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Something moving faster than light should have imaginary mass, like photons have zero mass and thus travel at $c$.

I have read this article of physicist E. C. George Sudarshan.

He said taking mass to be imaginary we get real energy and momentum. (I think I have understood that.) However, if something moves at a speed greater than $c$, its proper time $\tau=t\sqrt{1-v^2/c^2}$ becomes imaginary (where $t$ is coordinate time).

Does this imply causality violation? Does imaginary time mean time is going backward?

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    $\begingroup$ You should give a specific reference to the article you refer to. $\endgroup$ – Emilio Pisanty Oct 20 '14 at 9:13
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    $\begingroup$ Does imaginary time mean time is going backward? No, a sign flip would mean that it was going backward. The fact that the equation for time dilation misbehaves for $v\ge c$ is an indication that it is not trivial to extend the Lorentz transformation to velocities $>c$. It can in fact be done in 1+1 dimensions (although not just be plugging $v>c$ into the same equations). However, there is a no-go theorem that shows it can't be done in $3+1$ dimensions: V. Gorini, "Linear Kinematical Groups," Commun Math Phys 21 (1971) 150 $\endgroup$ – Ben Crowell Nov 16 '14 at 19:12
  • $\begingroup$ Here the link quest.ph.utexas.edu/sudarshan_tachyons.html $\endgroup$ – Paul Nov 17 '14 at 2:35
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    $\begingroup$ @WillO - If you assume the relativistic momentum formula $\vec{p} = m\vec{v}/\sqrt{1 - v^2/c^2}$ still applies, then if a tachyon has an FTL velocity in the +x direction of our coordinate system's x-axis, it would have a real momentum in the +x direction if its mass m were $i$, but it would have a real momentum in the -x direction if its mass m were $-i$, which would presumably lead to different predictions about things like collisions/scattering. $\endgroup$ – Hypnosifl Dec 31 '14 at 15:32
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    $\begingroup$ @Hypnosifl : You are mistaken, because there is no canonical choice for $\sqrt{1-v^2/c^2}$. If you choose one square root then a mass of $i$ corresponds to motion in the $+x$ direction, but if you choose the other then a mass of $i$ corresponds to motion in a $-x$ direction. $\endgroup$ – WillO Dec 31 '14 at 17:25
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The conclusion derives from looking at how tachyon signals would behave as seen in slower-than-light inertial frames, not from trying to consider a tachyon's own "time" (if you can call it that, since a tachyon's worldline would have to be space-like, not time-like)--basically, it's a consequence of the relativity of simultaneity. It can be shown that any signal that moves even slightly faster than light in one frame would move instantaneously in some other inertial frame, and if the first postulate of SR applies to tachyons, then if it's possible to send a message instantaneously in one frame, this must be possible in all inertial frames. Likewise, if a signal travels instantaneously in one frame, there must be some other inertial frame where it actually travels backwards in time (i.e. the event of the signal being received occurs before the event of it being sent), and if that's possible in any frame, it must be possible in all frames too.

If you're familiar with the basics of SR spacetime diagrams and how a surface of simultaneity in one frame looks tilted in other frames, then you could take a look at the helpful explanation with diagrams on this page, which shows how the ability of two different slower-than-light observers to send signals that move instantaneously as seen in their own frames implies that they can bounce a message back and forth, and in each one's frame the other one's signal is going backwards in time, so that the message gets returned to the original sender at an earlier point on his worldline than the point where he sent the message, a clear causality violation in all frames. You could also take a look at the tachyonic antitelephone article which goes into more detail, with equations and a numerical example.

Note that this answer is really about why the ability to send FTL signals would violate causality--as other answers have noted, in quantum field theories there could be tachyons that would be impossible to use for transmitting information, in which case no causality violation would occur (the last section of this article has a helpful discussion about why tachyons in QFT wouldn't be usable for information transmission).

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In addition to the Hypnosifl's answer: in lots of modern theories tachyons do not violate causality.

For example, in QFT you can choose the mass-squared term in the action to have a negative sign, that would make your field's quanta tachyons. You can check it by looking at the propagator, which oscillates at space-like region and decreases exponentially inside the light cone.

In bosonic string theory, there are also two tachyons (the zero mode of the string spectrum, one for opened strings and one for closed).

These tachyons do not violate causality, because there is no way of using them for sending information with superluminal speed. However, they make the vacuum unstable (since there is no lowest-energy state anymore). It is the actual reason of theories with tachyons being considered unphysical.

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In the context of bosonic string theory, the ground state with no oscillator excited, has a mass,

$$M^2 = -\frac{1}{\alpha'}\frac{D-2}{6}$$

where $\alpha'$ is the Regge slope, satisfying $\alpha' = 1/2\pi T$, where $T$ is the tension of the spring, and $D$ are the spacetime dimensions. It seems it has an imaginary mass (providing $D\geq 3$). You may have also heard this particle is unnatural because it propagates faster than $c$.


Let's go back to quantum field theory for a moment. Generally, the mass squared is simply the term that appears in the quadratic part of the Lagrangian, i.e.

$$M^2 = \frac{\partial^2 V(\phi)}{\partial \phi^2} \biggr\rvert_{\phi = 0}$$

Hence, if $M^2 < 0$, we can interpret that as the fact that we are expanding around a maximum of the potential for a tachyon field (see second derivative test). With this perspective, the Higgs field can also be viewed as a tachyon. As D. Tong states, it is unfortunate that bosonic string theory sits at this unstable point in the potential of the tachyon field. To date, we still don't know of a minimum of $V(\phi)$. (One can compute higher order corrections, and find a minimum, but then the next correction destabilizes the minimum again.) So it seems the issue in string theory is not causality, rather they run much deeper.

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