  We show that the causal properties of asymptotically flat spacetimes depend
on their dimensionality: while the time-like future of any point in the past
conformal infinity $\mathcal{I}^-$ contains the whole of the future conformal
infinity $\mathcal{I}^+$ in $(2+1)$ and $(3+1)$ dimensional Schwarzschild
spacetimes, this property (which we call the Penrose property) does not hold
for $(d+1)$ dimensional Schwarzschild if $d>3$. We also show that the Penrose
property holds for the Kerr solution in $(3+1)$ dimensions, and discuss the
connection with scattering theory in the presence of positive mass.
