# Gravity time dilation in the Earth?

1. Oct 9, 2008

### Artlav

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:

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?

.

2. Oct 9, 2008

### George Jones

Staff Emeritus
3. Oct 9, 2008

### JesseM

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. Oct 9, 2008

### Jonathan Scott

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. Oct 12, 2008

### Naty1

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.

6. Oct 23, 2008

7. Nov 8, 2008

### Chaste

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. Nov 8, 2008

### Jonathan Scott

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.

Last edited: Nov 8, 2008
9. Nov 8, 2008

### Chaste

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. Nov 8, 2008

### George Jones

Staff Emeritus
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,

11. Nov 9, 2008

### A.T.

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)

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.

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

Last edited: Nov 9, 2008
12. Nov 11, 2008

### Artlav

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.

Last edited by a moderator: Apr 23, 2017
13. Nov 11, 2008

### A.T.

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. Jun 18, 2010

### Stoonroon

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.