Can a Tachyon Observe Light Traveling Faster Than Itself?

[dynkindiagram]

So, I was thinking about tachyons, and I started wondering about what it would be like to be one. How would the universe look from a superluminal vantage point?

Tachyons (should they exist) travel faster than the speed of light. But according to the principle of relativity, a tachyon in a superluminal inertial frame must observe light traveling at speed c. However, if the tachyon is traveling away from a light source at a speed > c, then surely the light can never reach it? What about if it was traveling towards a light source at > c - how would the received light appear to the tachyon? What kind of time dilation/length contraction effects would be observable in the tachyon's frame?

Also, I've heard that tachyons travel backwards in time. Or rather, that there is always some subluminal observer that observes the tachyon traveling backwards in time. Does the reverse hold true: In the tachyon's frame of reference, does the observer appear to be moving backwards in time? Would a tachyon on a fly-by past the Earth 'see' (hypothetically speaking) us going about our lives in reverse?

 

[bcrowell]

Welcome to PF!

This is a cool question.

John Baez has a nice article on tachyons here: http://www.lightandmatter.com/cgi-bin/meki?physics/relativity#tachyons
Baez gives references to three older papers, which may also be helpful.

In your question, you talk about a tachyonic observer observing a beam of light. However, there are fundamental reasons (described in the Baez article) why tachyons can't be charged. (And one of the conclusions theorists seemed to agree on during the CERN neutrino fiasco was that tachyons also can't participate in the weak interaction.) So I don't think a tachyonic observer would be able to detect electromagnetic fields at all, except perhaps indirectly.

Baez also explains why the tachyonic telephone doesn't work. If a tachyonic physicist could observe our universe's non-FTL particles at event A, then travel to event B and "output" his memory of A back to us, then it would constitute a tachyonic telephone. Since this is impossible, it seems that either the tachyonic physicist cannot communicate with the non-FTL part of the universe, or else a tachyonic system of particles can't propagate at FTL velocities while maintaining any memory of its previous state. Since memory is required for consciousness, the latter possibility would imply that you can't make an FTL observer out of tachyons.

If tachyonic observers are possible, it's not obvious to me that such an observer would have to have a psychological or thermodynamic arrow of time that matched ours. Normally the second law of thermodynamics prevents two different systems from having opposite thermodynamic arrows of time. But this assumes that they interact. Tachyons can't interact with non-FTL particles through the electroweak interaction, so it's not clear to me that they would pick up an arrow of time from the low-entropy state of the early universe in the same way that non-FTL systems do.

If we ignore all these issues and try to use the Lorentz transformation to connect an FTL frame to a non-FTL frame, we get results that are imaginary numbers. We avoid letting tachyons have imaginary energy-momentum through assigning them an imaginary mass, but that doesn't help with the frame of reference. This suggests to me that it doesn't make sense to talk about their frame of reference, and such a result would seem pretty natural. It would be very similar to the idea that we can't have a frame of reference moving at exactly c, because the Lorentz transformation wouldn't be one-to-one, but there is a logical requirement that observers in different frames agree on whether or not two events are the same. (Either the arrow hit the target or it didn't. Either the pool balls collided or they didn't.)

So, attempting to answer some of your questions:

However, if the tachyon is traveling away from a light source at a speed > c, then surely the light can never reach it?

You can draw a spacetime diagram in which a lightlike world-line intersects an FTL one. On the diagram, there is a region where we would describe the two world-lines as closing, and a region where we would describe them as separating. If we assume that FTL observers can exist, that they can somehow (indirectly) detect light rays, and that FTL observers agree with non-FTL ones on whether events are distinct, then the FTL observer will agree with us that there was an intersection between the two world-lines. The FTL observer, like us, will split the diagram into closing parts and receding parts, but I don't see any way to tell whether those labels would agree with ours or be flipped.

What about if it was traveling towards a light source at > c - how would the received light appear to the tachyon?

I suppose the FTL observer would have to describe the beam of light's (x,t) coordinates as imaginary numbers. That starts not making sense before we even get to the point of worrying about whether the beam's velocity appears to be c.

What kind of time dilation/length contraction effects would be observable in the tachyon's frame?

Formally, they'd be imaginary numbers. This suggests to me that FTL frames don't make sense.

 

[John232]

I am not convinced that tachyons do not exist. There was one experiment a long time ago where they created a water dome to try and detect them. The goal was to find an interaction between the electrons in the water being hit by them. But, knowing if they could be detected in this manner for sure was still unkown. And there is no way of knowing that tachyons where being generated near the location, if there should have even been any there at all even if they can exist. I think if the experiment was run again for a longer period of time that they could find them.

It could also be possible that they escape into a higher dimension, since they end up taking this imaginary route or even a parallel universe. Or, it could mean that describing them may have to take a totally different approach to relativity. It could prove more difficult to set up because the distance the tachyon would travel would be greater than the distance the photon had traveled, then their is the question of does it even travel forward in time and distance at all? It could take tachyons being generated by some time in the future to even detect them. Current technology doesn't allow for particles to be sent faster than light, or some would say any future technology. But if some future technology did exist that could generate tachyons, it would most likely make them a lot easier to pick up.

 

[Ilmrak]

I think the answer is simply that, since it doesn't exists a rest frame for a particle with $p^\mu p_\mu ≤ 0$, it doesn't make sense to ask what we would see in such a frame

Ilm

 

[bcrowell]

I think the answer is simply that, since it doesn't exists a rest frame for a particle with $p^\mu p_\mu ≤ 0$, it doesn't make sense to ask what we would see in such a frame

Ilm

What is your reasoning for believing that there is no such frame? Is it different than what I gave in #2?

 

[Ilmrak]

What is your reasoning for believing that there is no such frame? Is it different than what I gave in #2?

Well, the rest frame is defined as the one in which the particle has no spatial momentum, so for a space-like four-momentum there is not such a frame. No Lorentz transformation can take a space-like vector to a time-like vector (proof is quite trivial).

This obviously says nothing on the possibility of the existence of a tachyon.

Correct me if I'm wrong

edit: I don’t think using imaginary coordinates would make any difference here, even if they would make sense.

Ilm

 

[bcrowell]

Well, the rest frame is defined as the one in which the particle has no spatial momentum, so for a space-like four-momentum there is not such a frame. No Lorentz transformation can take a space-like vector to a time-like vector (proof is quite trivial).

I think it's clear that you can't use a Lorentz transformation to get from an FTL frame to a non-FTL frame. One way of seeing this is that you get imaginary coordinates. Another is your argument.

I don't think it necessarily implies directly that FTL frames don't exist, only that FTL and non-FTL frames could never be connected by the Lorentz transformation. However, this only leaves a couple of unsatisfactory options: (1) Maybe the FTL part of the universe doesn't interact with the non-FTL part, but then the FTL part would be undetectable, and its existence would be religion or philosophy, not science. (2) Maybe the two parts do interact, but Lorentz invariance is broken in some unspecified manner. This is pretty unsatisfactory, since the motivation for tachyons is to have FTL without violating Lorentz invariance.

 

[Ilmrak]

I don't think it necessarily implies directly that FTL frames don't exist, only that FTL and non-FTL frames could never be connected by the Lorentz transformation. However, this only leaves a couple of unsatisfactory options: (1) Maybe the FTL part of the universe doesn't interact with the non-FTL part, but then the FTL part would be undetectable, and its existence would be religion or philosophy, not science. (2) Maybe the two parts do interact, but Lorentz invariance is broken in some unspecified manner. This is pretty unsatisfactory, since the motivation for tachyons is to have FTL without violating Lorentz invariance.

I'm failing in see why, if a tachyon would exist and would interact with non-tachyonic matter, option (2) would occur.

Could you please write the basic ideas for this?

Ilm

 

[Hurkyl]

Geometrically, a tachyon in a universe with 3 spatial dimensions and 1 temporal dimension is the same thing as a tardyon in a universe with 1 spatial dimension and 3 temporal dimensions.

I have no idea how physics would look with 1 spatial dimension and 3 temporal dimensions.

 

[bcrowell]

I'm failing in see why, if a tachyon would exist and would interact with non-tachyonic matter, option (2) would occur.

Could you please write the basic ideas for this?

Suppose you're willing to propose some transformation that's not a Lorentz transformation that gets you back and forth between FTL frames and non-FTL frames. You can probably do this just fine, but it will violate Lorentz invariance, and without the constraint of Lorentz invariance it's probably non-unique.

 

[Ilmrak]

Suppose you're willing to propose some transformation that's not a Lorentz transformation that gets you back and forth between FTL frames and non-FTL frames. You can probably do this just fine, but it will violate Lorentz invariance, and without the constraint of Lorentz invariance it's probably non-unique.

I agree.
What I don't understand is why, if tachyons would interact with non-tachyonic particles, would we need to introduce such a transformation as a symmetry of nature?

Ilm

 

[bcrowell]

I agree.
What I don't understand is why, if tachyons would interact with non-tachyonic particles, would we need to introduce such a transformation as a symmetry of nature?

I don't think you need it just because they interact. You need it if they interact and you also want to define FTL frames of reference in which FTL observers can observe non-FTL particles. And I don't think this would mean introducing a symmetry, it would mean breaking a symmetry. It would break Lorentz invariance.

Geometrically, a tachyon in a universe with 3 spatial dimensions and 1 temporal dimension is the same thing as a tardyon in a universe with 1 spatial dimension and 3 temporal dimensions.

Maybe I'm wrong, but this doesn't seem right to me. The metric has to be a symmetric matrix, and any transformation that carries out a change of frame has to be invertible. So Sylvester's law of inertia holds, and the signature of the metric is the same in both frames. All of this applies regardless of whether the transformation is a Lorentz transformation or something else. (Also, I don't think there is any meaningful way of distinguishing between 3+1 dimensions and 1+3. When we talk about a timelike dimension, we just mean the one whose contribution to the signature is the opposite of the others.)

 

[Ilmrak]

I don't think you need it just because they interact. You need it if they interact and you also want to define FTL frames of reference in which FTL observers can observe non-FTL particles. And I don't think this would mean introducing a symmetry, it would mean breaking a symmetry. It would break Lorentz invariance.

Why should anyone want to replace Lorentz invariance with such a transformation?
Photons exist, interact with ordinary matter, but they have light-like four-momentum so no rest frame can be defined for them (preserving Lorentz invariance).
Still, no one is going to define a coordinate transformation such that a rest frame exist for photon to use instead of Lorentz transformations.
We can nonetheless observe photons ^^

(I think I misunderstood your point of view, maybe you simply tried to assume the OP question makes sense and then try to see what that would imply)

[...]Also, I don't think there is any meaningful way of distinguishing between 3+1 dimensions and 1+3. When we talk about a timelike dimension, we just mean the one whose contribution to the signature is the opposite of the others.

I think this is exactly what he did mean (or at least how I interpreted his words)

Ilm

 

[bcrowell]

Why should anyone want to replace Lorentz invariance with such a transformation?

The only reason would be if you wanted to define an FTL frame.

Photons exist, interact with ordinary matter, but they have light-like four-momentum so no rest frame can be defined for them (preserving Lorentz invariance).
Still, no one is going to define a coordinate transformation such that a rest frame exist for photon to use instead of Lorentz transformations.
We can nonetheless observe photons ^^

The analogous question isn't whether we can observe photons, it's whether observers can exist who have a frame of reference moving at c.

The analogy breaks down somewhat because the mathematical misbehavior is qualitatively different in the two cases. When you put v=c in a Lorentz transformation, it's not one-to-one. When you put v>c in a Lorentz transformation, it gives imaginary results.

 

[bcrowell]

Related thread: https://www.physicsforums.com/showthread.php?t=586409 "constraints on interactions of tachyons"

 

[John232]

I don't think tachyons would travel FTL at all. The lorentz makes the speed of light a cosmic u-turn in physics. You get negative values for time and space, but the relation of direction of left or right in not even considered. So then it would be as if once something passes the speed of light it just instantly turns around and starts going another way through hyperspace. But then what about actually reaching c to begin with? If you considered the frame of the object in question it could vary well say that it still measures the speed of light to be about 300,000 km/s faster than it even as it approuches the speed of light. So then to the tachyon we could all be tachyons... I think with the time being undefined as an object traveling at c it could really mean that the triangle that the lorentz relates two spaces and times does not exist. So then does that in turn mean that there are no relations between these events that exist? They where predicted to have little or no interaction with matter, it is a shame that they where cut short on funding to find this over a small amount of time. I believe they have no mass and don't travel FTL really, so I don't think they really violate anything. Consider the Great Uncle Paradox, (there is no such paradox I just made it up). Say you never met your great uncle and you never seen him before in your life and no one ever talked about him and he never did anything to even effect your life. Then one day you start working in time travel and think hey, i never met my great uncle. So then late one night you travel back in time and meat him. Then you get in an argument and start fighting because he doesn't believe you and you kill him. You come back and then you ask your mother about him and she says, oh he died a long time ago someone just got into his home and killed him, was strange they didn't even take anything...

 

[Ilmrak]

The only reason would be if you wanted to define an FTL frame.[...]

Ok, I did misunderstood you point of view, sorry. Now I see and absolutely agree with you

I don't think tachyons would travel FTL at all. The lorentz makes the speed of light a cosmic u-turn in physics. You get negative values for time and space, but the relation of direction of left or right in not even considered. So then it would be as if once something passes the speed of light it just instantly turns around and starts going another way through hyperspace. But then what about actually reaching c to begin with? If you considered the frame of the object in question it could vary well say that it still measures the speed of light to be about 300,000 km/s faster than it even as it approuches the speed of light. So then to the tachyon we could all be tachyons... I think with the time being undefined as an object traveling at c it could really mean that the triangle that the lorentz relates two spaces and times does not exist. So then does that in turn mean that there are no relations between these events that exist? They where predicted to have little or no interaction with matter, it is a shame that they where cut short on funding to find this over a small amount of time. I believe they have no mass and don't travel FTL really, so I don't think they really violate anything. Consider the Great Uncle Paradox, (there is no such paradox I just made it up). Say you never met your great uncle and you never seen him before in your life and no one ever talked about him and he never did anything to even effect your life. Then one day you start working in time travel and think hey, i never met my great uncle. So then late one night you travel back in time and meat him. Then you get in an argument and start fighting because he doesn't believe you and you kill him. You come back and then you ask your mother about him and she says, oh he died a long time ago someone just got into his home and killed him, was strange they didn't even take anything...

I think there are some point you are interpreting wrong.

Tachyons are defined to have four momenta $p^\mu p_\mu \equiv m^2< 0\,$* so if we assume $p^\nu \in ℝ$, then $|\vec p |^2 > E^2$, i.e. they travel faster then light.

I think the only reason in assuming they would interact weakly with ordinary matter is that they are never been observed.

The "negative direction" of time is considered in Standard Model (negative energy solution, or anti particle), but not related to FTL particle.
Actually every anti-particle could be interpreted as a particle going backwards in time, but they still have time-like four momentum, they don't travel FTL. From the point of view of an anti-particle (well defined because they have $m^2 >0$) we are all made of anti-matter ^^*this means they are not massless, at least in the usual sense.

Ilm

 

[bcrowell]

I found this paper, which discusses how to define FTL frames of reference. They claim to have a natural extension of the Lorentz group in 1+1 dimensions, which includes the Lorentz group as a subgroup. However, it doesn't work in 3+1 dimensions. So if you buy their reasoning, then there is no elementary argument that suffices to rule out FTL frames, but FTL frames are ruled in 3+1 dimensions.

Vieira, An Introduction to the Theory of Tachyons, http://arxiv.org/abs/1112.4187

 

[Ilmrak]

I found this paper, which discusses how to define FTL frames of reference. They claim to have a natural extension of the Lorentz group in 1+1 dimensions, which includes the Lorentz group as a subgroup. However, it doesn't work in 3+1 dimensions. So if you buy their reasoning, then there is no elementary argument that suffices to rule out FTL frames, but FTL frames are ruled in 3+1 dimensions.

Vieira, An Introduction to the Theory of Tachyons, http://arxiv.org/abs/1112.4187

I started to read that paper, but I already have a dubt.

In 1+1 dimensions they find new transformations such that

$x^\mu x_\mu \rightarrow x'^\mu x'_\mu = - x^\mu x_\mu \, .$

It seems to me that such a trasformation do not form a soubgroup of the "generalized Lorentz transformations". Infact, given 2 of those "new" transformation $\Lambda_1$ and $\Lambda_2$:

$x^\mu x_\mu = - (\Lambda_1 x)^\mu (\Lambda_1 x)_\mu = (\Lambda_1 \Lambda_2 x)^\mu(\Lambda_1 \Lambda_2 x)_\mu \, .$

So $\Lambda_1 \Lambda_2$ is in the soubgroup of the usual Lorentz transformations.

This seems odd to me because I'd expect that composition of two "boosts", each with $v>c$, could result in another "boost" with $v>c$.

Is this an issue to me only because I'm not used to those new transformations?

Anyway I'll continue reading the paper

Ilm

 

[John232]

Tachyons are defined to have four momenta $p^\mu p_\mu \equiv m^2< 0\,$* so if we assume $p^\nu \in ℝ$, then $|\vec p |^2 > E^2$, i.e. they travel faster then light.

I think the only reason in assuming they would interact weakly with ordinary matter is that they are never been observed.

I am not saying that any particle that comes to have a negative value travels back in time. The only thing I thought could be a wrong interpretation is them traveling through hyperspace, because i/i=-1. In tachyon physics the speed of light barrier is not broken. It assums that any value greater than c makes the particle travel back in time. The velocity of the particle itself becomes negative, but there was nothing in the equations to determine the velocity that shows that it would have had a change in direction. So then if anything travels faster than c it no longer travels forward through spacetime, if it no longer travels forward through spacetime then it will never travel further than light in a given amount of time. So then if you say bah humbug something can travel FTL, then you put it into the equation and graph it, the particle still doesn't travel FTL according to the amount of distance it has covered over time. The equation just doesn't allow it.

On another note, they where predicted to have little or no interaction with matter but this was because of the mathematics done for them. I wish I could give more sources, but I didn't learn physics on the internet. I think my only conspiricay theory is that finding this was a cover up done by the government. I think they canceled this early because they found them with little problems, and then it has created a hole in modern physics. I mean what physics experiment have they done to just suddenly say nope you can't find them and then just shut it down for it to never be reapeted? And, I think I have seen one of these before...

 

[bcrowell]

Here is my summary of what I learned from the Vieira paper and some other similar ones that try to define tachyonic frames.

If a tachyon is going to have a frame of reference, then a tachyonic observer sees bradyons as going FTL, i.e., bradyons appear like tachyons to tachyons.

This means that we can't possibly have Lorentz invariance in the sense of preserving inner products. Transforming between tachyonic and bradyonic frames must flip the sign of a vector's squared norm.

Therefore a bradyonic observer sees a tachyon as having |p|>|E|. For a fixed mass, the lowest-energy state has |E|=0. Since the energy-momentum four-vector is related to the velocity four-vector by p=mv, the lowest- energy state of a tachyon corresponds to a state with infinite speed. In quantum-mechanical language, the energy spectrum is bounded below, but the ground state is highly degenerate, because the world-line could be in any direction.

Suppose we want to transform from a bradyonic frame K to the rest frame K' of a particle that K sees as a zero-energy tachyon. Start by considering only 1+1 dimensions, and let K' be moving to the right relative to K. The transformation carries the +x axis to lie along the +t' axis. If we want to preserve the light cone, then it must also carry +t to lie along +x'. The usual arguments about transformations preserving area depend only on homogeneity, so they still apply to the extended Lorentz transformation (ELT). Combining this with linearity, we find that this transformation is simply a flip across the right-going lightlike line x=t. This is not the same thing we'd get by analytic continuation of the LT, which would give (t,x)->(ix,it), not (t,x)->(x,t).

We now have a group formed by taking the Lorentz group and adjoining the flips F+x across x=t and F-x across x=-t. The product F+xF-x is simply the total inversion PT, which corresponds to particle-antiparticle interchange. This gives four families of frames, corresponding to bradyonic, antibradyonic, tachyonic, and antitachyonic observers. Combination of velocities can be determined simply by this group structure. Any given vector traces out a branch of a hyperbola under LT's, and under ELT's we get a family of four hyperbolic branches. Any ELT can be described by the direction in the plane to which it sends a fixed vector such as (0,1). The directions along the light cone are forbidden because the corresponding ELT's wouldn't be one-to-one. (What would be the standard term for this group structure?)

Since by definition every ELT is built out of LTs and flips, which preserve area, every ELT preserves area. For example, the wedge W bounded by the unit circle and the future light-cone has its area preserved by a flip.

In 2+1 dimensions, by isotropy the tachyonic observer sees both x and y as timelike. This violates the principle that all frames are equivalent. The tachyonic observer sees the topology of the space of allowed velocity or momentum vectors as a single connected piece, not disconnected future and past light cones. So the ELT's aren't even a symmetry group in 2+1 dimensions, or in n+1 dimensions with n>1. (This paragraph is my own argument. I haven't actually looked at the corresponding part of Vieira's paper, but he does say that you can't extend ELT's to 3+1 dimensions.)

So it looks to me like tachyonic frames are kinematically impossible in 3+1 dimensions, and therefore of no interest. This seems to hold as long as you require homogeneity, isotropy, and equivalence of all frames of reference, and it holds even if you relax the requirement that the ELT's preserve both the sign and the magnitude of squared norms.

What baffles me is how people like Vieira and Recami can take this with equanimity and still go ahead talking enthusiastically about tachyonic frames. It seems like a total no-go theorem to me. Even if there was some error in my own no-go theorem, Vieira proves the same no-go theorem in his own paper, and presumably he believes his own proof...!?

 

[John232]

Maybe only the area of the tachyon and the bradyon is the same but exchanging them is not. Like the tachyon that assumes it is a bradyon would only show the same area being covered as the tachyon for the bradyon. So then if a tachyon assums that it is a bradyon then an observed bradyon would only show to cover the same area as a tachyon and so on. IDK it is just a hunch. I can't see how they could if after traveling greater than c the velocity is reveresed, and maybe you would have to know its true "tachyonic velocity" before putting it into the lorentz to begin with.

I think the main reason why they haven't been able to be shown in accelerators is because the force they use to push a particle with mass only propogates at the speed of light. So then how could you use a force that travels at c to in turn make it travel faster than c? Seems like it would need an added extra push from another source.