Imaginary Transverse Space of Superluminal Lorentz Transform

[greswd]

I was reading this paper: http://dinamico2.unibg.it/recami/erasmo%20docs/SomeOld/RevisitingSLTsLNC1982.pdf

It is on superluminal Lorentz transformations and is too advanced for me.

But anyway, take a look at equation(s) (11). For the y' and z' transformations, there is an imaginary unit coefficient. But for the x' transformation, there is none.

So the space in the x-direction is 'real', but the transverse space around it is 'imaginary'?

 

[PeterDonis]

the space in the x-direction is 'real', but the transverse space around it is 'imaginary'?

From what I can see on a quick skim of the paper, no. They are still talking about Minkowski spacetime, so all four of the spacetime directions are real. They are just using some strange conventions because they are trying to construct "inertial frames" in which an object "at rest" is moving on a spacelike worldline.

 

[greswd]

From what I can see on a quick skim of the paper, no. They are still talking about Minkowski spacetime, so all four of the spacetime directions are real. They are just using some strange conventions because they are trying to construct "inertial frames" in which an object "at rest" is moving on a spacelike worldline.

it seems like they are using a tachyonic Lorentz factor $$\frac{1}{\sqrt{\beta^2-1}}$$
and the only way to preserve the light cone between transformations (such that a photon does not end up with a superluminal velocity) is a rotation to the complex plane.

I've got some info from: http://rspa.royalsocietypublishing.org/content/early/2012/09/25/rspa.2012.0340
but they didn't tackle the transverse spatial dimensions.

 

[PeterDonis]

it seems like they are using a tachyonic Lorentz factor

Yes, but that just corresponds to using a weird mathematical convention to describe the same spacetime. Nothing about the underlying spacetime or its geometry is changed.

 

[greswd]

I'm wondering how to describe the transverse space in the frame of a tachyon.

 

[PeterDonis]

I'm wondering how to describe the transverse space in the frame of a tachyon.

Part of the issue that this paper appears to be trying to deal with is that there is no such thing as "the frame of a tachyon", if by "frame" you mean "something that works just like ordinary inertial frames". An ordinary inertial frame only works for objects traveling on timelike worldlines. A "frame" in which a spacelike worldline (that of a tachyon) is "at rest" is a fundamentally different kind of thing. (I'm not even convinced that it makes physical sense, but whether it does or not, mathematically it's a fundamentally different kind of thing.) So I would not expect any description of the transverse space to match your intuitions about what such a thing "ought" to look like. The imaginary numbers that are showing up in the paper for the transverse dimensions are, I think, a symptom of the fundamental difference.

 

[greswd]

Part of the issue that this paper appears to be trying to deal with is that there is no such thing as "the frame of a tachyon", if by "frame" you mean "something that works just like ordinary inertial frames". An ordinary inertial frame only works for objects traveling on timelike worldlines. A "frame" in which a spacelike worldline (that of a tachyon) is "at rest" is a fundamentally different kind of thing. (I'm not even convinced that it makes physical sense, but whether it does or not, mathematically it's a fundamentally different kind of thing.) So I would not expect any description of the transverse space to match your intuitions about what such a thing "ought" to look like. The imaginary numbers that are showing up in the paper for the transverse dimensions are, I think, a symptom of the fundamental difference.

I see. Cos someone was telling me that superluminal reference frames could be easily described. Not so easy it seems.