# More on how clocks are affected by gravity

A couple of weeks ago, I had a post about how the rate at which clocks run is affected by the strength of the gravitational field in which they are in. So a clock on the surface of the moon would run faster than an identical clock on the surface of the Earth because the gravitational field there is less than the field here.

Then a couple of days ago I watched an episode of the PBS TV series Nova where they ran again a 45-minute program titled Inside Einstein’s Mind that looked at how he used thought experiments to arrive at his ideas of relativity. At the 39:00 minute mark, they show an experiment involving two identical atomic clocks, each accurate to one billionth of a second. After synchronizing them to give identical times, one is kept at sea level while the other is taken close to the top of Mount Sunapee in New Hampshire, where the peak is at a height of 831 m. After four days, the clock at the top is brought down again and compared with the other and is found to be ahead of the clock at the bottom by 20 nanoseconds, which agrees with what the general theory of relativity predicts.

The show also said that if adjustments are not made for this effect, the clocks on the GPS satellites that are at heights of about 28,000 km above the surface of the Earth would differ from the clocks on Earth and thus introduce errors into distance estimates. Over a day, the resulting error could be as much as six miles.

You can see the program below or by clicking the link for it above.

1. DonDueed says

I saw that show too, and was surprised that the GPS correction due to GR was so significant. I knew about it but had supposed it to be a much smaller effect.

I wonder whether some of the correction is for the higher speed of the satellites, as well as what’s needed due to the lower gravity at their altitude. But even low-earth orbital speed is a pretty small fraction of c, so maybe that can be ignored.

2. Rob Grigjanis says

DonDueed @1: A satellite at an altitude of about 20,000 km will gain about 39 μs (microseconds) a day compared to an Earth clock. You can separate that into a curvature (strength of gravitational field) part and a speed part*. The curvature part contributes about 46 μs per day, and the speed part (mostly from the satellite’s speed) subtracts about 7 μs.

That is a small daily correction, but GPS requires a great deal of precision, so even small deviations can have significant effects.

*You can separate the two only because the gravitational field is weak (distances much greater than the Schwarzschild radius), and velocities are much less than c.

3. Mano Singham says

DonDueed,

I too was surprised by the large correction for GPS distances.

If you multiply the time difference due to the gravitational redshift by the speed of light, you get about 7 miles. It may be that this is the error they were referring to but I don’t know if that relates to actual distance errors as measured along the ground.

4. robert79 says

“the rate at which clocks run is affected by the strength of the gravitational field in which they are in”

It’s been quite a while since I studied this (about 20 years) but is it really the strength of the gravitational field?

Basically, if we had a planet one-thousandth the radius of the earth, and one-millionth the mass (and thus 1000 times the density… but this is a thought experiment) and so the same gravitational acceleration at the surface (since that’s a 1/r^2 law.) Would clocks run at the same speed?

For some reason my gut says that gravitational potential would be a more likely candidate. I’m trying to think of a nice counter example involving the earth surrounded by a very massive neutronium sphere (which would imply the same g at the surface…)

Or perhaps a black hole for which the gravitational field is the same as that on the surface of the earth, right on the edge of its Swarchschild radius.

5. Rob Grigjanis says

robert79 @4: A motionless clock at radial coordinate r (in Schwarzschild coordinates) from the centre of a mass M will run at a rate relative to Schwarzschild time t given by

Δτ/Δt = (1 -- 2GM/c²r)^(1/2)

So, yes, it varies according to the gravitational potential rather than the gravitational force/acceleration. So a clock on a planet with 1/1000 the radius and 1/1000 the mass of Earth, would run at the same rate.

A subtlety here is that r is not the physical radius, but for weak gravitational fields, it’s close.

6. Rob Grigjanis says

robert79: You do raise a good point about our use of the term “gravitational field”. In Newtonian mechanics, it is GM/r². In GR, the term “field” is used somewhat more loosely.

7. Mobius says

I have often posted on evolution threads and have often had creationists tell me how wrong scientists can be. In their view, Newton and Einstein are wrong. I point out that Newton works just fine for getting us from one planet to another. It is not so much that Newton, and Einstein, were wrong as it is that they were incomplete.

Newton gave us the solution to one big piece of the puzzle, though not the entire puzzle. Einstein gave us the solution to another huge piece. But there is more still to be done, and people are working on it. Lots of people and some very brilliant people. It is likely that someone will be able to unify gravity with quantum theory. Perhaps even soon. And, as with General Relativity, the insights that will open will likely by amazing.

BTW…Thanx Mano for bringing that video to my attention. It was very informative.

8. Jenora Feuer says

How big the GPS correction gets makes a lot more sense when you realize two things:
-- All GPS calculations are based on the small differences between multiple nearly-identical values. Literally you get your position because the satellites in theory all have synchronized clocks, but the satellites that are further away will have their signals take longer to get to you, so you’ll see an earlier timestamp on the clock.
-- 1μs at the speed of light is about 300m.
This is made worse by the fact that the satellites usually only send the last several digits of their timestamp rather than the whole thing, so large enough errors can start wrapping around.

Even if you don’t have to compare against a local reference (and you don’t as long as you can see four satellites), the gravitational field around the Earth isn’t a fixed static thing, and the satellites aren’t all in perfect circular orbits. Errors accumulate at slightly different rates for each of them depending on which parts of the earth they’re flying over at which parts of their orbits.

(I worked with somebody who did their thesis on gravitational anomalies in the Earth’s crust and the resulting perturbations on satellite orbits. Sure, most of the time it doesn’t make enough of a difference to worry about. GPS is one of the few cases where it does. There’s a reason the GPS system is considered by some to be the world’s largest real-time ongoing test of General Relativity.)

9. Rob Grigjanis says

I just realized that I never read the first post on the subject. No idea how I missed it. There’s a common misconception in one of the comments, about mass supposedly increasing with speed. I posted a comment on that, if anyone’s interested.