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.