By the proper Lorentz group I mean the path connected component of the Lorentz group (I am considering the usual 3+1 space-time case), i.e., the set of elements which can be continuously linked to the identity element.
I suspect that the only homomorphism from the proper Lorentz group to the $U(1)$ group is the trivial map.
Is it true? If so, how to prove it?