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I'm studying actions on Minkowski space ($\mathbb R^4$) in the context of the Lorentz transformations. In particular, the Lorentz group is the group of all linear endomorphisms on $\mathbb R^4$ that preserves the quadratic form $$(t,x,y,z) \mapsto t^2-x^2-y^2-z^2$$ $\forall x,y,z,t \in \mathbb R$. This group is shown to be equivalent to the generalised orthogonal group $O(1,3)$ where $(1,3)$ is the signature of the quadratic form. The identity component of the Lorentz group is denoted $SO^+(1,3)$ and is called the restricted Lorentz group. This group consists of the Lorentz transformations that preserve the orientation of space and the direction of time. What I am trying to understand is how these two groups are different. Which transformations would be in the group $O(1,3)$ but not in $SO^+(1,3)$? In particular, I am trying to construct a general isomorphism from the Mobius group to the Lorentz group but it seems that this is only possible for the restricted Lorentz group.

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Some elements not in $\mathrm{SO}^+(1,3)$:

  • $\mathrm{diag}(-1,1,1,1)$, reverses the direction of time
  • $\mathrm{diag}(1,-1,-1,-1)$ reverses the orientation of space
  • $\mathrm{diag}(-1,-1,-1,-1)$ reverses both

These represent the three classes of nontrivial elements in

$$\pi_0(\mathrm{O}(1,3))\cong \mathrm{O}(1,3)/\mathrm{SO}^+(1,3) \cong \mathbb{Z}_2\times\mathbb{Z}_2.$$

This is all on Wikipedia.

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