what does a tachyon observe?


by dynkindiagram
Tags: relativity, tachyon
Ilmrak
Ilmrak is offline
#19
Mar13-12, 05:27 PM
P: 97
Quote Quote by bcrowell View Post
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

[itex]
x^\mu x_\mu \rightarrow x'^\mu x'_\mu = - x^\mu x_\mu \, .
[/itex]

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 [itex]\Lambda_1[/itex] and [itex] \Lambda_2[/itex]:

[itex]
x^\mu x_\mu = - (\Lambda_1 x)^\mu (\Lambda_1 x)_\mu = (\Lambda_1 \Lambda_2 x)^\mu(\Lambda_1 \Lambda_2 x)_\mu \, .
[/itex]

So [itex]\Lambda_1 \Lambda_2 [/itex] 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 [itex]v>c[/itex], could result in another "boost" with [itex]v>c[/itex].

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
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#20
Mar14-12, 01:49 AM
P: 249
Quote Quote by Ilmrak View Post
Tachyons are defined to have four momenta [itex]p^\mu p_\mu \equiv m^2< 0\,[/itex]* so if we assume [itex]p^\nu \in ℝ[/itex], then [itex] |\vec p |^2 > E^2[/itex], 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 mathmatics 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
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#21
Mar14-12, 09:23 PM
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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
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#22
Mar17-12, 03:05 PM
P: 249
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.


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