Time does not "cease to exist" in a black hole, but the nature of time is profoundly affected by the extreme gravitational forces within a black hole. To understand this, let's explore the concept of time in the context of general relativity.
According to general relativity, the presence of massive objects curves the fabric of spacetime, creating what we perceive as gravity. In the vicinity of a black hole, the gravitational forces are incredibly intense due to the concentration of mass within a small volume. As you approach the event horizon—the boundary beyond which nothing can escape the gravitational pull of a black hole—the gravitational effects become increasingly extreme.
Near the event horizon, the gravitational time dilation becomes significant. Time dilation means that time appears to pass more slowly in regions of stronger gravitational fields compared to regions with weaker gravitational fields. As you move closer to the event horizon, the gravitational time dilation becomes more pronounced.
At the event horizon itself, the time dilation effect becomes infinite. This means that an observer located outside the event horizon would perceive time for an object near the event horizon as infinitely slowed down. However, this does not mean that time "stops" or "ceases to exist." It simply means that from an external observer's perspective, the object near the event horizon appears to be frozen in time.
Inside the event horizon, space and time swap roles, and all paths lead toward the central singularity of the black hole. At this point, the curvature of spacetime becomes so extreme that any object or information that enters the black hole will inevitably be drawn towards the singularity. As a result, it becomes impossible to describe the behavior of time or anything else within the black hole using our current understanding of physics. The singularity represents a breakdown in our current theories and the point at which our understanding reaches its limits.
In summary, time does not "cease to exist" in a black hole, but its behavior is significantly altered due to the extreme gravitational effects near and within the event horizon.