# Does relativity permit tachyons given that they have negative gamma?

1. Mar 28, 2013

### Prote

I've heard it said that relativity formally permits tachyons and that we don't believe they exist merely because there is no empirical evidence of them. However, since a tachyon's speed exceeds c, its γ must be imaginary. The t-component of the particle's 4-velocity is, depending on definition, either γ or γc, which in either case will be imaginary for a tachyon. This means that the 4-velocity is not a vector in ℝ$^{4}$. My understanding is that spacetime is ℝ$^{4}$ (equipped with a metric that solves the Einstein field equation). Formally, this seems to imply that a particle cannot have a 4-velocity with an imaginary component. Am I correct in concluding then that relativity in its present formulation doesn't in fact allow tachyons?

Extending the theory to make spacetime ℂ$^{4}$ seems like it would be a nontrivial change, as it would become effectively 8-dimensional.

2. Mar 28, 2013

### bcrowell

Staff Emeritus
The Dawe and Hines paper is publicly accessible and describes previous work in which you can have FTL frames of reference, and all measurable quantities remain real-valued. Vieira proves a no-go theorem that says you can't have FTL frames of reference in m+n dimensions unless m=n. Since our universe is 3+1-dimensional, you can't have FTL frames.

It's still logically possible to have tachyons without having FTL frames of reference.

There was a vast literature created during the CERN FTL neutrino debacle. Most of it assumed Lorentz violation. That debate clarified that there are major problems with Cerenkov radiation unless your tachyons were sterile, i.e., they can't interact with either the electromagnetic field or the weak force.

Dawe and Hines, "The Physics of Tachyons I," http://adsabs.harvard.edu/full/1992AuJPh..45..591D
Vieira, An Introduction to the Theory of Tachyons, 2011, http://arxiv.org/abs/1112.4187

3. Mar 28, 2013

### fzero

The most conservative interpretation of finding that $\gamma$ is imaginary for $v>c$ is just to conclude that there are no Lorentz transformations that map $v<c$ to a $v'> c$. This is perfectly consistent with other results in special relativity such as the velocity addition formula or the conclusion that it is impossible to accelerate a massive particle to the speed of light with a finite amount of energy.

I'm sorry, but I stopped reading the Vieira paper after he discovers his forward and backward reference frames. Specifically, he obtains equations (4) and (5), by using the specious argument that quantities that both $=0$ (null distances) should be proportional. He thereafter derives a result ($\lambda(v)\lambda(-v) =1$) by dividing by zero! I have no hope for the validity of his results. Actually Dawe and Hines seem to be doing the same thing around their eq (15).

In any case, I think the motivations of these papers is rather misguided. In a Lorentz-invariant theory, tachyons also satisfy the wrong mass-shell condition, $p_\mu p^\mu < 0$ in an appropriate signature. Therefore they appear to have an energy that is unbounded from below. They cannot describe a stable solution of the equations of motion, say, in QFT. Of course we know the solution from the physics involved in the Higgs mechanism. The tachyon represents the quantization of a field around a false vacuum. Eventually the system must decay to a true vacuum, at which point the tachyon condenses and the resulting spectrum only contains normal particles.

This perspective explains the conflict between the claimed FTL neutrinos and Lorentz-invariance. The tachyonic degrees of freedom represent an expansion around a false vacuum and not true superluminally propagating particles. So Lorentz-violation would have been the only way to explain them.

4. Mar 29, 2013

### bcrowell

Staff Emeritus
Yep, you're right about Vieira, although I don't think that necessarily invalidates the whole paper. It's just a review paper, not a paper presenting original results. If you look carefully, I don't think Dawe and Hines are doing the same thing. They give the motivation for the minus sign on p. 594. At eq (15) on p. 596, they're just checking that it gives the right result in a special case.

I don't think what you're saying here has much to do with the OP's question or with the Vieira and Dawe-Hines papers. The OP wanted to know about gamma, which is a feature of the Lorentz transformation. That is, the OP wanted to know how the Lorentz transformations apply (or break down and fail to make sense) when you transform between frames that are FTL relative to one another. This is the issue that the Vieira and Dawe-Hines papers address, and they end up by proving that the results do *not* make sense in 3+1 dimensions.

The OPERA results were simply a mistake, which makes it not surprising that it was hard to explain them. When people tried to explain them without Lorentz-violation, they failed. When people tried to explain them with Lorentz-violation, they failed. That's because the results were due to a loose cable.

If you want to see why the OPERA claims were incompatible with Lorentz-invariance, you don't need any fancy, uncertain arguments about field theory. If OPERA had been right, the energy-dependence of the velocity would have been in the wrong direction based on the comparison with the SN1987A data.

5. Mar 31, 2013

### bcrowell

Staff Emeritus
It turns out that the original no-go theorem in 3+1 dimensions was in a 1971 paper by Gorini, which is publicly available at the link below. A nice summary of the content of the Gorini theorem is given at the end of section 3.2 of Recami, which is also publicly available. It says that you can't extend the Lorentz transformations to superluminal velocities while still maintaining all the following properties:

i) 3+1 dimensions
ii) real
iii) linear
iv) isotropy of space
v) invariance of c
vi) some group-theoretical properties

Section 14 of Recami catalogs a whole bunch of possible ways of making viable theories by violating one of these assumptions. However, there doesn't appear to be anything in the catalog that is attractive or has borne fruit since 1986. In answer to the OP's question, Recami does demonstrate theories that violate (ii), but he also demonstrates some that obey (ii) while violating one of the other five requirements.

Gorini, "Linear Kinematical Groups," Commun Math Phys 21 (1971) 150, http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103857292

Recami, Riv Nuovo Cimento 9 (1986) 1, http://dinamico2.unibg.it/recami/scientific.htm