Monday, May 12, 2014

Lab Book 2014_05_11 Hyperbolic Derivation of the Thomas Precession

Lab Book 2014_05_11     Hamilton Carter
Most of the day was spent documenting work.  A few new articles inspired more work on the hyperbolic Thomas precession derivation that's been in the hopper for the last few months.

Most of my work today was summarizing the Letaw and Pfautsch vs. the number operator arguments I built yesterday and getting the results out to all the interested parties.

I also found a rather complete reference about the many ways to derive the Thomas precession.  The method that I hope to publish which involves using the Walter methodology to arrive at the Takeno metric was not included!  I should start writing this up, once again, soon.

There are a few other articles regarding Fulling-Unruh radiation in rotating frames that have been added to the reading queue.  The first of these is from Nokic, and the second is from Nelson.  Interestingly, Nokic also published a paper where he opined on the validity of the Rindler quantization.  I’m curious as to what aspects of the first Nokic paper, which is in fact interesting, placed it topically into Physical Review A.  Nelson’s article is quite detailed, and also involves the Thomas precession.  Once again, the hyperbolic derivation I’ve developed isn’t used.

Notes on the new Thomas precession derivation and the associated paper
The introduction to the Thomas precession paper should include the following points:

Ideas for the abstract
There is a quote from the Nelson paper that can be adapted to express the intent of my paper.  Nelson says:
“Historically, accelerated reference frames and the Thomas precession have been studied by approximate methods. 1.2 The coordinates of an accelerated, rotating observer have also been studied by the method of FermiWalker transport.5 In this paper an exact, explicit, nonlinear coordinate transformation that incorporates the Thomas precession and leads to the metric above will be given.”

This sentiment should show up in the abstract of the new paper as something like:
“Traditionally, the Thomas precession has been studied by either numerically or geometrically approximate methods.  We present an exact Lorentz transform matrix that leads directly to the Thomas/Wigner angle for a particle moving in a circular orbit.”

Ideas for the introduction
1.  A general reconstruction of Walter’s arguments in the introduction.  For justification of doing something this way, see Davies et als’ rehash of Letaw and Pfautsch.
2.  Show the different ways this can be viewed including:
2.a.  Laboratory velocity as an angle projecting a constant velocity vector into velocity in time and velocity in space.  The ratio of the two gives the hyperbolic tangent expression usually associated with rapidity.
2.b.  The derivation that shows why acceleration in the laboratory frame can be related to acceleration in the moving frame by gamma cubed.
2.c.  The derivation in the existing Rocketing to Rapidity paper that shows how to derive the gamma squared expression relating dv to dv’.
The body of the paper should perhaps contain the EM derivation performed yesterday that shows the absence of the effects of transverse forces.  This could be followed by why it’s possible to treat the Thomas precession problem as if it were one dimensional, (ala tangential velocity), as opposed to the usual treatment on a two dimensional circle.  This should be followed by the meat of the argument.

Thomas Precession Derivation

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