Lorentz Group

up:: 032 MOC Relativity

The Lorentz Group, denoted $O(1,3)$, is composed of all transformations in Minkowski space which preserve the Invariant Interval (Relativity).

That is, for any Minkowski Metric $g_{\mu \nu }$, we have that $\Lambda$ is a Lorentz Transformation if

$${\Lambda }^{T}g\Lambda =g⟺{\Lambda }_{\mu }^{\alpha }{g}_{\alpha \beta }{\Lambda }_{\nu }^{\beta }=g_{\mu \nu }$$

$O(1,3)$ is a group

Let ${\Lambda }_1,{\Lambda }_2\in O(1,3)$. Then

$$({\Lambda }_1{\Lambda }_2{)}^{T}g({\Lambda }_1{\Lambda }_2)=g$$

Trivially the identity matrix $1$ belongs to $O(1,3)$.

Given $\Lambda \in O(1,3)$, its inverse ${\Lambda }^{-1}$ also satisfies this condition:

$$\begin{array}{rl}({\Lambda }^{-1}{)}^{T}g{\Lambda }^{-1}& =({\Lambda }^{-1}{)}^T({\Lambda }^{T}g\Lambda ){\Lambda }^{-1}\\ & =(\Lambda {\Lambda }^{-1}{)}^{T}g(\Lambda {\Lambda }^{-1})\\ & =g\end{array}$$

Properties

• Rotations belong to the Lorentz Group, since we have that they preserve distances (since they are Orthogonal Transformations, satisfying $R^{T}R=1$) and, thus, trivially preserve the invariant interval
  • Thus, the dimension of $O(1,3)$ is $6$: $3$ independent rotations and $3$ independent boosts
• Proper Lorentz Group $SO(1,3)$
• Restricted Lorentz Group $S{O}^{+}(1,3)$

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References

• Special Relativity - David Tong