According to the Noether theorem, what quantity is conserved in a Lorentzian transformation?
Noether theorem
energy-momentum..
| Bikerman wrote: |
| energy-momentum.. |
Not quite, technically speaking.
A Lorentz transformation (also called a Lorentz boost) is of the form:
x -> gamma (x-vt) t - >gamma(t-x/v) sort of thing.
The thing is, the transformation applied here depends on time, so a bit complicated.
Where do conservation of energy and momentum come from in ordinary physics? They come from the invariance of the laws of physics under:
spatial translation x -> x + a (a constant)
and temporal translation t -> t + b (b constant)
Similarly, conservation of angular momentum comes from invariance under spatial rotations, a FIXED spatial rotation that does not depend on time.
We can hence see that the Lorentz transformation is a bit more complicated. Energy-momentum conservation says that it does not matter where you place the origin of coordinates (space and time), but different frames still at rest wrt each other, whereas invariance under Lorentz transformations says that physics is invariant between frames in relative motion.
Best I can do at the moment is point you to this page:
http://math.ucr.edu/home/baez/boosts.html
which is another discussion of the same topic.
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