What we will explain is why, for objects moving uniformly in a straight line, time runs slower. We’ll use no mathematical symbols, and won’t even need any pictures!
OK, here goes:
Before we begin, you have to accept two facts. The first is that if you are in uniform motion, all the laws of physics appear exactly the same as if you were standing still. Think about it: if you are in a smoothly moving airplane, you can pour coffee into a cup, etc., just as if you were sitting on the runway. In short: if you are in a windowless box, unless you are accelerating, going around a curve (which is the same thing), etc., you can’t tell if you’re moving or standing still. Agreed? This is called the Principle of Relativity. The first person to point it out (that we know of) was Galileo.
The second fact is that the speed of light through space is constant, for all observers. This is just a curious fact of the world, but it’s been shown to be true. If I’m standing still, and a beam of light whizzes past me, and you take off after it at a million miles an hour, we are still both going to measure the speed of the receding beam at about 186,282 miles per second — a value known as c. I won’t belabor this part — but, for example, if the speed of light weren’t constant, then light coming toward Earth from a faraway planet when it’s moving toward us in its orbit would travel toward us faster than light shining from the planet when it’s moving away, and we’d see everything all out of sync. For that not to be true, there’d have to be some medium (like air for sound waves) that held the velocity constant. But we know there isn’t, thanks to the Michelson-Morley experiment of 1887. Everybody, no matter how they are moving, always sees light moving at the same speed. That’s just the way it is.
The rest is really easy.
Imagine a special kind of clock. It consists of a mirror on the floor, a mirror on the ceiling, and a pulse of light that bounces endlessly back and forth between them. Call the time this takes one “tick”. Because the speed of light is constant, the length of the ticks are always exactly the same.
Now imagine mounting one of these in a railway car. Let’s also put a very accurate conventional clock on the wall of the car. They should run nicely in sync, because, after all, they are both very accurate clocks.
Now set the train in motion, whizzing by as you stand on the platform. (We’ll give the railway car a glass wall, so we can see in.) The train runs very smoothly and steadily. Inside the train is an observer (let’s call him Al) with his back to the window. The train runs so smoothly, he can’t even tell for sure if he’s moving at all.
For Al, each tick of the clock takes a constant time, which is the height of the car (i.e., the distance between the mirrors), divided by the speed of light.
But things are different for you, standing on the platform, watching the train go by. By the time the light travels the height of the car, the bottom mirror has moved down the track a little way (along with the rest of the train). And by the time it bounces back to the ceiling, the train has moved the same distance, again. This means that, as far as you are concerned, the light didn’t go straight up and down, but made a sort of zigzag down the track, which is a longer path. (To be specific, the length of each bounce is the hypotenuse of a right triangle that has the height of the car as one leg, and the distance traveled by the train as the other.)
Are we clear about this so far? Can you picture it?
What this means, then — because as far as you can see, the distance traveled by the light is longer on the moving train, and because the speed of light is always constant — is that each tick of the clock, as seen by you on the platform, takes longer than it does as seen by Al on the train. (Remember that Al doesn’t even know if he’s really moving or not, and so for him the light just seems to be bouncing straight up and down.) And if you think that this is just some jiggery-pokery involving weird light-clocks designed just to give this effect, remember that if the light-clock were to get out of sync with the wall clock, Al would notice, and would therefore be able to tell he was moving! But we already know he can’t possibly be able to tell in any way whether he’s moving or not, thanks to the Principle of Relativity.
So there it is: since time is what clocks measure, and since clocks tick slower on a moving train, then time runs slower on a moving train! Simple, right?
If you want to figure out just how much slower, it’s easy; it’s all just a bunch of right triangles, and a little algebra. You can do it. (Maybe I’ll do it for you in another post, but it’s really not hard at all.)
Last thing: that train gets shorter, too. If you want to understand that, just turn the clock so it’s lined up lengthwise down the train. I’ll leave that one to you.
So: want to have some fun? Teach a little kid how Special Relativity works. You’ll both feel great.
25 Comments
I think I’m of below-normal intelligence. Although I love the illustration, it seems to me that it’s showing that, from the perspective of the train-platform observer, things move faster within the same time frame: the “tick” takes the same amount of time whether one is inside the train or outside of it, but the distance that the light has to travel appears radically different depending on perspective. Not time, then, but motion–and it’s not a matter of seeming, either: the bouncing light really is covering two different distances depending on perspective.
Kevin,
Your mistake is that the “things” you presume are moving faster inside the train are strictly just the photons (which are the particles of light). But that conflicts with the universal law that “the speed of light through space is constant, for all observers”, which is probably why this constant speed is usually represented by the lower case letter “c”. I am not sure that the reason for this representation by “c” is historically correct, but that is really immaterial to the fact that light speed is a universal constant.
Malcolm, you lost me. Sorry. I must be like Kevin because all I can think is that Al may have his back to a window but can’t he look out of the other side of the carriage? I’ve never been on a train that doesn’t have windows on both sides. So I would see that the train was moving even without the sensation.
As for the aeroplane example, well, they always bounce around. I’m always aware that I’m moving, and praying that it’s not in a downward direction, unless we’re about to land.
You say that the train gets shorter. Does it stay that way after it’s reached it’s destination? If so, I should have some sympathy with the ever decreasing lack of space in the carriages. It’s not their fault, they’re just shrinking .
Sorry Malcolm, I know that you’re just trying to lift the mood. Do you know how much I love you?
Let me be a little bit indulgent here. I was on a train back to Cambridge, nearly home. The station before Cambridge is Ely. A robotic voice asks that you to please check the overhead storage spaces, take everything with you and “thank you for traveling with us today”. It is the same message at each stop on the line. The train proceeded to fly through Ely so fast that I most definitely could not read the sign. All the passengers who wanted to get off were standing next to the door. I did catch the eye of a studenty type person opposite who seemed to find it quite funny and we had a giggle together. Not so funny for the commuters who had to gallop off the train, run over the bridge and catch the train going back in the opposite direction.
Kevin, you’re not.
good source https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations
As the person observing the train, how can I tell it’s the train moving and not me? No fair using the earth as a reference point, either–in relativity, there is no fixed frame of reference. Everything is in motion.
That means the same observational effect must apply in either direction. BOTH observers feel as if their own local time frame is moving more quickly than for the other, and both of them will see the foreshortening effect when they look at the other.
And that’s where my brain starts to short out. It’s almost as if our whole conception of movement is the opposite of reality. Maybe it’s actually light which is “at rest” and is the fixed frame of reference. Everything else, including distance and time, appear to vary depending on what you’re looking at. But light is always the same. Then again, it can’t be at rest, because how can something which is “at rest” be radiating in all different directions at once?
Henry,
Thanks for the clarification. And indeed, Malcolm himself was at pains, in his illustration, to point out that the constancy of the speed of light in a vacuum was to be taken as a given. I suppose my point was that I felt something was missing from the explanation: the idea that “time runs slower on a moving train” was a deductive leap to which the story hadn’t exactly built up.
I suppose, if I were to attempt to fill that gap in, I’d have to resort to a wee bit of math–specifically, the formula “distance equals rate times time,” or “d=rt,” which I simply call “dirt” when tutoring algebra to secondary students.
With “dirt” in mind, we take as given that “r” (rate, i.e., velocity) is a constant for light, as long as it’s traveling through a vacuum (light slows down in glass, air, and water). In the light-clock example, if we’re supposed to assume that “r” is unmovable, then the other parts of the equation have to give. Assuming distance is also unmovable for our purposes (the train travels X meters whether I’m inside the train or observing it), then the only other factor that can change is “t,” i.e., time. This, I think, is what’s missing from the story: the insight about time is the inevitable conclusion based on both an observation of the light-clock and the assumption of light’s constant velocity. Velocity can’t budge; distance can’t budge; but since something has to budge, that something must be time.
Or maybe I’m just seriously lacking in reading-comprehension skills. (Thanks, Musey, for the reassurances.)
Kevin,
That’s exactly right. And because the speed of light is constant, that means that, from the perspective in which the distance is longer, the ticks are farther apart. In other words, from the persepective of the person on the platform, the clocks in the train are ticking slower than my clocks are.
But there’s nothing special about me on the platform; from Al’s perspective it might as well be him standing still and me moving. So he’ll see my clocks ticking slower, for exactly the same reason.
Kevin,
Since you are truly interested in details, I will point out that when speaking of a constant as well as an isotropic rate of distance with respect to time, that “rate” is termed a “speed” as opposed to a “velocity”. Velocity is a vector quantity, namely a speed in a specified direction. Speed, including light speed, is a scalar quantity.
Musey,
Sure, Al can look out the window if he likes. But he might just as reasonably assume that he’s standing still, and everything else is moving. (We’ve all had that experience of seeing the train next to us moving, and not being sure which of us was — or looking up at moving clouds and feeling as if the earth was moving.)
Perhaps it’s better to think of two spaceships way out in interstellar space. No bumps, no moving scenery.
ce999,
Well, right! You can’t. It’s all relative, you might say.
Exactly right.
As for the rest of your comment, we begin to get into the deeper waters of general relativity, and something called the space-time “interval”. In a very important sense, from a photon’s perspective, light is at rest. But that’s way beyond the scope of this post.
Something that is even more counterintuitive than Special Relativity is that whereas Malcolm is on the Atlantic coast I am on the Pacific coast. Nevertheless, we appear to be on the same daily schedule. It’s a conundrum.
:)
Kevin,
I’ve edited the post slightly to make that clearer. And it seems like you’ve got the idea!
I didn’t mention that light travels at different velocities in different media; it wasn’t really relevant.
I’m a night owl, Henry.
Yes, Malcolm. I have known that for quite a while. That was the purpose of my trailing smiley :)
You did have an appreciative reader here. Martin said, not expecting to be reported, “he explained that very well”. High praise indeed!
You’re going to be a fine granddad some day.
Thanks, Musey. (July 23rd of this year, if all goes according to plan.)
Kevin,
Well, not quite: from Al’s perspective, the train hasn’t moved at all; it is just as reasonable for Al to say the platform is moving. (If it were the train that was unambiguously moving, regardless of perspective, then that would violate the Principle of Relativity.)
So from Al’s perspective, the light travels a shorter distance per tick (just the height of the car) than it does from your viewpoint on the platform. Because the speed of light is constant for any observer, though, the ticks are farther apart when seen from the platform, because the light has farther to go. So to the man on the platform, the light-clock ticks more slowly on the moving train. And because the light-clock can’t get out of sync with the wall clock, or Al’s heartbeat, or any other process on the train (the Principle of Relativity again) then we can only conclude that time itself passes more slowly on the moving train.
And of course Al can draw the same conclusion about us!
I have a minor suggestion that may make the explanation even simpler: It’s not that the speed of light in space is a special constant, it’s that there exists some generic maximum speed c. In order for c to be the maximum speed limit, it must be constant for all observers. And as you already explained, everything else follows from that.
(Landau and Lifshitz pointed this out in their classic textbook “Classical Theory of Fields.”)
I’m only making this comment because in my teaching experience, describing c as a generic maximum speed limit instead of as “speed of light in vacuum” removes a lot of the mysticism and confusion students have because they always wonder “what makes light so special?”
Science Teacher,
Yes, that’s an accurate and important point. The “speed limit” c is just a feature of spacetime itself; the spacetime interval between any two events on a “lightlike path”, i.e., any two events that could possibly be connected by anything traveling at the speed of light, is exactly zero. (That’s rather beyond the scope of this post, but the point you made isn’t, and I wish I’d made it myself.)
Thank you!
Jeeps Malcolm, I’m psychic! I keep telling everyone but they don’t listen!
Very beat wishes to the parents. And to you guys, Malcolm and Nina, you’re the best.
Beat was supposed to be best! I can’t change it.
I taught Einstein’s introduction to relativity last quarter and I actually had quite a bit of trouble puzzling out exactly what the principle of relativity is supposed to mean. There was an interesting wrinkle in Einstein’s presentation relating to Galilean systems (the stars shouldn’t move in circles, as far as I could gather), but it all got a little confusing. What exactly is supposed to be constant between systems moving uniformly with respect to each other and what isn’t? I ended up paraphrasing it rather lamely as, “Your description of how things are moving should be pretty much the same — with minor adjustments — i.e. there should be a way to translate the descriptions from system to system.”
Is the speed of light constant or do we observe it to be constant because our observations are based on light receptive cognition? If you were approaching a light emitting body at the speed of light would the speed which the light arrives to your eye still be the speed of light? Illogical it would have to be twice the speed of light but could your eye tell the difference? If while doing this you passed an observer who was standing still who was also observing the light his observation would be that it was moving at the speed of light. The two cannot both be true BUT the observation/interpretation can be true. The simple answer is that this theory of the speed of light being constant is incorrect but we cannot observe/detect the true answer.
The STR [Special Theory of Relativity] equations can seem meaningless to many a folk. However, as mentioned by the popular physicist Brian Greene, in his book called “The Elegant Universe”, pages 26 and 27, all objects are constantly on the move within the 4 dimensional environment called Space-Time, and they all do so at the speed of light, aka “c”.
In turn, if “c” represents the constant motion of all objects within Space-Time, suddenly the equations all become so simple to understand.
Personally, all I ever saw being taught concerning STR, was the bizarre outcomes predicted via STR, rather than the cause of STR having been exposed. Plus I knew nothing about Brian Greene’s work.
So I said to heck with all that, and resorted to do it yourself. I threw everything else out the window and started from scratch, and did so by starting with a step by step analysis of “motion”. The outcome of my analysis was the gaining of a full understanding of STR, along with having independently derived all of the STR equations, including the Lorentz transformation equations, and doing so in a manner that has not been done anywhere ever before. I then posted it on YouTube.
http://www.youtube.com/watch?v=KKAwpEetJ-Q&list=PL3zkZRUI2IyBFAowlUivFbeBh-Mq7HdoQ
Brian Greene wrote, “Einstein proclaimed that all objects in the universe are always traveling through spacetime at one fixed speed — that of light.” Greene, moreover, proceeded to clarify how Einstein’s seemingly preposterous “proclamation” via the Special Theory of Relativity (STR) makes sense.
The key to making sense of it is that the speed of every object is allocated in varying proportions along the 4-dimensional axes of spacetime (the 3 spatial dimensions that we are all familiar with, and the 4th dimension, time, that we ordinarily do not conceive of as a dimension). Thus, all objects that have non-zero mass will express most of their speed in the time dimension, and, with the exception of certain phenomena (such as ejecta from a supernova whose speeds are a substantial portion of light speed through space) very little in the three spatial dimensions.
Light, however, which comprises massless particles (photons), expresses all of its speed through 3-dimentional space and none of it through the dimension of time. As a consequence of light’s unique (except that “gravitons”, which have just recently been detected share this property) subdivision of its speed (i.e., through space only) light never ages! The microwave background radiation (electromagnetic radiation at very low temperature) that we detect is the same age it was when it first emerged, roughly 14 billion years ago.