the universe is a \(4\)-dimensional Lorentzian manifold \((\mathcal{M},g_{\mu\nu})\). i trace out a worldline, defined by a smooth timelike curve \(\gamma : \mathbb{R} \to \mathcal{M}\). i further assume that you trace out a worldline as well, which we shall call \(\eta : \mathbb{R} \to \mathcal{M}\). our goal is to construct a pair of points \(\tau, \tau^{\prime} \in \mathbb{R}\) such that \(\gamma(\tau) = \eta(\tau^{\prime})\)