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The future domain of dependence $D^{+}(\Sigma )$ consists of points through which every past-inextendible causal curve meets the partial Cauchy hypersurface $\Sigma$. Data on $\Sigma$ determine evolution there for suitable hyperbolic equations. Its future boundary is the Cauchy horizon $H^{+}(\Sigma )$; one precise definition is $\overline{D^{+}(\Sigma )}\setminus I^{-}(D^{+}(\Sigma ))$.

For a nonextremal Reissner-Nordstrom spacetime, $0<|Q|<M$, the metric function is $F=1-2M/r+Q^2/r^2$, with $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The outer event horizon lies at $r_{+}$. The future trapped region leads toward the inner null surface $r_{-}$, a future Cauchy horizon for bridge data on the illustrated $\Sigma$. Beyond it some past causal curves arrive from other analytic blocks or timelike singularities without meeting $\Sigma$. The CP diagram repeats these analytic blocks; $r=0$ is a timelike curvature singularity.

Near the inner horizon, arbitrarily late exterior radiation is compressed into finite infaller proper time. Its frequency is exponentially blueshifted, with scale $e^{|{\kappa }_{-}|v}$, so even a decaying flux can have diverging measured stress. Generic ingoing and outgoing perturbations produce mass inflation, growing internal mass and curvature. The smooth RN Cauchy horizon is expected to be unstable to back-reaction. This is a generic instability argument, not a universal theorem from every specially tuned test trajectory; the approximation of a harmless test particle becomes inconsistent.

Use the Null Raychaudhuri equation for $\ell =f^{-1}{\partial }_U$. Its null shear and null twist vanish, and $R_{ab}{\ell }^a{\ell }^b=f^{-2}{R}_{UU}$. Substitution of ${\theta }_{\ell }=2r_{,U}/(fr)$ givesThe quadratic terms cancel. Thus the double-null focusing identity isHere the last equality uses the Einstein field equations in units $G=1$; terms proportional to the metric vanish in this null component. The inequality follows from the null energy condition.

For any point of the future domain of dependence $D^{+}(\Sigma )$, the past continuation of its $\ell$ generator must meet the Cauchy hypersurface $\Sigma$: every past-inextendible causal curve from that point meets $\Sigma$. Along its future continuation, $U$ increases, and $f^{-1}{r}_{,U}$ is nonincreasing by the focusing identity. Its initial value is negative, so it stays negative. Since $f>0$, $r_{,U}<0$ throughout the regular part of $D^{+}(\Sigma )$. The argument is used only where the smooth double-null chart has $r>0$; it does not extend a congruence beyond a singular endpoint.

In a regular double-null region satisfying the null energy condition, an initially negative $r_{,U}$ on a Cauchy hypersurface remains negative in its future domain of dependence. If both spherical null expansions are negative at one sphere, the exchanged-coordinate focusing inequality preserves the other sign along its future outgoing null direction. Every later sphere that still exists there remains a trapped surface.