Daniel and Jorge Explain the Universe does a good podcast on the speed of light and the fact that it is arbitrary. Why is 299,792,458 the speed limit? We have no idea. Experiments with atomic clocks in airplanes do show that time slows down in the plane as it travels faster relative to one on the ground .. and you add in space observations and it does appear that the time slowdown is asymptotic to the speed of light.
The shifted time line isn't well explained since he's using a 2D model to try to explain it in 3D. Keep in mind, on Earth, we're currently traveling several thousand km/s. We don't feel it because this is a very large planet, with a lot of gravity, keeping as on it as it hurls through space. It's not that Earth is straight up and down on its world line ... it's straight up and down relative to the ship. For the ship, their world line is straight up and down (they should have animated a transition between the two I think).
The intuitive model I've seen used is that everything is always moving at constant velocity, divided up among the three spatial dimensions and the time dimension. So if you're moving very fast in space, you're not moving in time very fast. This nicely covers the way someone on a very fast space ship going for a short journey and back to Earth would find everyone very aged.
Yes, that's a good way to visualize it. I don't think the video's representation violates that model though.
But back to your original question "why light's speed is so special." That person on that very fast space ship going on a short trip; they would move slower through time (and faster through space) than everyone on Earth. An observer on Earth might watch them take over 100 years to make that short journey, but only a year might pass for the person on the ship. That difference between the two grows asymptotically as you approach the speed of light.
General relatively says as you get closer and closer to c, and you had infinite fuel, that difference would grow to the point where the relative time changes across the length of the ship would grow so great that final drop of fuel needed to reach c would never make it into the fuel tank.
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