On the conformal group of a globally hyperbolic spacetime
We study causal and conformal automorphism groups of globally hyperbolic spacetimes using an order-theoretic back-and-forth method on dense countable subsets. In two dimensions we show that any connected, globally hyperbolic spacetime with non-compact Cauchy surfaces that is directed is causally isomorphic to the Minkowski plane $\mathbb{M}^2$. Consequently, we obtain a partial classification of the causal and conformal automorphism groups of two-dimensional globally hyperbolic spacetimes, including the cases with compact Cauchy surfaces and non-directed causal order. The directed non-compact case is handled by refining the dense back-and-forth construction with the two intrinsic null orders, which record the two spacelike sides forgotten by bare causal incomparability. On the physics side, the resulting symmetry descriptions can be read as a factorized-versus-matched action of large reparametrization groups on null-type completion boundaries, illustrated by moving mirrors, conformal interfaces, and FLRW toy models.