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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?

marked as duplicate by Aaron Stevens, garyp, John Rennie special-relativity Oct 22 at 6:51

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    According to SR faster than light travel is impossible, so trying to draw reasonable conclusions from this equation is meaningless. – Aaron Stevens Oct 21 at 20:25
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    Tachyons are allowed in SR, and they do also have a proper time – Симон Тыран Oct 21 at 20:47
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    @СимонТыран I'll delete my comment if tachyons are shown to exist. – Aaron Stevens Oct 21 at 23:28
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    @AaronStevens SR doesn't rule out tachyons, but it does rule out boosts that traverse the lightspeed barrier, so non-tachyons cannot be smoothly accelerated to tachyons, or vice versa. Note that Feinberg, who coined the term, was never a big believer in tachyons, and in his later life was almost certain that they're impossible. – PM 2Ring Oct 22 at 0:09
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    Taking a step back: we are using equations to describe the world, not to prescribe it. Physical equations aren't cascades of symbols where one plugs in something and gets a truth about nature in return, their domain of validity has to be established by experiment. Even though there are cases where complex analysis reveals truths about the world also in the non-quantum world (dispersion relations come to mind), one cannot just sprinkle a few i's here and there until the equations match and get truths in return. – tobi_s Oct 22 at 2:16

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.

  • Tachyons are at rest in their rest frame and they do have a proper time. cyber.sci-hub.tw/MTAuMTEwMy9waHlzcmV2LjE4OC4yMjg3/… – Симон Тыран Oct 22 at 2:33
  • Only in the case of one spatial dimension. We have three. The paper doesn’t apply to this universe, it is just a mathematical exercise. Also, the y and z coordinates were just suppressed above, they are still there. – Dale Oct 22 at 3:08

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 $x_0$.

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.

  • You can travel to events in your forward lightcone. The interval between you and such events is timelike. – PM 2Ring Oct 23 at 3:41

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.

  • Yup. Or the fact that you need an infinite amount of energy for a massive object to even reach the speed of light, tells you that this is not possible. – Max Lein Oct 22 at 4:23
  • "It is better answed by something like this" is - even ignoring the typo - somewhat meaningless if I don't (or can't) open the link. It's SE preference that you explain briefly what the article you've linked says. That way, if the link changes/breaks, your answer still remains meaningful. If you edit your answer to add a brief summary, I'll cheerfully upvote you :-) – Chappo Oct 22 at 4:46
  • Backward time travel does not necessarily violate causality. Positrons are electrons traveling back in time (or vice versa) without any causality issues. A time loop simply prohibits free will. If you travel to yesterday to talk to the younger you and say exactly what you remember hearing yesterday from the older you, then the causality is preserved. – safesphere Oct 22 at 6:36

In fact, faster than light travel is theoretically possible, and one argument for that goes like this:

The important thing is for T to be a real number, and here we have three cases for that to happen:

  1. Both T0 and sqr(1-v^2/c^2) are real numbers, in this case we must have v

  2. Both T0 and sqr(1-v^2/c^2) are pure imaginary and in this case v>c. Don't be discouraged by the fact that T0 is imaginary because these particles cannot be at rest, in fact they always move faster than the speed of light. Such hypothetical particles are called tachyons.

  3. Both T0 and sqr(1-v^2/c^2) are zero. In this case v=c. And such particles are bound to always move in the speed of light. Such particles are called massless particles.

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    You didn't address the issue of traveling backwards in time. – D. Halsey Oct 22 at 0:11
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    I don't see how defining some particle to move faster than the speed of light means it is theoretically possible. Unless you have a very very lose definition of theoretically possible of "if I make a theory about it, then it must be theoretically possible." Usually theoretically possible means that we could achieve it, but there is some obstacle we currently can't overcome. Tachyons haven't been shown to exist, so there really isn't any argument to be had that it could be possible to travel faster than light. – Aaron Stevens Oct 22 at 0:11
  • @AaronStevens it seems that we have different understandings of "theoretically possible". I use it to mean that it is not ruled out by the current laws of physics. Tachyons are of this type, although their existence will create real problems for causality. – Arthur Oct 22 at 0:20
  • @D.Halsey if tachyons exist, then they travel with space-like intervals in spacetime, which makes them able to travel both forward and backwards in time. – Arthur Oct 22 at 0:24

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