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The Lorentz group has four connected components that can be characterized as follows:

  1. $\det A = 1$
  2. $\det A = -1$
  3. $A^0_0 = 1$
  4. $A^0_0 = -1$.

I think I understand the third and fourth components well, these are just the transformations that either preserve or reflect the arrow of time. I am trying to better understand the first and second component.

Do transformations 1 and 2 only affect the spatial coordinates? If so, then 2 just says that, along with a possible reflection, a parity transformation occurred and 1 says there is no parity. Can they be any more general than this? I'm confused because the identity matrix has determinant 1 but also satisfies $A_0^0 = 1$, so it would belong to both components 1 and 3 but this cannot happen since they are disconnected.

If 1 and 2 do indeed only affect spatial coordinates, how does this follow from the sign of the determinant?

Qmechanic
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CBBAM
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1 Answers1

6

You've incorrectly written down your components. They are:

  1. $\det \Lambda = +1$ and $\Lambda^0{}_0 \geq 1$: These are the proper orthochronous Lorentz transformations.
  2. $\det \Lambda = +1$ and $\Lambda^0{}_0 \leq -1$: Reverses time AND orientation of space.
  3. $\det \Lambda = -1$ and $\Lambda^0{}_0 \geq 1$: Reverses orientation of space.
  4. $\det \Lambda = -1$ and $\Lambda^0{}_0 \leq -1$: Reverses time.
Prahar
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