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This region $0\le r<R$ of de Sitter spacetime is static for the central observer, with a cosmological Killing horizon at $r=R$. The central proper time fixes the normalization of ${\partial }_t$. Static observers have acceleration $r/(R^2\sqrt{1-r^2/R^2})$. Horizon-crossing null coordinates remove the coordinate singularity, while the global extension has compact spherical spatial slices and spacelike past/future conformal infinity.

A timelike geodesic has timelike tangent and vanishing covariant acceleration in an affine parameter. Choosing proper time gives $g(\dot{\gamma },\dot{\gamma })=-1$.

The twin paradox compares proper time along the different world lines of a travelling observer and a stay-at-home observer who reunite. Reciprocal inertial time dilation produces an apparent contradiction only if the traveller's change of inertial frame is ignored.