# Possible mechanics based on the known symmetries in the nature (investigating rumor)

Somewhere I've heard about a relative new mathematical result regarding mechanics. Specifically, there is a list of the known symmetries of mechanics (both Newtonian and relativistic), i.e. different invariances such as rotation and translation symmetries. These symmetries specify the possible laws of mechanics.

There is a relative new (maybe as young as 10-20 years) result, which shows, that based on the group of the currently known symmetries of Nature, only two possible mechanics can exist: the Newtonian and relativistic. As I understand, it is only a rumor. Is it true? Where can I read more about this?

• I vaguely remember this from my Mechanics class 3 years ago. I recall that the derivation was mostly known for many years. At some point, we had the time reversal function, with the property $f(f(t)) = t$. It used to be set that $f(t) = -t$, but in the later years they proved that that is the only coherent function. May 23, 2014 at 12:56
• I'm skeptical of any result that says "the only thing that's possible is this thing that I already knew about"; smells to me of a fatal lack of creativity.
– rob
May 23, 2014 at 13:12

If I remember correctly, assuming only a homogeneous and isotropic spacetime, on top of an arbitrary group structure, the only 4D spacetime symmetries that are allowed are either

1. galileo group
2. SO(3,1) (that is Lorentz group)
3. SO(4) (that is euclidean 4D rotations).

The relevant references where this was shown are (according to link below)

• J-M. Levy-Leblond, American Journal of Physics, 44 (1976) 271.
• B. Preziosi, N.Cim. 109 B (1994) 1331.

A simpler (1+1)-dimensional version of the proof is given in the first 5 pages of these notes on general relativity and beyond: http://www.df.unipi.it/~menotti/appunti.pdf

I recently read a paper on the possible kinematics: http://scitation.aip.org/content/aip/journal/jmp/9/10/10.1063/1.1664490 It states that under the 3 assumptions they made, there were more then 10 possible Lie-Algebras (while they discarded one by heuristic arguments)