# How to obtain Thomas Precession from lie algebra of the Lorentz group?

+ 3 like - 0 dislike
62 views

it seems to be possible that you can get the Thomas Precession just through the commutation relations of the Lorentz group. With Thomas Precession i mean, that in general the product of two boosts is a boost with a rotation. The exercise 15b) in this book Lie Groups, Lie Algebras, and Some of Their Applications formulates my Problem pretty good.

I get into some details. Let $\mathsf{O}(n;k)$ be the general orthogonal/ pseudo orthogonal group with Lie algebra $\mathsf{so}(n;k)$. I already esatblished a decompositon:

$\mathsf{so}(n;k) = \mathsf{so}(n) \oplus \mathsf{so}(k) \oplus \mathsf{b}(n;k)$ with $\mathsf{b}(n;k)$ beeing the symmetric elements of the lie Algebra, thus the matrices of the form: $\begin{pmatrix} 0 & B \\ B^{tr} & 0 \end{pmatrix} \ \text{with} \ B\in \mathbb{R}^{n \times k}.$ I also showed $[\mathsf{so}(n),\mathsf{so}(k)] = 0$, $[\mathsf{so}(n),\mathsf{b}(n;k)] \subseteq \mathsf{b}(n;k)$, $[\mathsf{so}(k),\mathsf{b}(n;k)] \subseteq \mathsf{b}(n;k)$, $[\mathsf{b}(n;k),\mathsf{b}(n;k)] \subseteq \mathsf{so}(n) \oplus \mathsf{so}(k)$. I also esablished the fact that the exponential map is bijective from $\mathsf{b}(n;k)$ into the sets of boosts(symmetric, positive elements of $\mathsf{O}(n;k)$).

I want to show with that knowledge that the product of two Boosts is a Boost followed by a rotation. My first try was to write the boosts as exponentials of elements in $\mathsf{b}(n;k)$ and then use BCH formula like:

$e^{A}e^{B} = e^{A + B + \frac{1}{2}[A,B] ... }$, but i can't see how the commutator relations from above provide the desired result.

This post imported from StackExchange Physics at 2015-11-16 14:26 (UTC), posted by SE-user Ursus
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.