How does time behave in overlapping gravitational fields?

[Micheth]

That is to say, how does time behave in a region of space where multiple gravitational fields cancel out their accelerating effects?

I understand that time "slows down" in a gravitational field, but does this depend on the actual presence of the field, or instead the actual net acceleration caused by different fields? The most simple example of this case might be placing a "clock" at the center of a massive body such as the Earth, where I suppose you have no net acceleration, since it is being equally accelerated outward in all directions. So, in such a case, is time still going to be dilated, or would time flow be at the same "rate" as that of a body in empty space with no acceleration?

(If the latter is true, I imagine time dilation being maximum at the surface of the massive body, and gradually reducing to zero toward the center. If not, I imagine it gradually increasing from the surface toward some maximum at the center. Which would be the actual case?)

 

[Ibix]

Time dilation depends on the gravitational potential, not the acceleration. You can just add gravitational potentials (they are scalars, not vectors, and are always negative so never cancel - always reinforce) as long as you don't get too close to a black hole. So clocks at the centre the Earth will run slow compared to ones at the surface, even though they would be weightless.

 

[ISamson]

Time dilation depends on the gravitational potential, not the acceleration. You can just add gravitational potentials (they are scalars, not vectors, and are always negative so never cancel - always reinforce) as long as you don't get too close to a black hole. So clocks at the centre the Earth will run slow compared to ones at the surface, even though they would be weightless.

That is to say, how does time behave in a region of space where multiple gravitational fields cancel out their accelerating effects?

I understand that time "slows down" in a gravitational field, but does this depend on the actual presence of the field, or instead the actual net acceleration caused by different fields? The most simple example of this case might be placing a "clock" at the center of a massive body such as the Earth, where I suppose you have no net acceleration, since it is being equally accelerated outward in all directions. So, in such a case, is time still going to be dilated, or would time flow be at the same "rate" as that of a body in empty space with no acceleration?

(If the latter is true, I imagine time dilation being maximum at the surface of the massive body, and gradually reducing to zero toward the center. If not, I imagine it gradually increasing from the surface toward some maximum at the center. Which would be the actual case?)

Yeah, like Einstein said that time on the top of a skyscraper runs slower than on the ground. So it is true that time would be very slow nearer the centre of the Earth.

 

[Ibix]

So it is true that time would be very slow nearer the centre of the Earth.

The effect is still tiny for something as small as the Earth. Just slightly less tiny at the core than the surface...

 

[Orodruin]

Yeah, like Einstein said that time on the top of a skyscraper runs slower than on the ground. So it is true that time would be very slow nearer the centre of the Earth.

Also, the time on top of the skyscraper runs faster than on the ground.

 

[Micheth]

Time dilation depends on the gravitational potential, not the acceleration. You can just add gravitational potentials (they are scalars, not vectors, and are always negative so never cancel - always reinforce) as long as you don't get too close to a black hole. So clocks at the centre the Earth will run slow compared to ones at the surface, even though they would be weightless.

Thank you.
For me that (grav. fields not canceling out but net acceleration being zero) is kind of counterintuitive, but that's the way it goes I guess.

Kind of a weird thought experiment, but if a sentient entity were somehow living in a void at the center of the Earth (somehow not destroyed by the pressure and heat), while realizing that it was weightless, would it still be able to determine it was being affected by different gravitational potentials?

(Subsequent edit:
For example, a scientist living in some sort of temperature/pressure-resistant sphere set exactly at the Earth's center, maybe. Being weightless, he might assume the sphere to be in empty space with no gravitational field, but could he perform an experiment to find out that's not the case?)

 

[ISamson]

Also, the time on top of the skyscraper runs faster than on the ground.

Yes.

 

[Ibix]

Kind of a weird thought experiment, but if a sentient entity were somehow living in a void at the center of the Earth (somehow not destroyed by the pressure and heat), while realizing that it was weightless, would it still be able to determine it was being affected by different gravitational potentials?

Not easily. Time dilation just means that different clocks run at different rates, so you'd need to be able to see a clock out in space to observe that the rates are different.

That said, I think modern atomic clocks are precise enough to see time dilation between a clock on the floor and one on a desk. So you could compare a clock at the centre of the space and a clock at the edge and detect a rate difference. Since the whole volume is in zero gravity, I can't say if that would be enough to lead to a full theory of gravity, but it would be a clue that something was up (no pun intended).

 

[jartsa]

Not easily. Time dilation just means that different clocks run at different rates, so you'd need to be able to see a clock out in space to observe that the rates are different.

That said, I think modern atomic clocks are precise enough to see time dilation between a clock on the floor and one on a desk. So you could compare a clock at the centre of the space and a clock at the edge and detect a rate difference. Since the whole volume is in zero gravity, I can't say if that would be enough to lead to a full theory of gravity, but it would be a clue that something was up (no pun intended).

All points are at the same potential in that center of Earth lab, so all clocks run at the same rate in the lab.

 

[Ibix]

All points are at the same potential in that center of Earth lab, so all clocks run at the same rate in the lab.

D'oh! Correct, assuming it's a spherical hollow. I think potential can vary for hollows of other shapes. For example, if they drill into the walls, they'll start to notice time dilation and a force pushing objects back towards the hollow.