= Solution
The positive <conformal factor> leaves the <causal structure> unchanged. In the strip $-\pi/2<\eta<\pi/2$, a future-directed <causal curve> obeys
$$
\left|\frac{d\xi}{d\eta}\right|\leq1,
$$
and <null geodesics> have slopes $\pm1$. Its past and future conformal boundaries $\eta=\mp\pi/2$ are spacelike. A finite <conformal time> interval limits the range of light signals: the observer at $\xi=0$ can receive a signal from $(\eta,\xi)$ before future infinity exactly when $|\xi|<\pi/2-\eta$. Equality describes its observer-dependent <cosmological event horizon>.
For a finite <Penrose diagram> of the unwrapped spatial coordinate, set $U=\arctan u$, $V=\arctan v$, $X=(U+V)/2$, and $Y=(V-U)/2$. Then the metric is conformal to $-dY^2+dX^2$; light rays remain at $45$ degrees, and the boundaries are the images of $v-u=\pm\pi$. The figure also shows the spatially periodic de Sitter cylinder so that the global distinction is explicit.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-58-causal-diagram.png]
{title=Causal compactification of the unwrapped de Sitter strip and the spatially periodic de Sitter cylinder}
\b[Both the covering spacetime and the de Sitter cylinder are globally hyperbolic.] To see this directly in the covering spacetime, suppose an inextendible future-directed <causal curve> had a future limiting conformal time strictly below $\pi/2$. The slope bound makes $\xi$ Cauchy as that time is approached, so the curve has a limiting point inside the spacetime and can be extended, a contradiction. The past endpoint similarly approaches $-\pi/2$. Thus every constant-$\eta$ line is crossed exactly once and is a <Cauchy hypersurface>. On the cylinder the same argument uses compactness of the spatial circle. Equivalently causal diamonds between finite events are bounded and closed within the strip, hence compact.
Initial field data on a complete <Cauchy hypersurface> determine a unique solution for a well-posed hyperbolic <Cauchy problem>, with the relevant constraints imposed for constrained systems. Spacelike conformal infinity does not require additional timelike-boundary conditions. An observer's cosmological horizon limits which data that observer can see; it does not destroy global uniqueness from data on the whole hypersurface.
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