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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?

Qmechanic
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Guthrie Douglas Prentice
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4 Answers4

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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.

Dale
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I don't know what you mean by "some sort travel in a complex plane". Faster than light travel is by definition some object that changes position from $x_0$ to $x_1$ in such a way that $\dfrac{x_1-x_0}{\Delta t}>c$, where $\Delta t$ is the elapsed time. There is no time travel involved when this happens, but causality will take a blow if events at $x_1$ depend on events at $x_0$.

Cuspy Code
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You should not think in terms of the dilation, but in terms of “distance” in Minkowski space (or its generalization in general relativity): the “distance” between two (different!) points here can be positive, zero and and negative. For light-like separation the distance is zero, for space-like separated events the distance is positive and for time-like separated events their distance is negative.

Space- and light-like separated points in space-time can be traveled to with speeds $v \leq c$. For time-like separated events, you need a time machine, which is forbidden because you cannot move faster than light according to the theory of special and general relativity.

Max Lein
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Backward time travel is widely held as impossible due to violation of causality. And as has been hinted above, what time scale might you use to do so?

As for faster than c, entanglement seems to imply such a thing. It is better answed by something like this.

JohnnyFever
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