“Time slows down when you’re near the speed of light” is often how you might see the phrasing on science discussion boards around the ‘net, but I think we should be more precise and careful. There are a number of really interesting aspects of Special Relativity I’ll eventually want to write about, but I think the most effective first one to address is this one — this will build the foundation that makes exploring other related topics much easier.
Train Frame
Ok, let’s get a simple movie in our heads. Here are two people (in physics, it’s always Alice and Bob) tossing a baseball back and forth — they’re 10 meters apart and throw the ball with a speed of 10 m/s, so let’s think of the ball traveling between the people as the ticking of a clock, 1 second between ticks. And, using fairly standard notation, we’ll indicate the time they measure with the Greek letter tau (this is also called “proper” time, or the time measured in the rest frame of this “clock”).
Now, they could be on the ground, or they could be in a train car moving at some arbitrary speed. Still, as long as they’re both in the same inertial (non-accelerating) frame, they would record 1 second between “ticks”. This, by the way, is one of the principles that’s important to keep in mind: in their frame (their “rest” frame), nothing unusual happens, no matter how fast their frame is traveling relative to someone else. If they’re in a sealed, quiet train car, there is no experiment they could perform to determine how fast they’re going; turning this around, all experiments they could perform would be indistinguishable from those performed at rest.
Ground Frame
Ok — this other frame is just one that’s in relative motion with respect to the Train Frame. Usually we think of the train as “really” moving, but there’s no point in trying to figure out which one is really moving; all that’s important is that one is moving at a constant velocity with respect to the other. Were you sitting on the train and looking out the window, for example, it sure seems like the world is moving backwards past you. To make things simple, let’s suppose the Train Frame above is moving to the right with a speed of 5 m/s relative to the Ground, and we’re in the Ground Frame watching it go by. So now the ball not only has a component of velocity between the two people (let’s call this “v-prime”, the speed of the ball that would be measured in that other frame, which we said was 10 m/s), but also it has a component of velocity along the train’s motion (the speed of one frame relative to another is usually named “beta”):
So what do we observe from the ground? The ball leaves Alice at the bottom location, let’s say at Ground coordinate x=0, and Bob catches it at x=5 because of the motion of the train during the ball’s flight. If we are to agree that the time between catches is 1 second in our frame also (using the “regular” ∆t for time), then the mathematics of right triangles tells us that the ball covers a distance of 11.2 meters in that second, so we on the ground would report that the ball is moving at 11.2 m/s. There’s nothing really weird about that — it’s common for observers in relative motion to disagree about the speed of things. The usual way to calculate the velocity of an object in one frame as seen from another frame is to use the Galilean Transformation
or, rearranging,
Which you can just see is the right thing from the above vector diagram. In words, this says that “the velocity of the ball in our frame = the velocity of the ball in the other frame + the relative velocity of the frames”.
The Point of Relativity
Ah, now we come to it. What is the key difference between light and regular objects? Light is always seen by everyone to travel at one speed, c. This is the central and organizing principle of relativity, assumed by Einstein. It’s true no matter the relative motion of the two frames! But why should it be true? Where does it come from? I’ll have much more to say in a future post; for right now, let’s just all take it as given and look at the necessary consequences.
Note that this does not comport with everyday experience. If Alice throws the ball at 10 m/s in the direction of motion of the train (10 m/s), we on the ground would measure the speed of the ball to be 20 m/s. If the train instead were going at 0.5c and Alice emitted a pulse of light in the direction of motion of the train, we would measure the speed of that light to be exactly c, and not 1.5 c.
First of all, this implies something remarkable: if light is seen to move at the same speed in both frames, but covers a different amount of distance, then it has to be true that the measure of time is different in each frame! Speed is just displacement divided by time, after all, so if the displacements are different, then the times have to be also.
Let’s re-frame (ha!) the above example so that, instead of throwing a ball back and forth, there is a pulse of light bouncing between mirrors held by Alice and Bob. We’ll have to separate them by 300 million meters (which is 1 light-second of distance) so that they’d still register 1 second between bounces, but it’s a thought experiment, so we can do what we want. And then let’s say the train moves to the right (according to us) at 0.5c to keep the diagram at least schematically consistent:
Let’s make sure this makes sense: the vertical displacement is what is observed in the rest frame of the clock, so we use the letter “tau”. The other two displacements are observed in the ground frame, so we use the letter “t”. Ok then, this is just a right triangle, so let’s try to find the relationships between the times starting with Pythagoras:
Now a bit of an algebra break, rearranging to get
So apparently the times in the different frames are related by this factor that we’ll call “gamma” (it pops up so often in relativity that it deserves its own definition):
And finally,
Very cool. Our example had the train moving at 0.5c, so the “gamma factor” then would be about 1.155. What does this mean? For every second that goes by on the train (the rest frame of the clock), 1.155 seconds go by on the ground. So, in this sense, the clock on the train would appear to tick more slowly. You can verify that if we want the “gamma factor” to be 2, the speed of the train ought to be about 0.866c. And at 99% of the speed of light, gamma is about 7.1 (this factor increases rapidly the closer you get to the speed of light).
Back to the beginning, though, it’s worth emphasizing that this seemingly bizarre behavior of time (which is usually thought to be immutable and universal) in different reference frames is a direct consequence of the constancy of the speed of light. Why didn’t this happen for observations of baseballs? Because the speed of the ball changes between frames. For light, the speed is constant, so it’s the observed time that changes.
On deck:
The article I’m working on next is a pretty simple toy model of the Greenhouse Effect and the basic physics behind global warming.
SO EXCITING!!!
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