Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 37E (General Relativity)

For the Expanding flat slicing of de Sitter spacetime, take the geodesic LagrangianThe Euler-Lagrange equation for $t$ iswhile each spatial coordinate $x^i$ obeysComparing these equations with the geodesic equation gives the complete list of nonzero Christoffel symbols:

Because $\tau$ is proper time, the four-velocity has normalizationAt $t=0$, the initial data therefore giveBoth $t$ and $\tau$ are future oriented, so the positive root must be chosen:

The $x$ Euler-Lagrange equation has the first integralwhere the constant follows from the initial data. Combining this with proper-time normalization gives the Timelike geodesic in the expanding flat slicing of de Sitter spacetime:ConsequentlyIntegrating from $t=0$, where $\tau =0$, givesAs $t\to -\infty$, the first ratio tends to $1$, soThus $t$ tends to $-\infty$ at the finite proper time ${\tau }_{-}$.

This is the Past incompleteness of the expanding flat slicing of de Sitter spacetime. The coordinate patch covers only half of global de Sitter spacetime, and the surface $t=-\infty$ is its past boundary. The finite proper-time endpoint is a coordinate-patch boundary, not a curvature singularity: the timelike geodesic continues smoothly after passing to global de Sitter coordinates.