The *Lorentz Group* $O(1,3)$ (or O(3,1)) is the group of all linear transformations of $\mathbb{R}^4$ (or 4 x 4 real matrices) that preserve the Lorentz inner product

where $g = diag(1,-1,-1,-1)$.

The group $O(1,3)$ has four connected components, determined by the values of 2 homomorphisms from $O(1,3)$ into the 2 element group $\{\pm 1\}$:

(2)and

(3)**Definition**: The **kernel** of a group homomorphism $f: G\rightarrow G^{'}$ is the set of all elements of $G$ which are mapped to the identity element of $G^{'}$.

The kernel of the first of these is the *special Lorentz Group $SO(1,3)$* and the kernel of the second group is the orthochronous Lorentz group $O^{\dagger}(1,3)$ — the subgroup of $O(1,3)$ that preserves the direction of time.

The intersection

(4)is called the *restricted Lorentz group* or *proper Lorentz group* and is the connected component of the identity element in $O(1,3)$.

**Under Construction**