= Dependence of a Cauchy horizon on the initial hypersurface
{title2=$H^\pm(\Sigma)$}
A complete <Cauchy hypersurface> in <Minkowski spacetime>, such as $t=0$, has full <domain of dependence> and empty <Cauchy horizons>. This does not forbid horizons for a non-Cauchy initial surface in the same spacetime. The smooth edgeless spacelike hyperboloid $t=\sqrt{a^2+r^2}$, $a>0$, has domain of dependence $t>r$ and a past <Cauchy horizon> on the future light cone of the origin. Every causal curve starting inside that cone crosses the hyperboloid in the direction needed for dependence: $t^2-r^2$ passes through $a^2$, while a null generator of the cone never meets it. The time-reversed hyperboloid gives the analogous future horizon.
Back to article page
