Teleological response of a Vaidya event horizon ($\rho (v)=\frac{1}{2M_0}{\int }_v^{\infty }{e}^{-(s-v)/(4M_0)}\mu (s)ds$)

For a small bounded mass perturbation $m(v)=M_0+\mu (v)$ that becomes constant in the future, write the event horizon radius as $r_H(v)=2M_0+\rho (v)$. The outgoing radial null geodesics of the Vaidya metric give ${\rho }^{\prime }-\rho /(4M_0)=-\mu /(2M_0)$ to first order. Future stationarity selects the displayed solution and removes the exponentially growing homogeneous solution. If ${\mu }^{\prime }\ge 0$, integration by parts gives $\rho -2\mu =2{\int }_v^{\infty }{e}^{-(s-v)/(4M_0)}{\mu }^{\prime }(s)ds\ge 0$. Thus the event horizon can grow before local matter arrives, lying outside the instantaneous spherical apparent horizon $r=2m(v)$. This is a consequence of the event horizon's global definition, not propagation of a local signal into the past.