# Would time stand still at center of earth?

## Main Question or Discussion Point

would time stand still at center of earth?...or at least be really slow

## Answers and Replies

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Char. Limit
Gold Member

Not at all.

Time is not a function of distance from Earth's center, so that distance will not affect time.

However, a pet theory of mine is that you will not feel Gravity at the center of Earth.

Nabeshin

Time will indeed run slower at the center of the Earth than at the surface. This effect is due to the fact that large gravitational potential wells lead to a slowing of time with respect to other observers. For an object like the earth, however, one cannot expect the effect to be detectable by humans, but would be measurable by sensitive instruments.

Caveat: Presumably those on the surface of the earth are rotating with it, and this rotational velocity also leads to a slowing of time (Our bloke at the center is probably not moving). The two effects compete, then.

would time stand still at center of earth?...or at least be really slow
For a non rotating Earth, a clock at the centre will run slightly slower than a clock at the surface because the gravitational potential is lower at the centre of the Earth (and gravitational potential is a function of radius.) You can calculate how much slower by using the interior Schwarzschild solution. For a rotating Earth, the velocity of the clock at the surface will be slowed down by an additional factor due velocity time dilation. It would be interesting to work out which effect wins out, but that would require making some assumptions about the density distributon of matter within the Earth.

For a non rotating massive body of even mass density, time does not stand still at the centre until the radius of the body is 9/8 of its Schwarzschild radius. (ie almost a black hole.)

Char. Limit
Gold Member

Oh.

Never mind then.

At the centre of Earth the gravitational effect is null.
The field is maximum at surface.
google images for 'gravitational potential earth center'

Char. Limit
Gold Member

Nabeshin, I'm curious to know why that is.

Is that the point where M/r^2 in the gravitational equation is maximized, or where you have the maximum amount of mass per unit radius squared?

Dale
Mentor

It has to do with the fact that the earth is not uniform density. If the earth were a sphere with uniform density then what heldervelez said would be correct, however the earth bulges slightly and is much more dense at the core than at the surface. When you take that into consideration g increases below ground for some distance.

George Jones
Staff Emeritus
Gold Member

Time will indeed run slower at the center of the Earth than at the surface. This effect is due to the fact that large gravitational potential wells lead to a slowing of time with respect to other observers. For an object like the earth, however, one cannot expect the effect to be detectable by humans, but would be measurable by sensitive instruments.

Caveat: Presumably those on the surface of the earth are rotating with it, and this rotational velocity also leads to a slowing of time (Our bloke at the center is probably not moving). The two effects compete, then.
I made an idealized GR calculation of this in

I made an idealized GR calculation of this in

quoting from that post
"If an observer on the Earth's surface uses a telescope to look down a tunnel to a clock at the Earth's centre, he will see his clock running faster than the clock at the Earth's centre."

quoting from http://en.wikipedia.org/wiki/Gravitational_time_dilation" [Broken]
"The clocks that traveled aboard the airplanes upon return were slightly fast with respect to clocks on the ground."

As seen above to be at the center of the Earth is equivalent to be at an infinite distance of the Earth.

Do I see a contradiction?

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George Jones
Staff Emeritus
Gold Member

quoting from that post
"If an observer on the Earth's surface uses a telescope to look down a tunnel to a clock at the Earth's centre, he will see his clock running faster than the clock at the Earth's centre."

quoting from http://en.wikipedia.org/wiki/Gravitational_time_dilation" [Broken]
"The clocks that traveled aboard the airplanes upon return were slightly fast with respect to clocks on the ground."

As seen above to be at the center of the Earth is equivalent to be at an infinite distance of the Earth.

Do I see a contradiction?
There isn't a contradiction. These two situations aren't that similar, and stuff like this has to be calculated on a case-by-case basis. See

http://en.wikipedia.org/wiki/Hafele-Keating_experiment

for a more detailed analysis of the experiment done with clocks and planes. The Pound-Rebka experiment

http://en.wikipedia.org/wiki/Pound-Rebka_experiment

is closer to the situation that I described in my post.

In each case, general relativity makes a prediction that is verified by experiment.

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Dale
Mentor

As seen above to be at the center of the Earth is equivalent to be at an infinite distance of the Earth.
There is a difference between gravitational acceleration and gravitational potential. The time dilation (in a stationary spacetime) depends on the gravitational potential, not the gravitational acceleration. So the fact that the gravitational acceleration is zero at the center and at infinity is not relevant.

quoting from http://en.wikipedia.org/wiki/Gravitational_time_dilation" [Broken]
"The clocks that traveled aboard the airplanes upon return were slightly fast with respect to clocks on the ground."
That WP statement is not entirely true. See http://hyperphysics.phy-astr.gsu.edu/HBASE/Relativ/airtim.html

Predicted: Time difference in ns Eastward
Gravitational__ 144 +/- 14
Kinematic_____ -184 +/- 18
Net effect____ -40 +/- 23
Observed:____ -59 +/- 10

Predicted: Time difference in ns Westward
Gravitational __ 179 +/- 18
Kinematic _____ 96 +/- 10
Net effect ____ 275 +/- 21
Observed: ____ 273 +/- 21

The clocks going Westward were slightly faster, but the clocks going Eastward were slightly slower. Clocks flying in either direction speeded up due to being higher up in the gravitational field, but the slowing down of the clocks due to velocity time dilation exceeded the gravitational speed up for the Eastward clocks.

quoting from that post
"If an observer on the Earth's surface uses a telescope to look down a tunnel to a clock at the Earth's centre, he will see his clock running faster than the clock at the Earth's centre."
As mentioned earlier, the clock at the centre of the Earth would look slower if the Earth was not rotating. You have to take both gravitational and velocity effects into account.

If the Earth is modeled as a constant density, non-rotating sphere, then Schwarzschild's interior solution can be used.
It is not too difficult to work out the interior solution for a monotonically increasing density towards the centre, which might yield a better aproximation.

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