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Let's say there are 2 people, A and B. They both are at the sea level. A gets higher than B. Does time moves faster to A then B? (Does height -in other words gravity- causes Time Dilation?)

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As already said in the comments, yes. However, gravitational time dilation is described by general relativity. One will find that clocks closer to a massive object (such as the earth) will tick slower compared to clocks farther away. In other word, time "flows"1 slower closer to the surface of the earth.

There are two things however that I'd like to further elaborate on.

First, you cannot tell that time has "slowed down" for you in your own frame of reference. For you, time will always appear to pass at the same "rate".

Second, I'm not sure about whether you can say that height or gravity causes time dilation. In general relativity, gravity is the curvature of spacetime (thus doesn't cause it). However, this is more a question of language – I just wanted to clarify.


1 I put this in quotes because time cannot actually flow – see What is time, does it flow, and if so what defines its direction?. Though it should be clear what is meant by that.

jng224
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The title to your question "Does Gravity Cause Time Dilation" and the body of your question "Does height -in other words gravity- causes Time Dilation?" are slightly different and the difference is important.

It is generally considered that a massive object (like a planet) causes the warping of spacetime. In other words, the planet causes time dilation. And then the closer you are to the centre of the planet, the greater time dilation you will face. So you at sea level will indeed face slower relative time than your friend a the top of a mountain.

But this is where you need to understand the slight difference in interpretation. Many scientists believe that it is the gradient in time dilation that causes gravity. (And not the other way around that gravity causes time dilation) There is a saying: "The reason your bum is stuck to your seat is because time runs faster at your head than your feet." Things fall because of the gradient in time dilation. Things like to be where time runs slowest. So time dilation comes first and gravity comes second - but of course they are simultaneous.

foolishmuse
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Or you might consider that other solutions exist. For instance, that Time is NOT going to ever "dilate", for any reason, its a concept, so it not able to be affected by any Physical forces.

So Time as a concept, does not change with changes in gravitational force, but CLOCKS, (any type of physical process) can and must change under differing physical conditions.

Clocks can get out of sync with changing gravity, but TIME can not. Even the physical process in an Atomic Clock is affected by Gravity.