- #1
parton
- 83
- 1
Hi!
I was wondering why it is possible to write any proper orthochronous Lorentz transformation as an exponential of an element of its Lie-Algebra, i.e.,
[tex] \Lambda = \exp(X), [/tex]
where [itex] \Lambda \in SO^{+}(1,3) [/itex] and X is an element of the Lie Algebra.
I know that in case for compact, connected Lie groups, every element of the group can be expressed by the exponential of an element of the corresponding Lie algebra. But this is not necessarily true for non-compact groups.
In case of [itex] SO^{+}(1,3) [/itex] we are dealing with the connected component of the full Lorentz group. But it is non-compact. So why is it nevertheless possible to express every element of [itex] SO^{+}(1,3) [/itex] by an exponential of an element of the Lie algebra?
I hope someone could help me understanding that.
I was wondering why it is possible to write any proper orthochronous Lorentz transformation as an exponential of an element of its Lie-Algebra, i.e.,
[tex] \Lambda = \exp(X), [/tex]
where [itex] \Lambda \in SO^{+}(1,3) [/itex] and X is an element of the Lie Algebra.
I know that in case for compact, connected Lie groups, every element of the group can be expressed by the exponential of an element of the corresponding Lie algebra. But this is not necessarily true for non-compact groups.
In case of [itex] SO^{+}(1,3) [/itex] we are dealing with the connected component of the full Lorentz group. But it is non-compact. So why is it nevertheless possible to express every element of [itex] SO^{+}(1,3) [/itex] by an exponential of an element of the Lie algebra?
I hope someone could help me understanding that.