  Moitvated in part by [3], in this note we obtain a rigidity result for
globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a
timelike conformal Killing vector field. Specifically, we show that if M is a
Ricci flat, timelike geodesically complete spacetime with compact Cauchy
surfaces that admits a timelike conformal Killing field X, then M must split as
a metric product, and X must be Killing. This gives a partial proof of the
Bartnik splitting conjecture in the vacuum setting.
