# Thomas precession, Lie algebra of the Lorentz group and the conservation of energy

If you read this post Thomas Precession, you will see a very good answer by WetSavannaAnimal, on the subject of Thomas Precession, which I am currently working my through, in conjunction with some pretty basic texts on groups, Lie algebra and representations.

My problem stems from the fact I don't have the background, as yet anyway, to understand fully the statement below, in relation to conservation of energy:

Two boosts in different directions do not commute and is equivalent to a pure boost plus a pure rotation.

I will continue reading on the topic, but in the meantime, I will put my related (naive, I know) questions in terms of macro sized objects, and hopefully someone will set me straight on my reasoning, which I am painfully aware is incorrect in many areas.

All of the reading I have done so far is on the quantum level, but since Thomas himself seems to have used (Classical) GR concepts to explain the experimentally observed spin orbit interaction energy of an electron, I don't see why it can't apply to macro sized objects.

My Questions:

• If I take a spaceship and move it at any (relative to c) significant arbitrary velocity in a circular path, from the reference point of the coordinate observer, say on Earth, will this spaceship rotate around it's own axis?

• Or am I getting frames of reference mixed up, (or something more basic)?

• If the spaceship does rotate, how is conservation of energy maintained, re: the velocity of the spaceship?

I am trying to extend the spin-orbit precession idea, using the Lie algebra of Lorentz transformations, to the classical world.

Hopefully, I can figure out my mistakes myself and withdraw the question shortly.

But in the meantime, any help with the basic concepts would be appreciated, if Thomas Precession, or rather it's classical analog of pure rotation , is truly applicable on a macro, rather than simply quantum, level.

Telling me I have basic concepts completely mangled is fine since, so far on this website, the greater the level of "wrongness" my questions have contained, the more I have learned from them.

• Thomas precession is very much applicable to classical objects. It is a very easy calculation to show this; in fact, almost every good SR/GR book talks about Thomas precession in the classical context; c.f. chapter 6 of MTW. Moving on, what exactly is the issue with conservation of energy? – FenderLesPaul Aug 3 '15 at 16:27
• @FenderLesPaul my apologies, I only saw your comment today, I was confused as to where the energy to spin the classical object came from, because I was assuming the linear kinetic energy of the boosts remained the same. There is no need to reply thanks, I will read up more up this myself, as obviously enough, I am lacking the basics to ask this question properly. Thanks for your time. – user81619 Aug 18 '15 at 17:08