Center of Earth effects Time [duplicate]

Question

I was watching this documentary about time:

https://www.youtube.com/watch?v=v99-S4_IvVg

It shows that gravity does effect time, and the higher the force the slower time will pass and slower you will age.

On earth we are also pulled by gravity and the closer we get to the center of the earth the stronger the force of gravity. Therefore if you were to swim really deep in ocean would that have the same effect on time as it would do for spaceship orbiting black hole?

I know that there is lava down there but in general would swimming around the center of earth have an effect in time?

Answer 1

Lets see how the gravitational field is actually looking inside Earth. In this situation the distribution of mass is spherically symmetric (or at least very nearly so) in which case one can actually calculate the gravitational field as if all the mass was in the center as long as you are outside the Earth (it is possible to show that this is true mathematically). When you get inside the Earth you can apply the same trick, but only to the mass that is located inside a sphere centeret at the center of the Earth and with a radius equal to your distance to the center of the Earth (the sphere will just reach you). All the mass outside this sphere will have no effect, since it will cancel itself out (which can also be shown mathematically). The consequence of this is, that the further you move inside the Earth, the closer you get to the center, but lesser mass will contribute to the gravitational field, and thus you won't suddenly feel like being near a black whole. Actually, getting all the way to the core you would feel no force at all (since you are pulled equally in all directions).