Gravity & Time: Clocks at Sea Level vs Mountain

[Ahmed Samra]

If I get two clocks. One at a very high mountain and the other is at the sea level. Why does the clock moves faster than the ones which is at the sea level?

 

[Crazymechanic]

Actually it is the other way around , the clock on the mountain top would be slower than the one at sea level , that is because you are further away from the center of Earth on a mountain top so because Earth is a sphere your tangential velocity is higher and relativity says that the faster the speed of something the slower time goes for that thing or object or whatever.

In rotating balls or spheres like Earth with fixed velocity the further you are from the center the faster you move and while you move time goes ahead time also goes ahead for the one who is closer to ground or not so far from the center of the rotating body but looking from a speed point of view he is traveling slower because his tangential velocity is slower.

If you have ever used a "merry go round" you know that if you are closer to the center you see everything going right side you slower the more you move at the outer the more faster everything goes by your eyes.

 

[A.T.]

I think OP is asking about gravitational time dilation.

 

[Ahmed Samra]

You mean if I am away from the center of the Earth time will go slower? But I read in Wikipedia that time will move faster when I am away from the center of the earth. I am totally confused

 

[PeterDonis]

You mean if I am away from the center of the Earth time will go slower? But I read in Wikipedia that time will move faster when I am away from the center of the earth.

There are two effects, which work in opposite directions. What you read in Wikipedia is talking about gravitational time dilation, which A.T. referred to; clocks run faster at higher altitudes above a gravitating body.

However, if the gravitating body is rotating, as the Earth is, then you have to take that motion into account as well; moving clocks run slower than stationary clocks because of ordinary, kinematic time dilation. If you are at rest relative to the rotating Earth, then the higher up you are the faster you are moving (because you have to cover a larger distance around a circle in the same time, 24 hours), and the faster you are moving, the slower your clock will run. This is what CrazyMechanic referred to.

An approximate formula that includes both of these effects is:

$$\frac{d\tau}{dt} = \sqrt{1 - \frac{2 G M}{c^2 r} - \frac{v^2}{c^2}}$$

where $\frac{d\tau}{dt}$ is the clock rate relative to a clock that is at rest "at infinity" (i.e., very, very far away from the rotating Earth and not moving with it), $G$ is Newton's gravitational constant, $M$ is the mass of the Earth, $r$ is the distance of the clock from the Earth's center, and $v$ is the clock's velocity, again relative to an observer that is at rest at infinity. Increasing altitude means increasing $r$, which decreases the second term under the square root and makes the clock rate faster; increasing $v$ obviously increases the last term under the square root and makes the clock rate slower.

For example, a clock on Earth's surface at sea level at the equator, and at rest relative to the (rotating) surface, would have an $r$ of about 6.4 million meters and a $v$ of about 450 meters/second. This gives a clock rate of about 1 - 6.96 x 10^-10, i.e., about 7 parts in 10 billion slower than a clock at rest at infinity.

A clock at the top of Mount Everest would have an $r$ about 9000 meters larger and a $v$ about 1 meter/second larger; if you run the numbers, you get a clock rate of about 1 - 6.95 x 10^-10, i.e., about 1 part in a trillion faster (faster because we're subtracting a slightly smaller number from 1) than the clock at sea level at the equator. (I've cheated a bit by assuming Mount Everest is on the equator.)

So CrazyMechanic was actually not correct; there is a slowing effect due to increased velocity for a clock on top of a mountain, but it's more than compensated for by the speeding up effect of the increased altitude.

 

[Ahmed Samra]

So the clock on the mount Everest will move slower than the ones at the sea level

 

[PeterDonis]

So the clock on the mount Everest will move slower than the ones at the sea level

No, it will go faster. Read what I posted carefully.

 

[Crazymechanic]

Well the thing I said is correct just that mount Everest is not high enough for this phenomenon to kick in , so the distance from the center of Earth there is further away than the speed at which it is greater than the one who would stand at sea level.
I am too lazy to do the maths but I think that going higher up and the tangential speed factor would overcome the speeding effect of less gravity and the clock would get slower.The question then become at what distance r from Earth's center the velocity time dilation overcomes the speeding effect of decreasing gravity.

Or ofcourse we could just go nowhere stand right where we are and speed up Earth's rotation :D

 

[PeterDonis]

The question then become at what distance r from Earth's center the velocity time dilation overcomes the speeding effect of decreasing gravity.

And to answer it you would have to do the maths.

 

[Nugatory]

The question then become at what distance r from Earth's center the velocity time dilation overcomes the speeding effect of decreasing gravity.

And to answer it you would have to do the maths.

Or look through some of the older threads in this forum - one of the SAs worked it out and posted the derivation fairly recently.

 

[Crazymechanic]

anyway I think this is not the main concern as the OP got an answer to what he was asking.