Proper Lorentz Group

up:: Lorentz Group

Note that Lorentz Transformations $\Lambda$ satisfy

$$\Lambda \in O(1,3)⟹\det (\Lambda {)}^2=1$$

The transformations with positive determinant belong to the proper Lorentz Group $SO(1,3)$. This amounts to saying, for a $x$-boost, that

$${\gamma }^2-{\gamma }^2{\beta }^2=\pm 1⟺{\gamma }^2=\frac{\pm 1}{1-{\beta }^2}$$

whose negative version is not physically sound.

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References

• Special Relativity - David Tong