Gravity time dilation in the Earth?

In summary, in general relativity, the closer a clock is to a massive body, the slower it will go due to the gravitational pull. The dilation is dependent on the curvature of spacetime, not just the gravity acceleration. An experiment with two atomic clocks showed a difference in time rate when one was raised to 10 km. If one clock is placed at the center of the Earth and another on the surface, the time rate will be slower at the center due to the "density" of time being higher there. This is similar to the concept of a "gravitational well". GPS satellites must constantly correct for this time delay in order to maintain accurate positioning. The source for the Schwarzschild interior solution is still unknown.
  • #1
Artlav
162
1
Gravity of a massive body slows time rate, the closer the clock to it, the slower it goes, according to general relativity.
What i didn't quite understood is what is responsible for the dilation?
That is, is it dependable on the gravity acceleration, curvature of space, or something else?

There was an experiment with one atomic clock being raised to about 10Km and the other staying back on Earth, with the up going one then showing the difference.

Now, if we put one clock at the center of the Earth, and one on the surface, what will be the result?

What i mean:
gravques.png


There is no gravity acceleration in the center of the Earth due to it's own mass and "the space is flat" for what i can comprehend, so will the time rate be the same as in relatively flat outer space near Earth, faster than on the surface, same, slower, or entirely different?

If it will be slower than on the surface, then what is physically different in the center of the Earth (harsh pressure/temperature conditions aside) that causes the dilation?


.
 
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  • #3
Relating to your question about space being flat, according to GR you would experience gravitational time dilation relative to a distant observer even if you were in a hollow spherical shell where there'd be no gravitational forces inside, see the post by pervect at the end of this thread, or this paper on arxiv.org. So it seems like the gravitational time dilation between two clocks depends on the curvature of all the spacetime between them, you can't get it just from the local curvature near each one.
 
  • #4
Artlav said:
Gravity of a massive body slows time rate, the closer the clock to it, the slower it goes, according to general relativity.
What i didn't quite understood is what is responsible for the dilation?
That is, is it dependable on the gravity acceleration, curvature of space, or something else?

There was an experiment with one atomic clock being raised to about 10Km and the other staying back on Earth, with the up going one then showing the difference.

Now, if we put one clock at the center of the Earth, and one on the surface, what will be the result?

What i mean:
gravques.png


There is no gravity acceleration in the center of the Earth due to it's own mass and "the space is flat" for what i can comprehend, so will the time rate be the same as in relatively flat outer space near Earth, faster than on the surface, same, slower, or entirely different?

If it will be slower than on the surface, then what is physically different in the center of the Earth (harsh pressure/temperature conditions aside) that causes the dilation?


.

The time rate is effectively fractionally slowed by the Newtonian gravitational potential, so when the rest energy of an object is multiplied by the fractional time rate change, that gives the fractional change in the gravitational potential energy of the object as seen by an observer at a fixed potential. It is in a sense determined by the "density" of locally observed time compared with the time rate at a distant point in a flat background coordinate system; near massive objects, space-time effectively becomes more "dense" (as well as slightly curved), with clocks running more slowly and rulers "shrinking".

Outside a central mass, the Newtonian potential is -GM/rc^2 where M is the total mass and r is the distance from the center. Below the surface of the mass, the potential keeps decreasing, but as you move inside a "shell" of matter, the potential due to that shell stops decreasing, so the only further decrease is due to the remaining matter inside that shell. As you get close to the center, the decrease levels off to a halt and the time rate reaches a minimum at the center. If the Earth were hollow, the whole of the hollow would have an equal minimum time rate, but it would still be lower than the time rate outside.

In other words, if you are outside some spherically distributed shell of mass M, the term it contributes to the potential potential is -GM/rc^2 where r is the distance to the center of the shell, but when you are inside it, the potential is -GM/rc^2 where r is the radius of the shell (and as can be seen, these two are equal at a point on the shell).

The gravitational field is the gradient of the potential, so it does not affect the time rate. Instead, it describes how much the time rate differs between nearby points.

Note that the Wikipedia entry on "Gravitational Time Dilation" is somewhat confused, in that it appears to imply that the time dilation effect decreases inside a sphere. That is wrong.
 
  • #5
It's interesting to note that the global positioning system (GPS) must constantly correct for such time delay changes to retain distance accuracy within a few meters here on earth. Clock differences between the satellites and Earth surface is significant when determining positions.
 
  • #7
For time dilation in a gravitational well, which is analogous to the doppler effect, time runs slower at a stronger gravitational potential(bottom) and if we used a beam of light to measure the time, we would observer that the frequency of the light wave at the top is lower than it is compared to the bottom.

The thing I don't understand is why time run slower at the bottom due to the rate at which wavelength peaks pass, which is HIGHER in frequency then the top.

This higher frequency made me think that time passes faster, since there are more wavelength produced over a certain period of time.
 
  • #8
Chaste said:
For time dilation in a gravitational well, which is analogous to the doppler effect, time runs slower at a stronger gravitational potential(bottom) and if we used a beam of light to measure the time, we would observer that the frequency of the light wave at the top is lower than it is compared to the bottom.

The thing I don't understand is why time run slower at the bottom due to the rate at which wavelength peaks pass, which is HIGHER in frequency then the top.

This higher frequency made me think that time passes faster, since there are more wavelength produced over a certain period of time.

Think more carefully about it - it's not difficult.

If two observers have clocks running at different rates, but both observe a common signal of some form which defines a reference frequency (such as the same light beam, or something blinking, or someone waving a flag), then the one with the slower clock will observe more oscillations of the common signal in a given amount of time measured by their clock.
 
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  • #9
Jonathan Scott said:
Think more carefully about it - it's not difficult.

If two observers have clocks running at different rates, but both observe a common signal of some form which defines a reference frequency (such as the same light beam, or something blinking, or someone waving a flag), then the one with the slower clock will observe more oscillations of the common signal in a given amount of time measured by their clock.

So, it's the COMMON signal that the both clocks compare that determines who is faster.

Stating your example, considering a person is waving a flag on the surface of the Earth with observer A beside him, and there's another observer B in a satellite in space.
Observer B would notice the person waving the flag fewer times compared to observer A, right?
 
  • #10
stevebd1 said:
Hi George

I'd be interested to know what your source was for the Schwarzschild interior solution in post #8 of that thread as I've looked extensively on the web and can't seem to find anything specific.

Steve

Sorry, I didn't see this until now.

I took it from Gravitation by Misner, Thorne, and Wheeler, but it is given in a number of other books. For example, General Relativity: An Introduction for Physicists has it,

http://books.google.ca/books?id=xma...X&oi=book_result&resnum=2&ct=result#PPA294,M1.
 
  • #11
Artlav said:
Now, if we put one clock at the center of the Earth, and one on the surface, what will be the result?

What i mean:
gravques.png
The upper picture (gravitational potential) is more related to gravitational time dilatation.
If you are interested in more accurate visualizations of curved spacetime inside and outside of a sphere mass:
http://fy.chalmers.se/~rico/Theses/tesx.pdf (chapter 2)
http://www.adamtoons.de/physics/gravitation.swf

Artlav said:
There is no gravity acceleration in the center of the Earth due to it's own mass and "the space is flat" for what i can comprehend
If the Earth is not hollow, neither space nor spacetime are flat in the center. But that is not relevant for the gravitational time dilatation as JesseM pointed out.

stevebd1 said:
I'd be interested to know what your source was for the Schwarzschild interior solution in post #8 of that thread as I've looked extensively on the web and can't seem to find anything specific.
Someone should put it into the English-Wiki:
http://en.wikipedia.org/wiki/Schwarzschild_metric
The German version has it:
http://de.wikipedia.org/wiki/Schwarzschild-Metrik#Innere_L.C3.B6sung
 
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  • #12
A.T. said:
The upper picture (gravitational potential) is more related to gravitational time dilatation.
If you are interested in more accurate visualizations of curved spacetime inside and outside of a sphere mass:
http://fy.chalmers.se/~rico/Theses/tesx.pdf (chapter 2)
http://www.adamtoons.de/physics/gravitation.swf
Now that makes the best sense of all - i completely missed the point that time dimension is stretched by gravity too. Then, it is a question of difference of stretching between observers to get a time dilation, not the force of gravity of some kind.

Thank you all for help.
 
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  • #13
Artlav said:
Now that makes the best sense of all - i completely missed the point that time dimension is stretched by gravity too.
That's the key point of the model causing not only the time dilatation, but the attraction of masses itself. The stretching of the space dimensions (often visualized with the rubber sheet) causes only minor effects:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
 
  • #14
What's missing in all the above discussion is empirical evidence for the predicted effect.

The effect is too small to measure for massive bodies of any convenient size. And the centers of astronomical bodies are obviously inaccessible. But an indirect test could be done with a modified Cavendish balance.

The general relativistic prediction of a local minimum for the rate of a clock at the center has a dynamic effect which corresponds to the familiar Newtonian "hole to China" problem. If the rate of a clock at the center is in fact a minimum, then a test object dropped into a hole through the sphere will harmonically oscillate between the extremes.

This prediction has never been tested. But, as alluded to above, a Cavendish balance could be modified so as to perform the test in an earth-based laboratory.

Concerning the connection to the original question, it has always struck me as curious that the magnitude of the effect on clock rate and rod length (time curvature and space curvature) is the same outside matter (as per the coefficients in the Scwharzschild exterior solution); whereas inside matter the effect keeps increasing for clock rate but goes to zero for rod length (as per the coefficients in the Schwarzschild interior solution).

It's as though symmetry cancels the effect on space curvature, so why not also on time curvature? In any case, what is the physical mechanism for this? What does matter do to make a clock at the center run slow? "Gravitational potential" is a theoretical explanation, not a physical one.

Though the interior solution gravity experiment mentioned above would only indirectly test the clock-rate prediction, to my mind it would be very convincing. Since it would test a rather fundamental aspect of general relativity, one wonders why it has not yet been performed.
 

FAQ: Gravity time dilation in the Earth?

What is gravity time dilation?

Gravity time dilation is a phenomenon in which time moves slower in areas with stronger gravity. This means that clocks in these areas will run slower compared to clocks in areas with weaker gravity.

How does gravity affect time on Earth?

On Earth, the force of gravity is relatively uniform, so the effect on time is minimal. However, at higher elevations where the force of gravity is slightly weaker, time will move slightly faster compared to sea level.

What is the equation for calculating gravity time dilation?

The equation for calculating gravity time dilation is t' = t√(1 - 2GM/rc^2), where t is the time in a weaker gravity field, t' is the time in a stronger gravity field, G is the gravitational constant, M is the mass of the larger object, r is the distance between the two objects, and c is the speed of light.

How is gravity time dilation measured on Earth?

Gravity time dilation on Earth is measured using atomic clocks. Two identical atomic clocks are synchronized and one is placed at a higher elevation, while the other remains at sea level. The clock at the higher elevation will run slightly faster due to the weaker gravity, confirming the theory of gravity time dilation.

Can gravity time dilation be observed in everyday life on Earth?

Yes, although the effects are extremely small. For example, GPS satellites have to account for the slight time dilation due to their higher altitude in order to function accurately. Additionally, astronauts in space experience time dilation due to the weaker gravity in orbit compared to Earth's surface.

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