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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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    According to SR faster than light travel is impossible, so trying to draw reasonable conclusions from this equation is meaningless. – BioPhysicist Oct 21 '18 at 20:25
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    Tachyons are allowed in SR, and they do also have a proper time – Yukterez Oct 21 '18 at 20:47
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    @СимонТыран I'll delete my comment if tachyons are shown to exist. – BioPhysicist Oct 21 '18 at 23:28
  • Imaginary time is spacelike (and vice versa), sort of. But see https://physics.stackexchange.com/questions/121380/special-relativity-and-imaginary-coefficient-of-the-time-coordinate and the various links on that page. – PM 2Ring Oct 22 '18 at 00:04
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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 '18 at 00:09
  • @PM2Ring I never said SR rules out tachyons. – BioPhysicist Oct 22 '18 at 00:14
  • @Aaron You said "According to SR faster than light travel is impossible". That sounds like ruling out tachyons to me. If that's not what you meant, please clarify. (FWIW, I'm pretty certain that tachyons are unphysical). – PM 2Ring Oct 22 '18 at 00:19
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    @PM2Ring I see what you mean. I meant SR that we have experientially verified. – BioPhysicist Oct 22 '18 at 00:25
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    Why in the world are there close votes on this question? It's a perfectly reasonable question. I've always heard it that Faster Than Light travel is supposedly backwards time travel. This is just silly, for the reasons you point out in the question. –  Oct 22 '18 at 01:36
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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 '18 at 02:16

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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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  • Tachyons are at rest in their rest frame and they do have a proper time. https://cyber.sci-hub.tw/MTAuMTEwMy9waHlzcmV2LjE4OC4yMjg3/parker1969.pdf – Yukterez Oct 22 '18 at 02:33
  • 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 '18 at 03:08
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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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  • You can travel to events in your forward lightcone. The interval between you and such events is timelike. – PM 2Ring Oct 23 '18 at 03:41
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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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  • 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 '18 at 04:23
  • "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 Hasn't Forgotten Oct 22 '18 at 04:46
  • 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 '18 at 06:36
  • @safesphere Two selves? Double the amount of person? – Wookie Jul 29 '23 at 19:24
  • @safesphere Perhaps I'm reading your comments wrongly. I mean that when I set my weaponry down I weigh about 100 kilograms (not counting hair). At the same time tomorrow I'll weigh about the same, give or take what I eat etc. How are we going to have 200 kilograms of me in the same place at the same time yesterday? – Wookie Feb 28 '24 at 13:53
  • @safesphere The electron-positron annihilation is the kind of time travel I can understand more easily, where people might de-age, but I'm still not really getting how one electron might go to yesterday where there'd then be two positrons having some kind of interaction. Likewise, there's 100kg of me typing this today and there'd have been 100kg of me typing something else yesterday, but my weight has been gradually accreted: how does it make sense that there could be 100kg + 100kg of me interacting with each other yesterday? Surely the person who goes back in time only meets a thin remnant. – Wookie Mar 14 '24 at 10:28
  • @safesphere Got it. We're good to go. See you yesterday then! – Wookie Mar 15 '24 at 10:01