Matt Strassler 11/27/2011
The principle of relativity comes up regularly in the context of space travel, and this week’s launch of NASA’s “Curiosity” rover mission was no exception. The BBC has a pretty nice article about it, but as happens so often in press articles, it stumbles in a big way at one point. Quoting from the article: “By the time the encapsulated rover was ejected on a path to the Red Planet, it was moving at 10 km/s (6 miles per second.)”
Oh my heavens. 10 kilometers per second. That sounds very fast; highway speeds are 100 kilometers per hour.
But the statement is completely meaningless.
In fact, right now, as you sit in your chair, reading this nice little article, you are moving at 30 kilometers per second. In a sense, anyway. Yet nobody is about to issue you a traffic ticket, or give you an award for traveling faster than a speeding bullet (less than a kilometer per second.)
Do you know that Einstein did not invent the principle of relativity? The original principle of relativity — which includes the statements that the laws of nature do not permit you to determine if you are stationary, and thus your speed (and that of anything else) always has to be stated as relative to another object — goes back at least to Galileo, who is credited by scientists as having formulated the relativity principle. Einstein modified the details of this principle, in a striking and radical way, but without discarding Galileo’s basic realization that all speeds have to be measured as those of two objects relative to one another.
Galileo recognized that if you are on a ship sailing in calm waters, and you are inside your windowless cabin, there is nothing you can do to tell how fast the ship is moving. If you are able to toss a ball back and forth with a friend when you’re on shore, you will be able to throw and catch the ball just as easily if the ship is moving at five kilometers per hour, or fifteen, through the water — as long as it is moving in a straight line and not being jostled by wind or wave. An extreme version of this is that you can easily toss a ball back and forth on a jet plane moving at hundreds of kilometers per hour, as long as the ride isn’t turbulent. (I’m not sure you should try this in the aisles, but you might try jumping up and down, as I did when I was nine years old to see what would happen; you’ll find it feels exactly the same as when you’re on the ground.) And it’s a good thing!! If Galileo’s relativity principle weren’t true, we’d have a tough time eating, drinking, and walking on airplanes; the fasten seat belt sign would be on all flight!
What does the speedometer in a car actually measure? It measures the speed of the car relative to the ground. Of course, if you are traveling by car, that’s typically the only relevant speed — you want to know how long it will take to get from your starting point to your destination, and since the two locations are stationary relative to the ground, the speed of the car tells you how long it will take to travel between them.
But for an airplane, there are two measures of speed that matter. One of them is “ground speed”, and the other is “air speed”. Ground speed determines how quickly you are covering the distance between your point of departure and your point of arrival. Air speed, however, measures how quickly the air is flowing over the wings of the plane. It is air speed that determines how and whether the plane flies. Also, the maximum speed of a plane is a maximum air speed, not a maximum ground speed, because the engines have to work against wind resistance, which depends on air speed only.
If there were no wind, then the air and the ground would both spin around the earth’s axis exactly once each day, and ground and air speed would be identical. But the atmosphere does have strong winds, so air and ground speed can be quite different. In the mid-latitudes where most of us in North America and Europe and much of Asia live (as well as in southern South America, southern Africa and Australia) the winds at jet altitudes blow west to east. Much of this air flow occurs in the “jet stream”, which is up at the altitudes where planes fly. This “river” of air can move at roughly 100 to 250 kilometers per hour (roughly 50 to 150 miles per hour) relative to the ground. And what that means, in turn, is that a plane with an air speed of 800 kilometers per hour may have a ground speed of perhaps 700 kilometers per hour if it is traveling to the west, and perhaps 900 kilometers per hour if it is traveling to the east. This in turn explains (roughly) why a flight from Europe to the United States can take as much as a couple of hours longer than a flight from the United States to Europe; the plane’s air speed is the same in the two cases, but its ground speed is not. (The same principle makes a boat trip longer when you are traveling up a river, against the current, than when traveling down the river; the boat’s engine allows it to travel at a fixed speed relative to the water, and this is not the same speed relative to the shore for up- and down-river travel.)
Of course, when you’re in the plane (or boat), you don’t feel any speed at all; it’s the same to you if the air speed is 800 kilometers per hour or 500, because you and the plane (and the air inside the plane) are stationary relative to one another. In other words, you don’t have a speed. You have speeds, plural, relative to other things, plural: these include speed relative to the plane (zero), speed relative to the outside air (800 kilometers per hour) and speed relative to the ground (faster or slower than air speed depending on where you are going.) Which version of your speed is better? Well, that depends on what you want to know; ground speed is relevant for travel time, air speed is important for the integrity and flight characteristics of the plane, and plane speed is relevant to how long it will take you to get to the lavatory from your seat.
What about for a spacecraft? The new spacecraft with the Curiosity rover in it is traveling from Earth to Mars. It has a speed relative to the Earth. It has a different speed relative to Mars. It has a different speed altogether relative to the sun. Which one is relevant for the time of travel? None of them! An airplane’s starting and ending point are at a fixed distance from one another, but our spacecraft has a trickier problem, because Mars and Earth are moving relative to one another. And they will move a great deal relative to one another during the eighteen-month voyage of the spacecraft! Speed is not a simple thing in space, where everything is moving relative to everything else. That’s one of many reasons why rocket science does indeed take some serious training!
An aside for the future: In fact, because the earth is round and spinning, even a plane’s ground speed and air speed can get tricky if you think too hard about it. On top of that, planes don’t always take the shortest ground route when they fly, since sometimes they can ride a flow of air on a longer ground route to help them get a shorter flight time. The motions of planets and spacecraft, which travel in looping orbits around the sun, are complicated too. So if we go any deeper into this subject we’re going to have to go a lot deeper. But all that’s relevant here is that for short enough times all the motions are close enough to straight lines, and that allows us to appeal to Galileo’s principle of relativity. Let’s step back from the brink of this very long discussion for now, because there are other issues to deal with.
This brings us back to you, sitting in your chair. You may feel immobile, but you’re not. First, the earth is carrying you as it spins round its axis at something like 1000 kilometers per hour, depending on your latitude. Still more dramatically, the earth is also traveling around the sun, and we are all carried along with the earth — at about 30 kilometers per second relative to the sun. You don’t feel it, for two reasons. First, you only feel what you touch, and obviously you aren’t in contact with the sun. You’re in contact with your chair, and with the air in the room, and since you’re stationary with respect to them, you feel no motion. And second, your motion is in almost a straight line (it isn’t straight, but it bends very slowly) so Galileo’s principle of relativity applies to you, your chair and your room.
Meanwhile, the sun orbits the center of our galaxy — the Milky Way, that great city of stars in whose suburbs we live — at a speed of 220 kilometers per second. Whither thou goest, I will go — the earth travels with the sun, so our own speed relative to the galactic center is of a similar size. And the galaxy moves relative to other galaxies at even higher speeds… none of which we feel.
Finally, back to the BBC article. The spacecraft is traveling, according to the BBC, at 10 kilometers per second. Relative to what? I would guess that the speed quoted is probably the speed of the craft relative to Earth. But the article needs to say so! Otherwise the statement has no content. In fact, since Mars is further from the sun, and travels more slowly in its orbit than Earth (about 24 kilometers per second relative to the sun) it is quite possible that the spacecraft, despite having fired its rockets, has actually slowed down relative to the sun! That is, though it started off, like the rest of the earth, moving at 30 kilometers per second relative to the sun, it may now be moving at a slower speed (from the sun’s point of view) in order to make it easier for the craft to match Mars’ orbital motion down the line. That would be interesting to know, but unfortunately the BBC was silent on the matter.
And if you were on the spacecraft? Now that the rockets have stopped firing, and the spacecraft’s motion is steady, you would feel no motion at all. In accordance with Galileo’s principle of relativity, you would not know which direction you were headed or how fast relative to any planet or star, unless you very carefully measured the changing positions of the planets in the sky and watched the sun become gradually smaller. And were it not for your trust in the engineers and scientists who assured the rocket would send you in precisely the right speed and direction relative to Mars and Earth and the sun, you would have no idea whether you would someday approach Mars at all, or whether instead you would simply drift for eons as one more micro-planet among the multitudes.