What is the generic form for a proper Lorentz transformation?

The Proper Lorentz transformations are all matrix $\Lambda$ such that $\eta=\Lambda^T\eta\Lambda$ and $\det\Lambda=1$ with $\eta=\text{diag}(-1,1,1,1)$ the Minkowskian metric. $\textit{i,e}$, $\Lambda$ are the group $SO(3,1)$. This is a continuous group with six parameters and hence six generators. Three for the Lorentz boosts and three for spacial-rotations. Let $\vec B$ the generators for the boosts and $\vec R$ the generators for rotations.

A generic rotation in an axis $\hat n$ and an angle $\alpha$ is given by $$\Lambda(0,\vec\alpha)=\exp(\alpha\hat n\cdot \vec R)=\pmb I+\sin\alpha(\hat n\cdot \vec R)+(1-\cos\alpha)(\hat n\cdot \vec R)^2.$$

For a generic boost to a inertial frame with velocity $\vec v$ is given by $$\Lambda (\vec\beta,0)=\exp(-\tanh^{-1}(\beta)\;\;\hat\beta\cdot \vec B)=\pmb I-\beta\gamma (\hat\beta\cdot \vec B)+(\gamma-1) (\hat\beta\cdot \vec B)^2$$ with $\vec\beta=\beta\hat\beta=\vec v/c$.

My question is what the generic matrix for both, rotations and boost, is ? $$\Lambda(\vec\beta,\vec \alpha)=\exp(-\tanh^{-1}(\beta)\hat\beta\cdot\vec B+\alpha\hat n\cdot \vec R)=?$$

The first and second expressions are proved showing that $(\hat n\cdot \vec R)=-(\hat n\cdot \vec R)^3$ and $(\hat\beta\cdot \vec B)=(\hat \beta\cdot B)^3$ respectively.

The main problem is that $$e^{\pmb A+\pmb B}\not =e^{\pmb A}e^{\pmb B},$$ when $[\pmb A,\pmb B]\not =0$. And $[R_i,B_j]=-\epsilon_{ij}^{\;\; k}B_k\not=0$, can you help me?

• Yes, the commutation relations are less trivial and so writing expanded but simple expressions as you have for boost or rotations becomes more convoluted. But it is possible to write a generic proper Lorentz transformation as the exponentiation of the six generators multiplied by parameters in a similar fashion to what you wrote and you can also reduce it to the composition of a boost in some direction times a rotation if you constrain the parameters in a specific way. Take a look of, for instance, equations (10.1-15) and (10.2-13) in Tung's book Group Theory in Physics. Jan 12, 2018 at 15:05
• It looks like you're looking for a particular generic form, and not just any. You can produce a generic form, for example, by just multiplying a generic boost by a generic rotation matrix. Jan 12, 2018 at 15:06
• FWIW, the exponential map for the restricted Lorentz group is surjective, cf. e.g. this Phys.SE post. Jan 12, 2018 at 16:27
• For a similar discussion related to $\mathrm{SO}(n)$ see physics.stackexchange.com/q/377001 and links therein. I link this because of the way that you can get Lorentz boosts by making one angle imaginary. Jan 12, 2018 at 16:36