Superluminal Inertial Frames in Special Relativity?

[rajama]

A short while ago, in responding to another thread, I constructed a spacetime diagram and formula for the radial Doppler shift of a superluminal object.

 

This v > c diagram mirrors two similar but entirely conventional v < c diagrams found in the text ‘Introducing Einstein’s Relativity’ by Ray D’Inverno.

 

 

This diagram differs from that required for v < c in that here the photon reflected at event P crosses the t axis a period T before the path of the second inertial frame, rather than after it.

 

The result is the inverse of the usual term for v in terms of k, and the value for k will be real only if |v| > 1, as required.

 

How does this relate to the Lorentz Transform (LT)?

 

Firstly, let’s try to design an experiment to formally derive the LT for v > c.

 

The Lorentz Transform is most often derived via a thought experiment in which the (proper) time of flight of a light signal travelling to and reflected from a specific event is measured using (initially synchronised) clocks in two inertial frames that are in motion relative to each other.

 

The experiment requires that, in each inertial frame, a light signal passes through a detector in twice for time of flight to be measured, allowing the co-ordinates of the event to be found and in principle compared between frames.

 

It should be obvious that this experiment cannot be performed if the relative speed of the two frames is greater than that of light.

 

It seems reasonable therefore that a thought experiment deriving the LT for v > c must use two related signals that allow the position of an event to be determined by both frames.

 

In this example, the relevant event sits in the future of the superluminal frame S’ (it could equally well have sat in the past). In the S frame, it is at the convergence of two light signals originally emitted simultaneously at Q.

 

 

Comparing the co-ordinates of the reflection points along the line of simultaneity in S with the points of arrival in S’, and using the k term once again we have:

 

x’ – c.t’ = k(x – ct)

 

x + c.t = k(x’ + c.t’)

 

After some re-arrangement this gives:

 

 

This is just the standard LT written in terms of the radial Doppler shift.

 

But substituting k for v > c rather than k for v < c produces

 

 

 

Notes

 

1. The k term can be used to derive the SR composition law for velocities:

 

kac = kab.kbc

 

where a, b, and c are inertial frames

 

Using k v < c this normally gives:

 

vac = (vab + vbc) / (1 + vab.vbc)

 

Given two values for k, there are now 8 possible routes to vac

 

As k v > c = - k v < c these outcomes can be divided into two sets which produce either the conventional value for vac given above or its inverse.

 

This supports two views of the three frames, depending on whether the observer of these three frames is moving with v < c or v > c relative to ‘a’.

 

 

2. If you use the LT for v > c to derive the SR composition law, it gives the inverse of the normal result.

 

This appears consistent with the note 1 above…

 

 

Any comments?

[timo]

Since you were already asking for a reply somewhere else: I wouldn´t have time to go through your post anyways, but if I had, I´d first ask you to make a bit clearer what this is all about. I have the tendency to read the introductionary sentences and then the last sentences first to get an idea of what I have to expect and what the post is all about. However, neither your introductionary sentence which sais "this is a spacetime diagram according to some book you don´t own" nor your last sentence saying "this is consitent with the note 1 above" gives me a hint on what your point is.

In short: Perhaps write a short three or four sententence comment on what it´s all about, what you want to tell us (or ask us, I don´t even know that). While making physics claims without any math to back it up might be unprofessional, just throwing out a few calculations without saying what they are supposed to show is confusing.

[rajama]

In short: Perhaps write a short three or four sententence comment on what it´s all about, what you want to tell us (or ask us, I don´t even know that). While making physics claims without any math to back it up might be unprofessional, just throwing out a few calculations without saying what they are supposed to show is confusing.

 

Sorry, Atheist, yes in trying to cut down the length of the post I totally lost its context...

 

• I found the Relativistic Radial Doppler Shift ‘k’ for a superluminal inertial frame differs from the sublight value - is this correct?

• If so, given that the LT can be written entirely in terms of ‘k’, substituting k for v > c we get a different LT than for v < c...

• I assume I made a mistake or two as I haven't seen anything like this in SR and it implies a real rest mass for a tachyon?

 

Any expert input would be much appreciated...

 

 

:-) :-) :-) :-) :-)

 

 

And here's the long version:

 

A little while ago, a post was made on the Relativity board by CPL.Luke in order to discuss the appearance of a superluminal object.

 

In that thread, the model presented by Mowgli was, it seemed, non-relativistic: he maintained his theories would not work under special relativity (SR).

 

It occurred to me that this was probably not so.

 

As I have a text that derives the Relativistic Radial Doppler Shift without utilising the Lorentz Transforms (LT) in SR, I thought it possible to derive the Doppler shift without encountering any imaginary terms, so drew a thought experiment based on this, which produces a result different from the usual SR result…

 

My post of the diagram in that thread was either ignored / misunderstood by Mowgli et al (maybe I should have explained it more fully) but this brief piece of work led me to look at the whole ‘classical’ tachyon hypothesis again and I found myself railing at the idea of imaginary mass as it comes an experiment that cannot be conducted for v > c.

 

The textbook I mentioned derives the LT from this ‘k’ value… so in deriving a different value for ‘k’, do we get a different Lorentz Transform for inertial frames where v > c ?

 

I put together a thought experiment (though you don’t actually need it) for v > c and I wrote the LT purely in terms of ‘k’.

 

But when you put in the ‘k’ term you get a different LT…

 

You’ll find a very brief attempt to discuss this in the thread tachyons.

 

In that thread I decided it would be better to write a speculative post…

 

So, a second LT?

 

Sounds dodgy, doesn't it..?

 

Any expert input would be much appreciated...