# Definition The [[Restricted Lorentz Group]] in 4 dimensions $SO(3,1)_o$ is the group of [[boost|boosts]] and [[Rotation|rotations]] which preserve the orientation of space and time (i.e. the restricted [[Lorentz Transformation|Lorentz transformations]]). It is defined by elements of $O(3,1)$ which additionally have unit determinant and $A_{44} > 0$, where we use the metric signature $(+++-)$ so that time is the fourth component. # Elements The most familiar [[group element|element]] of $SO(3,1)_o$ is: $ L=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \gamma & -\beta \gamma \\ 0 & 0 & -\beta \gamma & \gamma \end{array}\right)-1<\beta<1, \gamma \equiv \frac{1}{\sqrt{1-\beta^{2}}} $ which can be interpreted [[Active and Passive Transformations|passively]] as a [[boost]] to a new reference frame that is unrotated but moving with a velocity $\beta$ with respect to the old reference frame. We can rewrite this as: $ L=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cosh u & -\sinh u \\ 0 & 0 & -\sinh u & \cosh u \end{array}\right), \quad u \in \mathbb{R} $ where $\tanh u = \beta$ and $u$ is the rapidity. An arbitrary Lorentz transformation takes the form: $ L=\left(\begin{array}{cccc} \frac{\beta_{x}^{2}(\gamma-1)}{\beta^{2}}+1 & \frac{\beta_{x} \beta_{y}(\gamma-1)}{\beta^{2}} & \frac{\beta_{x} \beta_{z}(\gamma-1)}{\beta^{2}} & -\beta_{x} \gamma \\ \frac{\beta_{y} \beta_{x}(\gamma-1)}{\beta^{2}} & \frac{\beta_{y}^{2}(\gamma-1)}{\beta^{2}}+1 & \frac{\beta_{y} \beta_{z}(\gamma-1)}{\beta^{2}} & -\beta_{y} \gamma \\ \frac{\beta_{z} \beta_{x}(\gamma-1)}{\beta^{2}} & \frac{\beta_{z} \beta_{y}(\gamma-1)}{\beta^{2}} & \frac{\beta_{z}^{2}(\gamma-1)}{\beta^{2}}+1 & -\beta_{z} \gamma \\ -\beta_{x} \gamma & -\beta_{y} \gamma & -\beta_{z} \gamma & \gamma \end{array}\right) $ or generalizing the rapidity to $ \pmb{\beta} = \frac{\tanh u}{u} \pmb{u} $ then we get: $ L=\left(\begin{array}{cccc} \frac{u_{x}^{2}(\cosh u-1)}{u^{2}}+1 & \frac{u_{x} u_{y}(\cosh u-1)}{u^{2}} & \frac{u_{x} u_{z}(\cosh u-1)}{u^{2}} & -\frac{u_{x}}{u} \sinh u \\ \frac{u_{y} u_{x}(\cosh u-1)}{u^{2}} & \frac{u_{y}^{2}(\cosh u-1)}{u^{2}}+1 & \frac{u_{y} u_{z}(\cosh u-1)}{u^{2}} & -\frac{u_{y}}{u} \sinh u \\ \frac{u_{z} u_{x}(\cosh u-1)}{u^{2}} & \frac{u_{z} u_{y}(\cosh u-1)}{u^{2}} & \frac{u_{z}^{2}(\cosh u-1)}{u^{2}}+1&-\frac{u_{z}}{u} \sinh u \\ -\frac{u_{x}}{u} \sinh u & -\frac{u_{y}}{u} \sinh u & -\frac{u_{z}}{u} \sinh u & \cosh u \end{array}\right) $ Note that the set of [[boost|boosts]] is not closed under [[matrix]] multiplication, so it does not form a [[subgroup]]. However, as we already know, [[Rotation|rotations]] *are* closed under matrix multiplication, so they do form a subgroup. In fact, $SO(3) \subset SO(3,1)_o$. Moreover, any [[Group Element|element]] in $A \in SO(3,1)$ can be decomposed as a rotation and a boost as follows: $ A = L R' $ where $ R' = \begin{pmatrix} R &0\\ 0& 1 \end{pmatrix} $ such that $R \in SO(3)$. # Lie Algebra Recalling that $\mathfrak{so}(3,1) = \mathfrak{o}(3,1)$ and from [[Lorentz Group#Lie Algebra mathfrak o n-1 1|this discussion]], we know that an arbitrary element can be written as: $ \begin{pmatrix} X' & \pmb{a}\\ \pmb{a}& 0 \end{pmatrix}, \quad X' \in \mathfrak{so}(3), a \in \mathbb{R}^3 $ Then we can write down the following six generators: $ \tilde{L}_i \equiv \begin{pmatrix} L_i & \pmb{0} \\ \pmb{0} & 0 \end{pmatrix} $ $ K_{1} \equiv\left(\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right), \quad K_{2} \equiv\left(\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right), K_{3} \equiv\left(\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right) $ where the $L_i$ are the angular momentum matrices described [[Special Orthogonal Group in 3 Dimensions#Lie Algebra|here]]. Observe the following [[commutator|commutation]] relations: $ \begin{aligned} &{\left[\tilde{L}_{i}, \tilde{L}_{j}\right]=\sum_{k=1}^{3} \epsilon_{i j k} \tilde{L}_{k}} \\ &{\left[\tilde{L}_{i}, K_{j}\right]=\sum_{k=1}^{3} \epsilon_{i j k} K_{k}} \\ &{\left[K_{i}, K_{j}\right]=-\sum_{k=1}^{3} \epsilon_{i j k} \tilde{L}_{k} .} \end{aligned} $ The $K_i$ can be interpreted as generating boosts along their axes. The last commutation relation means that, if we perform boost 1 then boost 2 (along different axes) or perform boost 2 then boost 1, the different between these 2 situations is a *[[rotation]]*. By the [[Baker-Campbell-Hausdorff formula]], one can conclude that the product of two boosts is not necessarily a boost (it is generally a combination of boosts and rotations).