# Relativity on Airplanes

Einstein’s “Theory of Relativity” is commonly divided into two parts: “special relativity” and “general relativity,” and each part results in different effects on your airplane ride. Broadly speaking, special relativity tells us that the laws of physics are the same in all reference frames in uniform motion. General relativity is the modern formulation of the law of gravity.

I. SPECIAL RELATIVITY

A. TIME DILATION

Special relativity predicts two effects that relate to your airplane flight. (I’m not going to derive why they are true here, but take my word that they are true.) The first is that moving clocks run slow. This being relativity, a good question to ask is “relative to whom and the answer is, “relative to a stationary observer.” So, for example, if you are standing on the ground, and I fly past you, you will observe my clock to be running slowly. That is to say, if an hour passes on your watch, you will perceive that less than one hour passes on my watch.

To me on the plane, however, time will appear to be passing normally. My hand doesn’t seem to move slowly as I reach for the soda in front of me, or anything like that. And this is true no matter how quickly the plane is moving, even if it is going far faster than any plane we could ever hope to make.

Furthermore, it would be perfectly appropriate for me to regard myself as not moving, and you on the ground as moving quickly away from me. And indeed, while I believe my watch is functioning normally, I will perceive YOUR watch as running slow. This may seem like a contradiction, but it is not. It is true that if we synced our watches when we passed each other, if we came together again we really would have to agree on whose watch was ahead, and whose was behind. But in order to do this I would have to turn around, and undergo acceleration – and so the symmetry of the situation would be broken. (Read up on the “twin paradox” for more on this.)

Observations have confirmed that this effect does really occur. In the upper atmosphere, a particle known as the “muon” is produced that lives, on average, for around 2ms (2 thousandths of a second) before decaying into other particles. Thus, even if these muons were moving at the speed of light (the maximum allowable speed); we would expect almost none of them to reach the surface of the Earth. But, in reality we see that many of them do, and the reason is that time is passing more slowly for the muons. While they “think” they only live for 2ms, we see them living much longer. (For a calculation, see here.)

B. LENGTH CONTRACTION

Special relativity also tells us that lengths are contracted in the direction of motion. So, for example, if I am holding up what I consider to be a 12 inch ruler in my airplane window when I fly by, you will measure my ruler to be less than 12 inches long. (Let’s not worry too much about the practical details!) The situation is also perfectly symmetric – I will measure your ruler to be too short. If I stop and we come together, so that there is no longer any relative motion between us, and we hold our two rulers up to one another, we will find that they match.

To see experimental evidence of this, consider again the muon problem. From our perspective, muons are able to reach the surface because they live longer than 2ms, because their “clock” is running slow. By what about from the muon’s perspective? Well, they think they die after 2ms. So how do they make it the 20 kilometers to the surface? Ah, but from their perspective, they are stationary, while the atmosphere and the surface of the Earth, is rushing up quickly toward them. Therefore they see this distance, between themselves and the surface, as contracted. We think they have 20 kilometers to move through, but they see it as much, much less. So from their perspective, while they don’t live nearly as long, they don’t have to move nearly as far, and so they still reach the surface.

Note too that the observation, “muons reach the surface,” is agreed on from both perspectives, but is explained differently.

II. GENERAL RELATIVITY

General relativity tells us that clocks closer to the center of the Earth (or any gravitational source) run more slowly than those farther away. So, if I work on the 50th floor of a building, and you work on the 10th, if one hour passes for me, slightly less than one hour will pass for you.

This has also been observed experimentally. In fact, perhaps you have a GPS system in your car, or on your phone. Well, if the GPS satellites didn’t take this effect into account, the position they reported to you would be off by many miles!

III. HOW BIG IS THE EFFECT?

If you’ve read this far, you may be thinking, “I’ve never noticed any of these effects, ever.” And that’s true, and the reason is that they are very small. (If they weren’t very small, we probably would have noticed them before the 20th century.) To estimate just how small they are for a typical airplane flight, we’re going to have to do a little math. So, imagine you are on a flight from St. Louis to Washington, DC, onboard a Boeing MD-80. The flight takes two hours (as measured by someone on the ground), and you average 390 miles per hour, or 174 meters per second.

How much time passes for you? Well, let’s consider the two effects, from special and general relativity, separately. From special relativity, the time will be contracted for you by an amount known as the Lorentz factor, equal to 1/SQRT (1-v^2/c^2), where v is your speed relative to ground, and c is the speed of light. For this trip, the Lorentz factor equals 1.0000000000001682. So, only 1.9999999999986 hours will pass for you. This is a difference of approximately 5 nanoseconds (5 billionths of a second).

While we are on the subject, note that a ground observer would see your plane as contracted by this same factor. So, let’s say your plane is 150 feet long. Then, while it is in flight, a ground observer would see it as being only 149.999999999975 feet long. This is a difference of about 0.3 billionths of an inch.

General relativity predicts that gravity will tend to cause your clock to run faster than those on the ground. Near the surface of the earth, the DIFFERENCE in time measured between a clock on the plane, and one on the ground, is approximately equal to (T*g*h)/(c^2), where h is the altitude from the plane above the surface, g is the acceleration due to gravity at the surface (9.8 meters per square second), and T is the time that would pass for an observer very far from the earth. For a 2 hour flight at 30000 feet (9144 meters), this comes to a time difference of 1.99 picoseconds (2 trillionths of a second).