# Definition The [[Lorentz Group]] in 4 dimensions is the set of all $O(3,1)$ [[Lorentz transformations]], including ones which change the orientation of space and time. # Elements We use the metric signature $(+++-)$ so that time is the fourth component. We have the [[Inversion Operator|spatial inversion]] operator given by: $ P = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & +1 \\ \end{pmatrix} $ and the [[Time-Reversal Operator|time-eversal]] $ T = \begin{pmatrix} +1 & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{pmatrix} $ If we add these two elements to the [[Restricted Lorentz Group in 4 Dimensions|restricted Lorentz group]] $SO(3,1)_o$, we recover the entirety of $O(3,1)$. However, there must be multiple disconnected components. For starters, for an element $A$ we have a partition based on the [[determinant]] of $A$, where it's $+1$ or $-1$. However, this is not sufficient in this case as it was for $SO(3,1)$. We cannot group together the components with $I$ (i.e. $SO(3,1)_o$) and the component with $PT$ (which reverses the direction of both space and time), even though they both have $\det(A) = +1$. We must also divide based on the sign of $A_{44}$. Thus, we get: ![[Pasted image 20210703140723.png]] # In Physics Physicists often use a different metric signature, $(+---)$ is prevalent in high-energy physics for example. In this case, the Lorentz group is denoted $O(1,3)$ and things have to be shuffled a bit, but the fundamental properties clearly remain the same.