Say you were walking north. For every step you take forward, you move one step north. If you were walking northeast, you’re still taking one step forward, but you’re only moving north at half the speed. Your overall speed hasn’t changed, but the amount of you move north has.
To transfer the same concept to time dilation, instead of north and east, there is space and time. Everything moves through spacetime at the speed of light, just like moving one step at a time. If you are physically moving at the speed of light, you will still be taking one step forward through spacetime, but all of your movement will be through space, and you won’t be moving at all through time. This is like walking all north and not at all to the east. On the other hand, if you are physically at rest, you’re still moving through spacetime at the speed of light, but you will only experience moving through time. In the north and east analogy, you’re still walking one step at a time, but you only experience moving east, not north.
While we commonly say you can’t travel faster than the speed of light, it’s more accurate to say you can’t physically travel faster than the speed of light. In spacetime, everything moves at the speed of light. But since most things physically travel much slower than the speed of light, these same things experience most of their journey in the dimension of time, not space.
If you want to understand how matter effects time, all you have to do is remember that matter warps spacetime. Look up the photograph we recently took of a black hole. It looks like a black spot surrounded in light. The spacetime of the black hole is so warped, light can’t escape. Just outside of the area where light can’t escape is an area where spacetime is still severely warped, but not so much as closer the to center of the black hole. Light moving through this area have to travel a longer distance than light further away, so time experienced in that area will slow down.
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