An excerpt from my lecture notes on relativity (translated from Dutch):
“Special (special in the notes indicates that the determinant of the representation matrix equals +1) Lorentz transformations with arbitrary velocities don’t form a group. Special Lorentz transformations form a group (an Abelian subgroup of the Lorentzgroup) only when the boosts are parallel.”
I don’t see this. Why don’t arbitrary boosts in arbitrary directions form a group? What criterion (closed under the operation, existence of the inverse, containing neutral element etc) of forming a group is not satisfied here?
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asked Jan 19 at 21:40
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Why don’t arbitrary boosts in arbitrary directions form a group?
They don’t form a group because the composition of two non-parallel boosts is not a boost. Instead, the composition is a boost combined with a rotation. This rotation is known as a Wigner rotation.
See my answer to a related question for an explicit example.
The best way to understand the origin of the Wigner rotation is by considering infinitesimal Lorentz transformations and their generators. As mentioned here, the commutator of two boost generators is a rotation generator.
The “special Lorentz transformations”, which are those having a determinant equal to 1, include boosts, rotations, and compositions of these, and do form a group.
answered Jan 19 at 21:52
G. SmithG. Smith
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