It is commonly believed that a black hole is a merciless cosmic vacuum cleaner, sucking in everything indiscriminately. But new research flips the picture: right at the horizon, curvature transforms from a trap into a trampoline. Analysis of the Fokker–Planck equation in the Schwarzschild–de Sitter metric revealed an "anti-capture" paradox: a dust grain is more likely to head for the cosmological horizon than to fall into the black hole.

The secret lies in how the "fabric" of the radial coordinate is curved. Near the horizon, the metric factor f(r) tends to zero, and radial cells stretch enormously, as if each step of the particle becomes a giant leap. This gives the dust grain more space for random walks, and geometric drift often steers it into a safe orbit. Even starting from a distance of five gravitational radii, the particle has only a 20% chance of capture. In flat space, the same starting position would mean almost certain doom.

The accretion orchestra also changes its tune. Numerical simulation of millions of trajectories launching from the Sunyaev–Shakura disk showed that time dilation acts as a nonlinear filter, making the power spectrum remarkably flat at low frequencies (slope around −0.2) and steeply cutting off at high frequencies (down to −3). In a flat model, the slope stays at −2 everywhere. This matches perfectly the observed "red noise" of Cygnus X-1, where the low-frequency index is indeed shallower than minus one.

The geometric approach turns curvature from a passive background into an active conductor of a stochastic symphony. It provides a new diagnostic tool: from time series, one can test strong fields without directly resolving the horizon. In the future, extending to the Kerr metric will add frame-dragging, and including magnetic fields will allow building a universal accretion model for any compact objects. The irony is that the strongest gravity in the Universe does not swallow matter but repels it — perhaps nature thus limits the growth of black holes, avoiding singularities. This is no longer just accretion physics: it is a way to peer into the very essence of how spacetime governs matter.