Earlier quoted context omitted.
If the spacetime is static, then yes, the trajectory of a light beam (or anything else) is reversible. The question you pose is an excellent one and is one of the reasons that many physicists believed that black holes could not exist after Karl Schwarzschild discovered the Schwarzschild metric that describes spacetime around a spherically symmetric black hole. So what happens if you direct a beam of light radially in…
You have this backwards. To a distant observer, light going into a black hole is red-shifted, not blue-shifted. If this were not the case, sending even a single photon into a black hole would produce infinite energy.
An analogous effect occurs for an observer stationed at rest, hovering above the event horizon. Such an observer will observe the photons from afar to be blueshifed in agreement with the distant observer. Moreover, as this stationary observer gets closer and closer to the event horizon, the photons will be blueshifted to infinitely large energies.