Does Time Really Slow Down Near Massive Bodies, or Is It Just Light?

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If time slows down near a massive body like the Earth, is that actually the slowing down of time our is it just the slowing down of light?
 
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"Light moves at the exact same speed in all reference frames."

^ Central tenet of relativity

Our clocks run more slowly. Interestingly, this must be accounted for in GPS satellites to properly "synchronize" their clocks with our ground-based ones.
 
So time actually does slow down, and it doesn't just seem like it's slowing down.