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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?

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  • $\begingroup$ 1 and 2 affect all coordinates. It means that the orientation (the same concept as in differential geometry) of your orthonormal basis was changed (in case 2.) or not changed (in case 1.) by the Lorentz transformation. 3. 4. mean that the "arrow of time" stays the same/ is reversed in the new orthonormal basis. $\endgroup$
    – jd27
    Dec 2, 2023 at 8:20
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    $\begingroup$ You also have the components wrong i think (which explains your last point). The components are 1. & 3., 1. & 4., 2. & 3., 2. & 4. according to this wikipedia article. $\endgroup$
    – jd27
    Dec 2, 2023 at 8:35
  • $\begingroup$ @jd27 Thank you! $\endgroup$
    – CBBAM
    Dec 2, 2023 at 18:00

1 Answer 1

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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.
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  • $\begingroup$ Thank you, this helped clear all my confusion. $\endgroup$
    – CBBAM
    Dec 2, 2023 at 18:00

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