Influence in spacetime. I'm not sure how knowledgable in physics you are, so this is going to be a long reply.
Think about two trucks, one has nothing in its bed, the other one has a bed full of bricks, both traveling at the same speed. The one with the bed full of bricks is going to get less miles per gallon because the engine has to work harder to move it to the same speed. This is because the heavier something is, the more energy it takes to move it.
So let's ramp it up. Suppose we have a motorcycle and the truck with bricks and now they're going faster. The truck is using even more energy than before to get to the higher speed, therefor getting even less miles per gallon than the motorcycle compared to the other truck before.
We start seeing a trend, the faster something goes, the more energy it takes to achieve that speed. The more energy it uses, the worse its performance is compared to something that weighs less. Let's ramp it all the way up, let's say it's being compared to something that doesn't weigh anything. It's performance is much worse compared to this weightless object, and this weightless object has the best performance you can achieve, as there's no such thing as being less than weightless.
So what does having the best possible performance translate to? Well, if we consider this object traveling through time, since it has the best possible performance, it has no weight, which means it takes no energy to move it, and it doesn't move through time. Energy is the ability to do something, so without any of this objects energy directed to moving through time, it doesn't. Therefor, since time passes, we can say this weightless object travels at the same speed that time passes, just like we say that two cars traveling 60 mph are going 0 mph relative to each other, and 60 mph relative to the road.
So in nature, the weightless object most people are familiar with is a photon, a particle of light. The rate at which it travels is the speed of light. The speed of light is the speed at which time passes, so a photon does not experience the passage of time (best possible performance). Something which is going slower that this speed, experiences the passage of time more (lesser performance).
Thus, we can say a couple things:
A moving clock ticks slower than a standstill clock, as in, experiencing more of the passage of time means time passes faster.
A person flying in a space ship will travel through time slower than someone on Earth, as the person in the space ship is moving faster than the person on Earth.
The more massive you are, the more energy it takes you to move, the more energy it takes you to move through time, the faster time passes. Since time is always moving, we have to think of time like a river with a current. A bolder will affect the current more than a pebble will. So we can say the more massive you are, the more you affect the time around you as well as your own passage of time. (This is gravity)
Putting all of these things together, we can say, the more energy it takes to move you, the more you influence the passage of time. Something with a lot of mass influences the passage of time more than something with less mass. Something with no mass doesn't influence the passage of time. Everything is influenced by the passage of time around them, and so two objects experiencing different rates of time passing will have aged differently, or in other words, time dilation.
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