When we say time slows down for an observer in the context of time dilation, we mean that the rate at which time passes for that observer appears to be slower compared to an observer in a different frame of reference. This phenomenon arises due to the relative motion between the observers and is a consequence of the principles of special relativity.
According to special relativity, the speed of light is constant in all inertial frames of reference. However, the concept of simultaneity is relative, meaning that events that appear simultaneous to one observer may not be simultaneous to another observer in relative motion. This leads to the dilation of time.
Let's consider a scenario where two observers, one stationary (called the "proper frame") and another in motion relative to the first, are observing a clock. From the perspective of the stationary observer, the clock in motion is ticking more slowly. This is because, from the stationary observer's frame, the moving clock has to cover a greater spatial distance for each tick, as it is moving away.
From the perspective of the moving observer, however, they are at rest in their own frame of reference, and it is the stationary observer who appears to be moving. Thus, the moving observer perceives their own time as passing normally, while the stationary observer's time appears to be dilated or slowed down.
The key point is that both observers will agree on the measured speed of light, but they will disagree on the passage of time. This discrepancy arises due to the relative motion between the observers and the fact that the speed of light is constant for all observers.
In summary, when we say time slows down for an observer, it means that their perception of the passage of time is different from that of an observer in a different frame of reference, due to the effects of relative motion and the principles of special relativity.