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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

> 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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