Cauchy horizon

The future or past Cauchy horizon of a partial Cauchy hypersurface is the boundary of the region whose physical evolution is determined by data on that hypersurface. Crossing it can destroy uniqueness of the maximal globally hyperbolic development.

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.

Near a nondegenerate outgoing inner Cauchy horizon, a regular null coordinate behaves as $V_{-}\propto -e^{-{\kappa }_{-}v}$, where ${\kappa }_{-}>0$ is the magnitude of the inner-horizon surface gravity. Transforming stress components gives $T_{V_{-}{V}_{-}}\propto e^{2{\kappa }_{-}v}{T}_{vv}$. Even a small power-law tail can therefore cause an unbounded locally measured flux, motivating mass inflation.

Mass inflation is the growth of the interior effective mass and curvature near an inner Cauchy horizon under suitably perturbed black-hole data. It can obstruct regular spacetime extensions that exist for an exact Reissner-Nordstrom spacetime. The extension regularity matters: curvature blowup alone does not rule out every continuous-metric extension.