= Teleological response of a Vaidya event horizon
{title2=$\rho(v)=\frac1{2M_0}\int_v^\infty e^{-(s-v)/(4M_0)}\mu(s)\,ds$}
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 $\rho'-\rho/(4M_0)=-\mu/(2M_0)$ to first order. Future stationarity selects the displayed solution and removes the exponentially growing homogeneous solution. If $\mu'\geq0$, integration by parts gives $\rho-2\mu=2\int_v^\infty e^{-(s-v)/(4M_0)}\mu'(s)\,ds\geq0$. 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.
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