[JabavuAdams]

The velocity-time relationship is explained by Special Relativity (SR).

The Sun example is a bit more complicated, because there's the velocity-time effect due to SR, but also the fact that you're deep in a big gravity well, which has effects on your measured time due to General Relativity (GR).

SR says that "moving clocks run slow", but deep in the Sun's gravity well, we should also be running slow relative to the Earth. Not sure what the relative size of the two effects is.

EDIT> I assume the SR effect is larger than the GR effect, simply because SR was obvious before GR.