We present the results of an investigation of the speed of a radially infalling test particle crossing the event horizon of a black hole within a Schwarzschild spacetime. One finds that the speed as measured by a special class of observers, at rest outside the horizon and static inside the horizon, increases when the test particle approaches the horizon but decreases inside the horizon. The corresponding situation regarding black holes possessing both outer and inner horizons is also briefly discussed.

Three types of phenomena occurring on both sides of the event horizon of spherically symmetric black holes are analyzed and discussed here. These phenomena are: a light ray orbiting a photon sphere and its analogue, the motion of a uniformly accelerated massive particle and a generalized Doppler effect. The results illustrate how the anisotropic dynamics of the interior of black holes, distinct in the cases both with and without an additional internal horizon, affect non-quantum behaviour.

We consider gravity in the system consisting of the Bogomol’nyi-Prasad-Sommerfield (BPS) $D3$-brane embedded in a flat background geometry, produced by the solutions of supergravity. The effective action for this system is represented by the sum of the Hilbert-Einstein and Dirac-Born-Infeld actions. We derive the Wheeler-DeWitt equation for this system and obtain analytical solutions in some special cases. We also calculate the tunneling probability from the Planckian size of the $D3$-brane to the classical regime.

A black hole in a Schwarzschild spacetime is considered. A transformation is proposed that describes the relationship between the coordinate systems exterior and interior to an event horizon. The application of this transformation permits considerations of the (a)symmetry of a range of phenomena taking place on both sides of the event horizon. The paper investigates two distinct problems of a uniformly accelerated particle. In one of these, although the equations of motion are the same in the regions on both sides, the solutions turn out to be very different. This manifests the differences of the properties of these two ranges.