fraction, rather than negative or backwards time flow, due to the square root of a negative number being a complex one?

Wouldn't this then mean that faster than light travel does something weird, rather than backwards time travel? In other words, wouldn't what happens during faster than light travel be some sort travel in a complex plane and wouldn't that have radically different implications to backwards time travel, depending on the direction one took FTL?

Best Answer

When using formulas in physics it is important to keep in mind the assumptions that the formula is based on. In this case $T_0$ is the time on a clock in its rest frame. It is doubtful that tachyons exist, but if they do then they are not at rest in any inertial frame, so the time dilation formula simply does not apply.

However, the Lorentz transform does apply. So (in units where c=1) if we had a tachyon which moved at 2 c in our frame then it would have a worldline like $(t,x)=(\lambda,2\lambda)$ where $\lambda$ is an affine parameter and the y and z coordinates are suppressed. Now, if we do a Lorentz transform to a frame moving at 0.6 c relative to our frame then the worldline would be $(t’,x’)=(-0.25\lambda, 1.75\lambda)$.

Note that the worldline in the primed frame has the affine parameter increasing as time decreases whereas the affine parameter increases as time increases in our frame. In that sense it is traveling backwards in time in one frame or in the other.

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