# Minkowski as a Quotient Space

I've read many times, in many articles or books that one can obtain the four dimensional Minkowski space $\mathbb{M}^4$ as the quotient space $$ISO(3,1)/SO(3,1),$$ and equivalently de Sitter and Anti-de Sitter can be seen as $SO(4,1)/SO(3,1)$ and $SO(3,2)/SO(3,1)$, respectively. I.e. we can see these manifolds by taking their symmetry Lie group and quotienting by the normal Lorentz subgroup $SO(3,1)$.

This construction seems intuitive but I cannot find a decent reference where these steps are taken with a little more care, I don't mean "Math-rigour" but just "Theoretical Physics-rigour": I would like to see $\mathbb{M}^4$ emerging from explicit group theoretical calculations (I'm familiar with group actions, orbits and all that).

Can anyone provide a nice and specific explanation without referencing to general theorems but starting from general grounds? Or just a good, not too mathy but still clear, reference?

• This is just an application of the general theorem that, when given a transitive group action of a Lie group, the quotient of the Lie group by the stabilizer of a point is a manifold. Alternatively, you can think of the spaces you act on as being manifolds a priori, and obtain an equivariant diffeomorphism between $G/H$ and $M$, where $G$ is the Lie group, $H$ is the stabilizer of a point and $M$ is the manifold acted upon. The proof of this theorem is not easy; it relies on the closed subgroup theorem and related machinery. – Danu Aug 27 '17 at 11:12
• You can work out for yourself why $SO(3,1)$ is the stabilizer of a point in Minkowski space, probably. – Danu Aug 27 '17 at 12:37
• @Danu can you provide someplace to read these things? Thanks – Boltzee Aug 27 '17 at 17:46
• @bgr95 I learned it from here. – Danu Aug 27 '17 at 18:55
• Since you ask for a reference, try the old paper by E. Ivanov and J. Niederle, Phys. Rev. D 25 (1982) 976. They construct GR as an ISO(3,1) gauge theory on the ISO(3,1)/SO(3,1) coset space. – Tomáš Brauner Aug 30 '17 at 15:47