Wednesday, January 21, 2015

special relativity - Is there a general theorem stating why the restricted Lorentz group's exponential map is surjective?


The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group.


Is there a more general theorem that states that for some class of Lie groups or Riemannian manifolds (which includes the restricted Lorentz group), the exponential map is surjective?


There is a theorem stating that compact, connected Lie groups have surjective exponential maps. But as the restricted Lorentz group is not compact, this isn't applicable.



Answer




Comments to the question (v2):




  1. The consensus in the literature seems to be that the surjectivity of the exponential map exp:so(1,d;R)SO+(1,d;R) for the restricted Lorentz group for general spacetime dimensions D=d+1 does not have a short proof.




    • The case d=1 is trivial.




    • The case d=2 can be proved via the isomorphism SO+(1,2;R)SL(2,R)/Z2, cf. e.g. this Phys.SE post.





    • The case d=3 can be proved via the isomorphism SO+(1,3;R)SL(2,C)/Z2, cf. e.g. Wikipedia and this Phys.SE post.






  2. Already the exponential map exp:sl(2,R)SL(2,R) is not surjective, cf. e.g. this MO.SE answer and this Phys.SE post. Note that the Lie algebras so(1,2;R)  sl(2,R) are isomorphic, but only the Lie group SO+(1,3;R) for the lefthand-side of the isomorphism (2) has a surjective exponential map; not the Lie group SL(2,R) for the righthand-side. A counterexample such as (2) undoubtedly makes it delicate to try to formulate a generalization of (1) beyond the restricted Lorentz groups SO+(1,d;R) and case-by-case-proofs. See also this Math.SE post.





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