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Time dilation can be visualized when you imagine that light takes time to get from one point to another. An imaginary example would be:
1. Imagine a clock stationary in space. The time on the clock is 12:00:00 am at this instant
2. As one second passes, the clock now shows 12:00:01am, and the light from the clock at the instant 12:00:00am has propagated away from the clock a distance of 3.0E8 meters (299792458 meters to be precise) in all directions.
> The light from the clock at 12:00:00am can be visualized as a spherical wave emanating from the clock.
3. Its now 12:00:02am, the "spherical light shell" of 12:00:00am now has a radius of 6E8 meters, the "spherical light shell" of 12:00:01am has a radius of 3E8 meters.
4. Imagine now you are in a space ship located at 3 light seconds (9E8 meters) away from the clock that travels at 1.5E8 meters per second (half times the speed of light) and travels in a straight line radially away from the clock.
5. The "spherical light shell" at 12:00:00am from the clock catches up with you, u see the clock as 12:00:00am.
6. The "spherical light shell" at 12:00:01am takes 2 seconds to catch up with your spaceship, because your spaceship is going at 0.5 times the speed of light.
There's your time dilation. If you were stationary relative to the clock, the seconds of the clock would go on accurately. But if you are moving relative to the clock, the light at an instant of time of the clock has to catch up with you giving you a perception of a longer interval of time as measured by the clock.
Here's the kicker. If you had a watch with you on the spaceship, your watch would tell you that the clock is a slow clock! two seconds on your watch would have to pass by before the clock advances one second.
Time dilation is the phenomenon whereby an observer finds that another's clock which is physically identical to their own is ticking at a slower rate as measured by their own clock. This is often taken to mean that time has "slowed down" for the other clock, but that is only true in the context of the observer's frame of reference. Locally, time is always passing at the same rate. The time dilation phenomenon applies to any process that manifests change over time.
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