When an object, such as a photon, approaches the event horizon of a black hole, it experiences significant time dilation from the perspective of a distant observer. This means that, as observed from afar, the photon's perceived time appears to slow down relative to the observer's time.
However, it's crucial to understand that this time dilation does not imply that the photon itself is experiencing a change in its own reference frame. From the photon's perspective, it always travels at the speed of light and does not experience time in the same way as objects with mass do. Photons are massless particles that always travel at the speed of light in a vacuum.
When a photon crosses the event horizon, according to our current understanding, it is inexorably drawn toward the singularity at the center of the black hole. From an external observer's point of view, the photon's time dilation continues to increase as it gets closer to the event horizon, to the point where it appears to take an infinite amount of time for the photon to reach the observer's location. This is due to the extreme gravitational time dilation near the black hole.
However, it's worth noting that this perspective is based on classical general relativity, which does not account for quantum effects or the behavior of particles at extremely high energies and small scales. Our understanding of the behavior of photons and matter near the event horizon and within the singularity requires a theory of quantum gravity, which remains an area of active research and speculation.
In summary, photons, being massless particles, always travel at the speed of light and do not experience time in the same way as objects with mass. While an external observer perceives a photon's time as slowing down near the event horizon, from the photon's perspective, it continues to move at the speed of light.