Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 26I / c / ii / Solution

Consider the smooth mapIts derivative isAt $A\in O(1,3)$ write $H=AY$. ThenThis derivative is onto the ten-dimensional space of symmetric matrices: for a symmetric $S$, taking $Y=\frac{1}{2}MS$ gives $Y^{T}M+MY=S$. Hence $M$ is a regular value. Since $M_4(\mathbb{R})$ has dimension 16, the regular level set theorem gives

The tangent vectors are exactly those for which the derivative vanishes:Writing $Y$ in one-plus-three block form shows explicitly thatThe three entries of $v$ and three independent entries of the skew-symmetric matrix $\Omega$ again display dimension six. This is the Lie algebra of the Lorentz group.