Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 57 / 4 / a / Solution

A smooth principal bundle with structure Lie group $G$ can be specified by a locally trivial map $\pi :E\to B$ whose fiber coordinate changes have the form $(x,g{)}_i\mapsto (x,h_{ji}(x)g{)}_j$, with the usual transition functions of a principal bundle cocycle on triple overlaps. In each chart define $(x,g)\cdot k=(x,gk)$. On overlaps, $h_{ji}(x)(gk)=(h_{ji}(x)g)k$, so these local right actions agree. They therefore give a global smooth right group action, free and transitive on every fiber. Equivalently, this action together with equivariant local triviality is the definition of a principal bundle.

Use five-dimensional Minkowski spacetime with ambient metric $\eta =\mathrm{diag}(-1,1,1,1,1)$ and realize unit-radius de Sitter spacetime as $\eta (x,x)=1$. Its tangent space is $x^{\perp }$, of Lorentzian signature $(3,1)$. An oriented pseudo-orthonormal frame $(e_0,e_1,e_2,e_3)$ of $x^{\perp }$, together with the last column $x$, is precisely a matrixConversely the last column of such a matrix lies on the hyperboloid, and its other columns give that frame. ThusRight multiplication by $\mathrm{diag}(h,1)$ fixes $x$ and changes the tangent frame by $h$. It is free and transitive on frames over $x$, proving the principal bundle description. Smooth local choices of frames provide its local trivializations. The dimensions are $10=4+6$.

This is the oriented frame bundle of de Sitter spacetime. Allowing orientation reversal gives $O(4,1)$ and structure group $O(3,1)$; imposing a time orientation as well selects $S{O}_0(4,1)$ and $S{O}_0(3,1)$. For radius $a$, take the last column $x/a$.