Computing Thomas Precession: Part 2 | Francesco

In summary: This is a summary of the content of the conversation. In summary, the author is describing how to derive the Thomas precession and explains that it is hard to calculate. He provides a step-by-step explanation of how to do it and notes that there is a more in-depth treatment available in Jackson's book. However, he does mention that Lie algebra is not necessary to understand the concepts behind the Thomas precession.
  • #1
Coelum
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TL;DR Summary
While computing the transformation matrix associated to Thomas precession - as described by Goldstein (7.3) - I cannot reproduce a step in the derivation. This is a step later than the one described in my previous post "Computing the Thomas precession".
Dear PFer's,
I am reproducing the steps to derive the expression for the Thomas precession, as described in Goldstein/Poole/Safko "Classical Mechanics". Hereafter an excerpt from the book describing the step I am currently working on.
Screenshot from 2022-08-30 10-18-29.png

Based on the text, the transformation S_3 -> S_1 should be the composition of a boost on the x" axis

1661847678210.png

and a boost on the y" axis

1661848318941.png

which, when composed assuming γ'=1, yield

1661848394308.png
.
The differences with (7.20) are:
  1. the '-' sign in elements [1,2], [2,2], [2,1], [3,1]
  2. the '0' in position [3,2]
  3. the element in position [2,3], which can be approximated as (dropping the " for readability):
    βxβyγ = γ(βy/βx)βx2 ≈ γ(βy/βx)β2 = γ(βy/βx)(1-1/γ2) = (γ-1/γ)βy/βx. Close, but not identical.
Can anybody help me understand?

Thanks,

Francesco
 
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  • #3
vanhees71 said:
The Thomas precession is hard to calculate. Here's my attempt (Sect. 1.8):

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf
Hi vanhees71,
thanks for your answer! I am not familiar enough with Lie algebra to follow your exposition. But I understand I will need to learn the mathematical tools you are using if I want to dig deeper.
 
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  • #4
I think you don't need much more about Lie algebra than what's in this manuscript.
 
  • #5
vanhees71 said:
I think you don't need more about Lie algebra than what's in this manuscript to understand Fermi-Walker transport and thus the Thomas precession.
 
  • #6
Not to sound like a wise guy, but I've always thought the treatment of J.D. Jackson (I have a translation of the 1975 edition) to be the best: lucid and straight to the point.
 
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  • #7
Hi dextercioby, thanks for the pointer! In fact, the Thomas precession is discussed in section 11.8 of the second edition of Jackson's, though it requires reading the previous section on infinitesimal transformations.

As an update: I fixed all the issues except the '0' element in position [3,2]. That looks very strange: that element is certainly zero!
 
  • #8
Just an idea: to first order in β, γ"-1 ≈ 0. It can replace 0 in position [3,2]...
 

1. What is the purpose of computing Thomas Precession in Part 2 of Francesco's research?

The purpose of computing Thomas Precession in Part 2 of Francesco's research is to analyze the effects of relativistic corrections on the precession of a spinning particle. This can provide a deeper understanding of the behavior of particles at high speeds and in strong magnetic fields.

2. How is Thomas Precession calculated in Part 2?

In Part 2 of Francesco's research, Thomas Precession is calculated using the Thomas precession formula, which takes into account the velocity and acceleration of the spinning particle. This formula is derived from the relativistic effects on the motion of a spinning particle.

3. What are the implications of Thomas Precession in particle physics?

Thomas Precession has important implications in particle physics, particularly in understanding the behavior of particles at high speeds and in strong magnetic fields. It also plays a role in the development of theories such as the Standard Model and quantum mechanics.

4. How does computing Thomas Precession in Part 2 contribute to the overall understanding of the topic?

By computing Thomas Precession in Part 2, Francesco is able to analyze the effects of relativistic corrections on the precession of a spinning particle. This provides a more comprehensive understanding of the topic and can lead to further advancements in particle physics.

5. Are there any practical applications of Thomas Precession?

Yes, there are practical applications of Thomas Precession, particularly in the fields of particle accelerators and magnetic resonance imaging (MRI). Understanding the effects of relativistic corrections on the precession of particles is crucial in these technologies.

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