Gorini vs. Jin: Superluminal Frames in 1+1 Dimensions

In summary, Gorini 1971 presents a no-go theorem that disproves attempts to extend the Lorentz group to superluminal velocities in n+1 dimensions, for n\ge3. His assumptions are stated in abstruse notation, but he provides a translation into plainer language, which includes the axiom that time has a unidirectional flow and that a common time standard is used, and that space is isotropic and that the localization of events in space is given by orthogonal coordinate systems of the same parity. He also assumes linearity, that the transformations are one-to-one, and that the transformations are real. Gorini gives explicit counterexamples in 1+1 dimensions, including the "Parker group,"
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bcrowell

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Gorini 1971 (reference below) proves a no-go theorem for attempts to extend the Lorentz group to superluminal velocities in n+1 dimensions, for [itex]n\ge3[/itex]. His assumptions are stated in abstruse notation, but he gives the following translation into plainer language:

> Axiom: [...] time has a unidirectional flow and that a common time standard is used, c) that space is isotropic and that the localization of events in space is given by orthogonal coordinate systems of the same parity.

(He also assumes linearity, that the transformations are one-to-one, and that the transformations are real.) This is a very weak set of assumptions. He doesn't actually assume that it has the Lorentz group as a subgroup, or even that it's a Lie group. The basic idea of the proof is that to a hypothetical tachyonic observer, bradyons would appear to move faster c, and tachyons slower than c. This effectively interchanges the roles of timelike and spacelike vectors, but it's not possible to do such an interchange in more than one spatial dimension, because then the transformation couldn't be one-to-one.

He gives some explicit counterexamples in 1+1 dimensions to show that his theorem really doesn't hold for n<3. One of these is a group that he calls the "Parker group." The Parker paper is paywalled, but from Gorini's description the idea simply seems to be that you take the Lorentz group and adjoin an operator (I'll call it Q) that does (t',x')=(x,t).

There is a more recent paper by Jin and Lazar, which makes some claims that seem to contradict Gorini's. I'm trying to figure out whether they really are contradicting each other, or whether there is some difference in their assumptions that I'm not understanding.

Jin seems to be assuming linearity, reality, invariance of c, and homogeneity of spacetime. From these assumptions he does a 1+1-dimensional derivation of the Lorentz transformations, extending them to superluminal velocities. He doesn't claim this is new, and he gives extensive references to the literature, although he seems unaware of Gorini and Parker. Based on these assumptions, he claims to arrive at a uniquely determined form for the transformation, which is [itex]\Lambda=\gamma(...)[/itex], where "..." is the matrix you would expect for ordinary Lorentz transformations, and [itex]\gamma[/itex] is:

(I) [itex](v^2/c^2-1)^{-1/2}[/itex] for [itex]-\infty < v < -c[/itex]

(II) [itex](1-v^2/c^2)^{-1/2}[/itex] for [itex]-c < v < c[/itex]

(III) [itex]-(v^2/c^2-1)^{-1/2}[/itex] for [itex]c < v < +\infty[/itex]

This function is neither even nor odd. "In the standard derivations of Lorentz transformations, the Lorentz factor [itex]\gamma[/itex] is assumed to be an even function of v... [T]his assumption is based on isotropy of space, i.e. space is non-directional, so that both orientations of the space axis are physically equivalent." Therefore he claims that extending the Lorentz transformations to superluminal velocities automatically violates isotropy of space.

As far as I can tell, Jin's group seems to be exactly the same as Parker's, and Gorini claims that Parker's group *does* respect isotropy. So what's up here?

The reason I say that I think the two groups are the same is that if you work out the limiting form of Jin's transformations for [itex]|v|\rightarrow\infty[/itex], you get exactly the operator Q described above.

My suspicion is that Jin is simply misinterpreting his result. I think his result is probably consistent with isotropy of space, and his argument quoted above is just wrong. The group he's talking about is not topologically connected to the identity. I think he's imagining that an observer could simply measure the value of [itex]\gamma[/itex] for motion in the positive direction, check whether it's positive or negative, and thereby distinguish the physical properties of the positive spatial direction from the negative one. It's certainly true that one can measure [itex]\gamma[/itex] for subluminal velocities. E.g., the classic experiments with cosmic-ray muons measured a time-dilation effect, which is equivalent to measuring [itex]\gamma[/itex] for motion downward toward the surface of the earth. A violation of isotropy might then have shown up in a 24-hour variation of the experimental results due to the rotation of the earth. But it's far less obvious to me that this works if the particles are tachyons.

There are some other logical points in the paper that bother me, but they seem more tangential. (In particular, I don't understand p. 4, "Eq. (11) shows that gamma(v) is positive when" v is <-c.)

Can anyone provide any insight here?

References

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

Jin and Lazar, "A note on Lorentz-like transformations and superluminal motion," http://arxiv.org/abs/1403.5988

Parker, 1969, Faster-Than-Light Intertial Frames and Tachyons, Phys. Rev. 188, 2287, http://journals.aps.org/pr/abstract/10.1103/PhysRev.188.2287
 
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One thing that occurs to me now is that Jin seems to be constructing a new group of symmetries G that has the Poincare group P as a subgroup. But if P is a subgroup of G, then G has a higher degree of symmetry than P, not a lower one. P includes rotations, and therefore so does G. Does my reasoning make sense?
 
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Jin's conclusion also seems to be contradicted by Andreka, ' http://arxiv.org/abs/1211.2246 , which states that the 1+1-dimensional case is OK.
 

1. What is the concept of superluminal frames in 1+1 dimensions?

Superluminal frames in 1+1 dimensions refer to the theoretical possibility of objects moving faster than the speed of light in a two-dimensional space. This concept is based on Einstein's theory of relativity, which states that nothing can travel faster than the speed of light in a vacuum.

2. How is the Gorini vs. Jin experiment related to superluminal frames?

The Gorini vs. Jin experiment is a thought experiment that explores the concept of superluminal frames in 1+1 dimensions. It suggests that by manipulating the frames of reference in a two-dimensional space, it may be possible to create a situation where an object can appear to move faster than the speed of light.

3. Is there any evidence to support the existence of superluminal frames?

At this time, there is no empirical evidence to support the existence of superluminal frames. It is a theoretical concept that has not yet been proven through experimentation. However, the Gorini vs. Jin experiment and other thought experiments continue to be studied and discussed by scientists.

4. What are the implications of superluminal frames in 1+1 dimensions?

If superluminal frames were to exist, it would challenge our current understanding of the laws of physics, particularly the concept of causality. It could also have significant implications for space travel and the possibility of faster-than-light travel. However, until there is concrete evidence for the existence of superluminal frames, these implications remain purely theoretical.

5. How does this research impact the field of physics?

The study of superluminal frames in 1+1 dimensions is a topic of ongoing research and discussion in the field of physics. It pushes the boundaries of our current understanding of the universe and challenges scientists to think creatively about the nature of space and time. This research has the potential to lead to new discoveries and advancements in our understanding of the laws of physics.

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