Persistence of spherical trapping

Consider a regular four-dimensional region with spherical symmetry and two null coordinates,with both ${\partial }_U$ and ${\partial }_V$ future directed. Assume the Einstein field equations and null energy condition, so $T_{UU},T_{VV}\ge 0$.

Suppose $r_{,U}<0$ on an initial Cauchy hypersurface. In the portion of its future domain of dependence whose past constant-$V$ null segments reach that hypersurface within the regular coordinate region, let their initial coordinate be $U_{\Sigma }(V)$. The double-null focusing identity integrates toThus the ingoing spherical null expansion remains negative there.

Now take a sphere $(U_0,V_0)$ with the other expansion negative as well. Along its future outgoing null hypersurface $U=U_0$,Both spherical null expansions in double-null coordinates are therefore negative on every surviving sphere of that outgoing null hypersurface, so those spheres remain trapped surfaces. The result applies while the stated regularity and domain conditions hold.

Each focusing identity controls variation along its own null direction. A claim about every sphere in a whole two-dimensional future region needs an additional argument controlling the transverse direction; the two separate inequalities above establish the stated generator-by-generator persistence.