Computing the Thomas Precession: Help from PFers Needed!

In summary, the conversation discusses the steps for deriving the expression for the Thomas precession and the discrepancy in results obtained for the approximation of the matrix L''. The speaker is seeking help in identifying their mistake and understanding how γ can approximate γ^2.
  • #1
Coelum
97
32
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.
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 2022-08-29 17-35-19.png

I have been able to compute the matrix L'' (eqn. 7.18). However, computing its approximation as described above (eqn 7.19 and γ' = 1), I get a different result:

Screenshot from 2022-08-29 17-38-37.png
.
As you can see, I get γ^2 rather than γ in the third row. I cannot find my mistake and I cannot see how γ can approximate γ^2. Any hint?
Thanks a lot in advance,

Francesco
 
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  • #2
Just an idea: to first order in β, γ^2 ≈ γ since γ is quadratic in β. Sort of dirty trick, but formally correct?
 

FAQ: Computing the Thomas Precession: Help from PFers Needed!

What is the Thomas precession and why is it important in computing?

The Thomas precession is a relativistic effect that occurs in rotating frames of reference. It is important in computing because it affects the behavior of particles in high-speed motion and is necessary for accurate calculations in fields such as particle physics and astrophysics.

How is the Thomas precession calculated?

The Thomas precession is calculated using the Thomas rotation matrix, which takes into account the velocity and acceleration of the rotating frame of reference. This matrix is then applied to the equations of motion for the particle being studied.

What is the role of the PFers in computing the Thomas precession?

The PFers, or the Poincaré group of transformations, are a group of mathematical transformations that are used to describe the symmetries of special relativity. They play a crucial role in computing the Thomas precession by providing a framework for understanding the effects of rotation on particles.

What challenges are faced when computing the Thomas precession?

One of the main challenges in computing the Thomas precession is accurately accounting for the effects of acceleration and rotation on the particle being studied. This requires advanced mathematical techniques and can be computationally intensive.

How is the Thomas precession used in real-world applications?

The Thomas precession is used in a variety of real-world applications, including particle accelerators, GPS systems, and satellite navigation. It is also important in understanding the behavior of high-speed particles in astrophysical phenomena such as black holes and neutron stars.

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