# What is the experimental evidence for gravitational time dilation?

There are many experiments done to prove time can be changed. I don't know lots of them, but I think the most popular (most cited) is to put a clock on a fast moving object (airplane), fly around and then compare it with a similar clock on the ground. The flying clock did slow a tiny bit.

Here I doubt the credibility of that method, or actually any method that uses clocks, mechanical or atomic or whatnot. How can we be so sure that the flying clock was running correctly, i.e. it was measuring time correctly while flying; but not because high speed has bent space and thus altered the clock's mechanics a tiny bit? If gravity can even make the angles of a triangle add up differently than 180', there's no reason it can't mess with the way a clock works. To put it in another way, we are relying on mechanical things, which are prone to failure by gravity, to measure a (possibly stable and can't be stretched) phenomenon which is time, and thus we're in an illusion that time changes, too.

The main issue leads to a side question: how can we ever know what is a 'stationary' clock and what's moving fast? All clocks are on Earth, which is rolling on itself like crazy. It's orbiting the Sun, which is orbiting the MW center, which is dashing for Andromeda, which are all falling toward the Great attractor... So in the mentioned experiment, a clock on a plane flying East or South or whatever direction may actually have a lower velocity than the 'stationary' clock on the ground, if we use a cosmic eye to look at them. In that case, even using mainstream physics, shouldn't the flying clock run faster than the ground one?

• You appear to be mixing up special relativity's time dilation due to speed, with general relativity's time dilation due to gravitational potential. – PM 2Ring Sep 9 at 14:08
• You should look for some reading about the general relativistic corrections to the clocks on GPS satellites. Without accounting for general (and special) relativity, the GPS system wouldn't work at all. – rob Sep 9 at 16:11
• Time dilation used to extend a weekend camping trip on Mount Ranier by 22 nanoseconds (2005): leapsecond.com/great2005 – rob Sep 9 at 16:15
• @PM2Ring: The OP seems to have heard about the Hafele-Keating experiment, in which both effects were present. – Ben Crowell Sep 9 at 21:28

## 2 Answers

The original classic demonstration of this effect was not the 1971 Hafele-Keating experiment with atomic clocks aboard airplanes, it was the 1959 Pound-Rebka experiment, which involved nuclei emitting and absorbing gamma rays at the top and bottom of a tower.

How can we be so sure that the flying clock was running correctly, i.e. it was measuring time correctly while flying; but not because high speed has bent space and thus altered the clock's mechanics a tiny bit?

The Pound-Rebka experiment did not use clocks, at least by the layperson's definition of "clock." Really, any repetitive process can be considered a clock. The fact that all clocks agree on the results makes it hard to explain relativistic effects based on the details of the clocks' mechanisms.

The main issue leads to a side question: how can we ever know what is a 'stationary' clock and what's moving fast?

The Pound-Rebka experiment was stationary relative to the earth. But here you are basically asking about the twin paradox, which is special relativity, not general relativity. I'm sure the site already has questions and answers about the twin paradox.

• Actually time corrections both for special and general relativity effects are necessary for the GPS system to work accurately. physicscentral.com/explore/writers/will.cfm – anna v Sep 9 at 16:04
• As the OP seems unusually skeptical, I should add one detail about GPS, which is in use on the tramway system rebuilt in Bordeaux: Its cars are fed power thru the permanently-exposed surface of short lengths of third rail set in the pavement of city streets. GPS signals electrify each length of 3rd rail only when a car is directly above it, and turn off the current to that length when the car clears it: So, GPS is what keeps pedestrians crossing Bordeaux's downtown streets from getting fried whenever they step on the rail! – Edouard Sep 9 at 19:19
• @Edouard really? That seems extremely dangerous. It seems like many things could fail. – Javier Sep 9 at 22:24
• No accident's caught the attention of the French news that I watch almost daily. The long, articulated cars are always over at least two lengths of temporarily "live" rail, so that the setup has a bizarre resemblance to the SR gedanken of lightning bolts striking opposite ends of a train in disputed simultaneity. GPS works! – Edouard Sep 10 at 4:31
• tk you. I didn't know of 1 key fact: in physics, time is what a clock reads. With this definition there's no window for argument at all. @Edouard: not really skeptical, I was ignorant ;) – longtry Sep 10 at 6:08

So it’s important to understand that there are two different ideas here, as PM 2Ring points out in comments.

# What special relativity predicts

Special relativity says that when you are accelerating, clocks ahead of you appear to tick a little faster, and clocks behind you appear to tick a little slower, in proportion to the distance that they are away from you. The tick rate in fact is $$1 + ax/c^2$$ where your acceleration is $$a$$, the clock is a coordinate $$x$$ in front of or behind you in the direction that you are accelerating, and $$c$$ is the speed of light. To you, your clock always appears to tick at a rate of one second per second; clocks that are way ahead of you appear to tick much faster, clocks way behind you appear to tick much slower. If you stop accelerating, then you see a characteristic offset; someone who was stationary for the whole time thinks that two clocks are in-sync but you think that they are out-of-sync. This offset is called the “relativity of simultaneity” if you want to google more about it.

This gives rise to a second-order effect as it inter-plays with the “classical” effect of acceleration, namely that things around you start to appear to gain a backwards motion proportional to the time that you spend accelerating. As those two first-order effects (simultaneity-shifting and backwards motion) start to interact with each other it emerges naturally that two observers who are moving relative to each other both think the other person’s clock is ticking a tiny bit slowly relative to their own. This is called “time dilation” and it goes hand-in-hand with another second-order effect called “length contraction.”

## Some well-known observations

The first-order effect of special relativity predicts things like, for example, that when you dump a ton of energy into a charged particle it does not go unlimitedly faster, as the classical formula $$v=\sqrt{2 E/m}$$ would suggest, but actually instead that it must top out at a maximum speed of $$c$$. This is no longer even reasonable to question. We build particle accelerators on the deep assumption that this is true, and if it were not true then they would shred themselves apart in short order: but they do not shred themselves apart and therefore we know they must not have breached the speed of light barrier. Like, when the OPERA experiment detected neutrinos moving faster than the speed of light through the Earth, it was quickly regarded as a bug in their experiment and indeed they eventually found a loose timing cable that, when more firmly plugged in, eliminated the error and the neutrinos were discovered to only travel at the speed of light. This is no longer physical theory but engineering reality. (I could rattle off some other examples of the same; for example those particle accelerators can be a great source of high-energy photons but only if you put the window a certain direction in the ring, this idea that light does not emit in all directions uniformly but “beams” in the direction of motion is a similar prediction of special relativity.)

The two second-order effects have a classical demonstration of it, which involves us observing muons at Earth’s surface. Muons are a sort of heavy electron that are created in the upper reaches of Earth’s atmosphere, maybe 15 km in the sky or so, but because they are so heavy they decay radioactively into electrons with a half-life of about $$1.5~\mu\text s$$ at rest. So the speed of light barrier would otherwise limit them to travel only about 450 meters, if you do not take these higher-order effects into account. That discrepancy of 33x between the distances (they travel 15km, they should only travel 0.45km), when you are talking about half-lives, means that only one in eight billion muons generated in the upper atmosphere should live by chance long enough to reach our detectors at the surface. This seems strange given that we can measure the energy of these muons and their rate; 10,000 of these impacting on every square meter per minute with energies of about 4 GeV, a total energy of $$0.1~\mu\text{W/m}^2.$$ The problem is that if you multiply by 8 billion you start to get into about $$800 ~\text{W/m}^2,$$ comparable to the total irradiance that the Sun provides which is only $$1366~\text{W/m}^2.$$ So without the second order effects you’re actually saying that the Earth does not get most of its energy from the sunshine hitting the ground and heating the atmosphere from the bottom up, but from these cosmic rays and muons heating the upper atmosphere. That 33x distance discrepancy is thus obviously experimentally wrong in no uncertain terms. Moving muons must live longer, radioactively speaking.

The second-order effects can be used to predict this either way; either you look in the ground frame of reference where the muon was generated $$15~\text{km}$$ in the sky, if muons are being generated with 4 GeV or so of kinetic energy then you would see their “radioactive clock” ticking slower by a factor of 40 or so, so they last for $$60~\mu\text s$$ and can travel $$18~\text{km}$$ before being attenuated by half, and we thus have only the tiniest bit of the solar irradiation going into creating these things and heating the atmosphere from the top down. Or, you look at it from the muon’s perspective where the Earth is flying towards it at the same near-$$c$$ speed, but the atmosphere is only $$15~\text{km}/40 = 375~\text m$$ and there is no reason that it cannot live for those $$1.25~\mu\text s$$. Either way, it works out perfectly nice.

# What general relativity predicts, and its observations

General relativity builds on special relativity by saying that you are accelerating away from a body whenever you keep what appears to be a constant distance away from its center. In fact the “natural” state is simply free-fall. Anything else is an acceleration.

As mentioned, observers at the “ground” must therefore see clocks “high up” tick faster, at a rate of $$1 + g~h/c^2.$$ This is called gravitational time dilation.

For gravitational acceleration at Earth’s surface this effect is weak, only $$10^{-13}/\text{km}$$. But, we have atomic clocks that are sensitive to maybe a little better than one part in $$10^{17},$$ so in theory they can resolve this effect over scales as little as $$0.1~\text m$$ or so. Searching to see whether anyone has ever gotten that level of precision elevation difference, the answer is yes: NIST in the US has observed the different ticking at an elevation difference of only 33cm.

With that said, of course, remember that this is a consequence of those first-order special relativity predictions combined with an assumption, called the principle of equivalence, about the nature of gravity. If that assumption about the nature of gravity is correct (as the NIST experiment proves) then all of our evidence for those first-order effects of special relativity also transfers here, and as I hinted above the first-order effects of special relativity are so commonplace that they are just an engineering reality.

The principle of equivalence can be alternately observed for example by the observations of gravitational lensing, and one can argue that then the special relativity observations of “you cannot accelerate a particle past $$c$$” should also be taken as observations that in a gravitational field the higher clocks should tick faster. It’s the same fundamental effect at that point.

• tks for the informative post! – longtry Sep 10 at 4:53