# How does gravitational time dilation work?

• I
I understand time dilation due to speed, but am confused on why time dilates depending on the distance from a mass. Help?

Nugatory
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I understand time dilation due to speed, but am confused on why time dilates depending on the distance from a mass. Help?
You can calculate this effect directly from the Schwarzschild metrics; Google will find many explanations including some in threads here.

A more intuitive picture comes from considering the behavior of two periodically flashing lights at opposite ends of a uniformly accelerating spaceship; the equivalence principle gets you from there to the equivalent gravitational situation. Again, Google will find many good explanations.

Either way, find a good explanation and try working through it. When you get stuck, come back with a more specific question and we'll be able to help you over the hard spot.

I understand time dilation due to speed, but am confused on why time dilates depending on the distance from a mass. Help?
I saw you marked this as intermediate however based on the question I thought you may be looking for a simple association for time dilation, mass and distance. That answer is, your distance to a mass is directly related to the strength you'd experience from it's gravitational field; ergo time dilation is related to the strength of that gravitational field. I hope this helped.

Orodruin
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I saw you marked this as intermediate however based on the question I thought you may be looking for a simple association for time dilation, mass and distance. That answer is, your distance to a mass is directly related to the strength you'd experience from it's gravitational field; ergo time dilation is related to the strength of that gravitational field. I hope this helped.
This is wrong. Gravitational time dilation is related to the gravitational potential, not to the field.

@Orodruin I understand that our theory has it based on gravitational potential as you say, but I was wondering if there have been any experiments to test this versus it being based on the gravitational field.

I have seen many experiments proving gravitational time dilation. Some even with only a few feet difference between their heights above sea level. But, have yet to see one proving it is gravitational potential versus the field.

jbriggs444
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I have seen many experiments proving gravitational time dilation. Some even with only a few feet difference between their heights above sea level. But, have yet to see one proving it is gravitational potential versus the field.
What you are up against is the clock postulate, which says that acceleration does not cause any additional time dilation.

This has been tested to as high as 10^18g. Usually it is done with radioisotopes of known half-lives on high speed centrifuges. As James R has noted, the time dilation always comes out to be equal to that caused by the tangential velocity. This is easily verified by varying the length of the centrifuge radius and its speed. This way you can get a number of different combinations of tangential velocity and g-force.

• Orodruin
PeterDonis
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have yet to see one proving it is gravitational potential versus the field.

Yes, you have. All of the experimental data show that the time dilation factor is ##\sqrt{1 - 2M / r}##. That means it depends on the gravitational potential. If it depended on the field, the dependence on ##r## would be different.

• Orodruin
Janus
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@Orodruin I understand that our theory has it based on gravitational potential as you say, but I was wondering if there have been any experiments to test this versus it being based on the gravitational field.

I have seen many experiments proving gravitational time dilation. Some even with only a few feet difference between their heights above sea level. But, have yet to see one proving it is gravitational potential versus the field.
That's what the Pound-Rebka experiment does. You send a signal from one height to another. The signal is Doppler-shifted by either climbing against, or falling with the gravity field. In other words, the amount of Doppler-shift is a result of the change in gravitational potential between the two heights. The shift is exactly equal to the gravitational time dilation predicted by GR. Put another way, the quantitative results from real life experiments are consistent with gravitational time dilation being due to gravitational potential, but can't be made consistent with it being due to difference in local field strength.

Jonathan Scott
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Yes, you have. All of the experimental data show that the time dilation factor is ##\sqrt{1 - 2M / r}##. That means it depends on the gravitational potential. If it depended on the field, the dependence on ##r## would be different.
Most experiments actually suggest that the time dilation factor is ##1 - M/r##, or in conventional Newtonian terms ##1 - GM/rc^2##, so the fractional time dilation between two locations for most purposes is simply the difference in ##-GM/rc^2## at those locations. Given the smallness of ##-GM/rc^2## in normal cases such as the solar system, this is consistent with the General Relativity theoretical prediction of ##\sqrt{1- 2M / r}##. Measurements of the perihelion precession of Mercury (and other planets) confirm that the square root form is more accurate, but this is a very small difference.

PeterDonis
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Most experiments actually suggest that the time dilation factor is ##1 - M/r##, or in conventional Newtonian terms 1−GM/rc21−GM/rc21 - GM/rc^2, so the fractional time dilation between two locations for most purposes is simply the difference in ##-GM/rc^2## at those locations.

This was true of the original Pound-Rebka experiment and similar ones, yes. However, I don't think it's true of GPS, or of the experiment described in this paper:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.45.2081

I think there is enough altitude difference in those cases that the simple approximation ##1 - M/r## no longer quite holds.

Jonathan Scott
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This was true of the original Pound-Rebka experiment and similar ones, yes. However, I don't think it's true of GPS, or of the experiment described in this paper:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.45.2081

I think there is enough altitude difference in those cases that the simple approximation ##1 - M/r## no longer quite holds.

To the next term, ##\sqrt{1 - 2M/R}## is approximately ##1 - M/R - \tfrac{1}{2} (M/R)^2##, so the difference is around ##\tfrac{1}{2} (M/R)^2##.

For Earth, I make it that the fractional difference in time dilation at the surface relative to infinity is about ##7 * 10^{-10}## (feel free to check it).

So even in that case, you would have to measure the fractional difference in the time rate to a few parts in ##10^{10}## to detect the difference; that is, you would need to measure the time rate itself to a few parts in ##10^{19}## (a few hundredths of a second in the age of the universe) to detect the difference by that means.

PeterDonis
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So even in that case, you would have to measure the fractional difference in the time rate to a few parts in ##10^{10}## to detect the difference

Yes, and GPS time measurements are that accurate.

you would need to measure the time rate itself to a few parts in ##10^{19}## (a few hundredths of a second in the age of the universe) to detect the difference by that means.

I don't understand where this number is coming from.

pervect
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I understand time dilation due to speed, but am confused on why time dilates depending on the distance from a mass. Help?

The simple SR explanation might be the best route to take. Consider an accelerating spaceship. In an inertial frame of reference, there is only time dilation due to speed. When we do the math to properly handle accelerating frames of reference, though, we find that the clocks in front of the spaceship tick faster than the clocks in the rear.

The detailed mathematical treatment can get complicated. I wish I had a better I-level reference for the math, but at the moment, I don't. The experimental consequences of this are observable and easy enough to understand intuitively. If one transmits a signal from the rear of the spaceship to the front of the spaceship, the spaceship accelerates while the signal is travelling and winds up red-shifted. Similarly, a signal emitted in the other direction gets blue-shifted, for the same reason.

Without going into the mathematical details, we can say that the speeding up of the clocks at higher elevations, and the slowing down of clocks at lower elevations, properly predicts the observed redshift and blueshift of signals as measured by clocks.

In the non-inertial accelerating frame of reference, the explanation we offer for this red and blue shifting is "gravitational time dilation". Here we make use the principle of equivalence to argue that the gravity on an accelerating spaceship (often called gravity on Einstein's elevator, where Einstein' elevator is essentially an accelerating spaceship) should be similar to the gravity of an actual gravitating body.

I do think the above argument is a bit weak without math, but I'm not sure how to do the math without introduce A-level components to the thread :(. But hopefully the general idea is clear, even if the specifics are not.

stevendaryl
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I don't understand where this number is coming from.

The difference between using ##\sqrt{1- \frac{2GM}{c^2 r}}## and using the approximation ##1 - \frac{GM}{c^2 r}## is on the order of ##(\frac{GM}{c^2 r})^2##, which is of the order of ##10^{-19}##. So he's saying that the approximation is good enough for practical purposes.

• Jonathan Scott
Jonathan Scott
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Yes, and GPS time measurements are that accurate.
Sorry if I'm not clear, but I was saying that the value of the DIFFERENCE between clock rates (or to put it another way the difference between two ##GM/rc^2## values) has to be measured to a few parts in ##10^{10}## so the overall time rates have to be measured to a few parts in ##10^{19}##.

As far as I know, the only experiment which gives direct experimental evidence for the PPN ##\beta## parameter, which determines the time dilation term of order ## \left ( \tfrac{GM}{c^2 r} \right ) ^2 ##, is the Mercury perihelion precession measurement (and the less accurate results for the other planets).

PeterDonis
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The difference between using ##\sqrt{1- \frac{2GM}{c^2 r}}## and using the approximation ##\frac{GM}{c^2 r}## is on the order of ##\frac{GM}{c^2 r})^2##, which is of the order of ##10^{-19}##.

I was saying that the value of the DIFFERENCE between clock rates (or to put it another way the difference between two ##GM/rc^2## values) has to be measured to a few parts in ##10^{10}##

Ah, got it. Yes, you're right, GPS and other near-Earth measurements aren't accurate enough for that.

I understand time dilation due to speed, but am confused on why time dilates depending on the distance from a mass. Help?

Time dilation in SR is rewritten in GR way as ##g_{00}=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}## > 1 thus proper time of moving body ##d\tau < dt## coordinate time.
You know in general ##g_{00}## decides ratio of proper time and coordinate time in GR.
Like relative speed in SR, being nearby mass affects ##g_{00}## and decides ration of proper time and coordinate time for bodies at rest.

PeterDonis
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Time dilation in SR is rewritten in GR way as ##g_{00}=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} < 1## thus proper time ##d\tau < dt## coordinate time.

This has nothing to do with GR. I also don't know what coordinate chart you think you are using here.

I thought of Rindler system and rotation system that are fully understood by SR and have metric ##g_{00}## not equal to one.

PS error in formula was corrected.

PeterDonis
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I thought of Rindler system and rotation system that are fully understood by SR and have metric ##g_{00}## not equal to one.

It's correct that ##g_{00} \neq 1## in Rindler coordinates, but what you wrote down is not ##g_{00}## in Rindler coordinates, or indeed in any coordinates I am aware of.

error in formula was corrected.

Your formula still looks wrong. I would strongly suggest not posting further in this thread since you are giving incorrect information.

• sweet springs
Yeah, I have mistaken in inverse and square root.

As for rotation system, ##g_{00}=1-\frac{r^2\omega^2}{c^2}## where ##r\omega=v## is tangent velocity of rotation in IFR where center of rotation is at rest.

PeterDonis
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As for rotation system, ##g_{00}=1-\frac{r^2\omega^2}{c^2}## where ##r\omega=v## is tangent velocity of rotation in IFR where center of rotation is at rest.

In Born coordinates, yes, this is correct.