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If we imagine a tachyonic particle travelling faster than the speed of light, can there be any reference frame which perceives a tachyon as travelling slower than $c$ by applying Lorentz contraction? Also, can a tachyon perceive a tachyon to be travelling faster than $c$? (I imagine that tachyon also sees the speed of light as constant).

PS: it is not a same question as "do tachyons really move faster than speed of light". Rather I want to know the speed of tachyons from tachyon's and normal observers. perspective please reopen this.

Qmechanic
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Shaurya Kad
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2 Answers2

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Tachyon geodesics are space-like.

Lorentz transformations always preserve the quality of a spacetime interval. Therefore, in any inertial frame, time-like geodesics are always time-like, light-like (null) geodesics are always light-like, and space-like geodesics are always space-like.

So shifting from one inertial frame to another, a normal matter trajectory remains time-like, a light geodesic remains light-like, and a space-like tachyon trajectory remains space-like.

Also, can a tachyon perceive a tachyon to be travelling faster than $c$

In one sense, a tachyon "sees" other tachyons as tachyons. Just as it takes infinite energy to speed up normal matter to $c$ (relative to your normal matter timelike frame), it takes infinite energy to slow down a tachyon to $c$. When normal matter loses kinetic energy, its speed approaches zero. But when a tachyon loses kinetic energy its speed approaches infinity.

However, it doesn't really make sense to ask what a tachyon sees, since observers by definition follow time-like geodesics.

In a comment you said:

maybe it can't be defined just like lights reference frame can't be defined

That's basically the case. From Would time freeze if you could travel at the speed of light?

As a young student, Einstein tried to imagine what an electromagnetic wave would look like from the point of view of a motorcyclist riding alongside it. But we now know, thanks to Einstein himself, that it really doesn't make sense to talk about such observers.

PM 2Ring
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The following assumes tachyons of the variety that travel at greater than the speed of light actually exist (no one has ever detected one). Some sources define a tachyon as a particle with imaginary mass that does not travel greater than the speed of light.

If they exist, they have to obey the relativistic velocity addition rule (using units such that c=1): $$ u = \frac{v + u'}{1 + vu'} \ ,$$

just like any other particle: If an observer S' is moving with velocity v relative to observer S sees a particle moving with velocity u' then the observer S will see the particle moving with velocity u. The above formula can be re-arranged to obtain the velocity u' from the point of view of of Observer S':

$$ u' = \frac{u-v}{1 - vu} \ ,$$

It is possible to show that u' is always less than c when both abs(u)>c and abs(v)>c. This means one tachyon always 'sees' another tachyon as traveling at less than the speed of light relative to itself.

However, the tachyonic observer would have to have imaginary mass from the point of view of observer S, but interestingly enough one tachyon has 'real' mass from the point of view of another tachyon, suggesting normal particles and tachyons live in different 'realms' and do not interact with each other.

If observer S sees a photon moving at u=c relative to himself, then a hypothetical tachyonic observer moving a v = 60c relative to S, would 'see' the same photon moving at:

$$ u' = \frac{1-60}{1 - 1\times60} = \frac{-59}{-59} = c, $$

so even tachyonic observers observe photons to move at c.

Let's say S sees tachyonic observer S' moving a v=70c and another tachyonic particle particle moving at u = 70c in the same direction, then S' would see the the particle moving at:

$$ u' = \frac{70-70}{1 - 70\times70} = \frac{0}{-4899} = 0, $$

More generally, if we take the limit as u goes to v the limit always goes to zero for v = u, suggesting one tachyon can in fact be rest wrt another tachyon.

Tachyonic observer S' would 'see' normal observer S as moving at greater than c and from the tachyon's point of view, the normal observer has imaginary mass.

From a hypothetical tachyons point of view, other tachyons always travel at less than c and photons always travel at c and other tachyons have real mass. As far as tachyons are concerned, we are travelling at greater than c and have imaginary mass. In other words, tachyons would perceive everything the same way as we do and think we are the hypothetical tachyons.

All the above suggests the speed of light is a two way barrier that cannot be crossed in either direction. A normal particle cannot be accelerated to greater than the speed of light and tachyon cannot be slowed down to less than the speed of light relative to us.

P.S. it occurred to me that we have not considered the relativistic Doppler effect:

$$\frac{\lambda_r}{\lambda_s} = \frac{f_s}{f_r} = \sqrt{\frac{1+\beta}{1-\beta}}$$

Where $\beta$ is positive if the emitter is going away from the receiver and negative if the emitter is coming towards the receiver. If $\beta> 1$ or $\beta< -1$ as in the case for a tachyonic emitter, then the wavelength and frequency of the received photon is imaginary, which presumably means we cannot actually see tachyons.

KDP
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