Maximality of the Einstein–Maxwell GHD does not assert geodesic completeness

For the maximal smooth globally hyperbolic development in Source-free smooth maximal globally hyperbolic Einstein–Maxwell development, maximality means that every development of the same complete source-free data admits a data-fixing development embedding into it, in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality. By The geodesic initial-value problem and the exponential map of the metric-selected connection, an affinely parametrized geodesic for its metric-selected connection is a smooth curve $\gamma :I\to M$ on an open interval $I\subseteq \mathbb{R}$ satisfying $D_{\lambda }\dot{\gamma }=0$; in coordinates $y^{\mu }$ this is ${\ddot{y}}^{\mu }+{\mathrm{\Gamma }}^{\mu }{}_{\alpha \beta }(y){\dot{y}}^{\alpha }{\dot{y}}^{\beta }=0$. That theorem supplies a unique maximal solution ${\gamma }_{p,v}:I_{p,v}\to M$ through each initial datum $(p,v)$, with $I_{p,v}$ an open interval; here inextendible means that the curve is such a maximal solution, so no geodesic on a strictly larger interval with the same affine parameter restricts to it. Geodesic completeness means that every such maximal geodesic has parameter domain all of $\mathbb{R}$. The maximality conclusion by itself asserts no geodesic-completeness property. This remark does not claim that a particular Einstein–Maxwell maximal development is geodesically incomplete.