Speaker 1:

Hey everyone. Welcome to this next episode of Your Daily Equation. Yesterday we focused on Einstein’s E equals MC squared. Today we’re going to focus on a different equation but related to equals MC squared. It’s the time dilation equation. It’s a wonderful profound equation that in fact allows us to seriously talk about a certain kind of time travel, not sort of sci-fi time travel, but unless the sci-fi film is really good, had good advisors perhaps, but real time travel written into the laws of physics. All right, let’s jump right into it. The environment, the situation is very similar to that which we discussed yesterday with equals MC squared. Albert Einstein is at the patent office, 1905, Bern, Switzerland thinking about the ways in which you might synchronize clocks that are at a distance from one another, thinking about light and its speed. He comes to this wonderful and strange conclusion that the speed of light is a fixed, unchanging constant number, 671 million miles per hour, 300 million meters per second, use whatever units you like. One light year per year. It doesn’t matter.

The point is that that number in whatever chosen unit is independent of who does the measurement, independent of the motion of the source, motion of the receiver, a fixed value for the speed of light. The speed of light is constant. Good. Okay. Now, why would this have any impact on our understanding of time or even space and time? Well, the answer intuitively is pretty straightforward. As I discussed yesterday, speed is nothing but a measure of how far you go compared to how long it takes you to get there, and how far you go, that’s distance, that’s a measurement of space. How long it takes you to get there is time, right? That’s the duration. It’s how much time elapses. If speed behaves weirdly at say huge speeds, the speed of light and space and time must behave weirdly too. They’re all intricately interconnected. What Einstein does in June of 1905 is he works out the implications of the constant nature of the speed of light on both space and time.

Today we’re going to focus upon the result he found for time. Now, there’s one other piece of background that’s really good to have in mind that Einstein makes profound use of, which is simply this. If you are moving at a fixed speed in a fixed direction, not speeding up, not slowing down, not going around a curve or a band or anything like that, then you have every right in the world to claim that you are at rest and it is everybody else you see who is moving with respect to you. The example that we often use in this case, imagine you are on a train. Okay? You don’t feel any bumps, you don’t feel any turns. The train is moving at a fixed speed and fixed direction. It could actually be in the station for all that you know, and you look out the window when you wake up and you see another train in motion. For a split second, you don’t know whether that train is moving, whether you are moving because all you sense is the relative motion between them.

From your perspective, you’re completely justified in saying that it is only that train that is moving because you look around in the train car, the seats aren’t moving relative to you, the passengers, the other passengers are not moving relative to you. From your perspective it is as if that train is stationary even if it happens to be hurling down the tracks at a hundred miles an hour. You have absolutely every right to claim that it is the other train. In fact, everything else in the world that you see out of your window that is moving relative to you. Fixed speed, fixed direction, you don’t feel the motion. No acceleration. You can claim to be at rest. Good. Okay, now, how do we make use of the constant nature of the speed of light and this obvious fact that you can claim to be at rest if you’re not accelerating? How do you make use of that to gain some insight into the nature of time?

All right. Now to do that, we need a clock, a device that measures how much time elapses in any given situation. I could introduce whatever clock I’d like. A grandfather clock, I could introduce a Rolex watch. We could use the time as measured by my iPhone. Any clock is really as good as any other as long as it’s well made. For the analysis that we’re about to undertake, there is a very special kind of clock that makes the mathematics and the physics and the analysis much easier. It is an unfamiliar clock. It is called a light clock. I don’t know the history of who introduced it. It’s a wonderful pedagogical device. Maybe it was Einstein, but I think it may have been Richard Feynman. Maybe somebody will tell me the intricate details of the history of the light clock. What is a light clock? It is simply two mirrors that are parallel to one another with a single ball of light bouncing up and down between the two mirrors.

When the ball of light goes up, we call it tick. When it goes down, we call it tock. Tick tock, tick tock. That is a clock because you can measure durations by how many round trip travel times, how many tick tocks go by on the light clock. Just so you have a little visual picture of that, I have an animation which I believe I can bring up. Good. Here it is. There are two light clocks side by side. Each has this mirror, these mirrors that are parallel to each other. Then you see a tick tock, tick tock, and the counter on top simply counts the number of tick tocks that go by. If you want to know how long it takes someone to whatever, eat a pizza, just count how many tick tocks on the light clock go by and that gives you a measure of how long it took that event, that eating of the pizza to take place. Although it is an unfamiliar kind of clock, it does what any other clock in the world does. It measures how much time goes by between the start of an event and the conclusion of an event.

Why do we introduce this particular strange clock? We introduce it because it’s so simple, right? It has no gears, no quartz crystal, no caesium atom vibrating, and because it’s so simple, its inner mechanism is so simple we can easily determine how motion affects the passage of time on a moving version of a light clock. We can figure out how motion affects the passage of time, and that’s ultimately what we are after. What we’re going to do in a moment, and I’ll show you an animation, but let me first describe it in a more homegrown way. I’m going to take one of these light clocks. I’m going to set it into motion, fixed speed and a fixed direction. That’s crucial. I’m going to figure out how the motion affects that tick tocking of the clock. Before I do that, there’s often a confusion that seeps in.

People often wonder, well look, if the clock is in motion, light clock is in motion, the ball of light people wonder might not simply go up and down. It might hit the sides, bounce around. Maybe I destroy the workings of a light clock by having it go in motion. If I was accelerating the clock, there would be some issues there. Imagine this clock has always and forever been going at a fixed speed and a fixed direction. Will the ball of light still go tick tock? Answer, absolutely yes. What’s the argument? Very simple. If the clock is going at a fixed speed and a fixed direction, then if I’m carrying the clock, for instance, I can claim to be at rest and all of you, I would say are moving relative to me. Since I’m at rest, the light clock will do what it always does, tick tock up and down. The ball of light will travel. How could your motion out there have any impact on the inner workings of the light clock?

Since I am not accelerating, fixed speed and fixed direction, claimed to be at rest, light clock does what it always does. Good. Okay. Now there is something that does change when you watch my light clock in motion, imagine it in your minds. I’ll show you an animation in just a second. The ball of light starts here and as I’m moving from your perspective, not from mine but from your perspective, the ball of light will go up and down, right? It goes up to meet the upper mirror and down to now reach the lower mirror. This clock is in motion. You see the ball of light go in a double diagonal up and down. The key thing is the trajectory, the path is longer for the double diagonal than it is for the straight up and down. Because the speed of light is constant, if the light has to travel a longer distance to go tick and a longer distance to go tock, then it wouldn’t be stationary clock, then it will go tick tock more slowly.

There’ll be fewer tick tocks, therefore on the moving clock compared to the stationary clock, less time will elapse on the moving clock compared to the stationary clock as you watch, say my moving clock go by you. You work out the number of tick tocks on your stationary one. It will be a bigger number of tick tocks, more time will elapse than on the moving clock. Let me show you that one to make it a little bit more clear. Here what I’m going to do exactly what I did with my hands. I’ve got these two light clocks. I’ll leave one stationary, the one on the left. The one on the right starts to move. Notice that the double diagonal trajectory is longer so the tick tocks are happening at a slower rate and look at the counter. Less time has elapsed on the moving clock from our stationary perspective compared to that on the stationary clock itself, time slows down on a moving clock is what this argument gives us.

Key to this is that the speed of light is constant because the light is traveling a longer distance at the same speed in order to go tick tock in the moving clock, and therefore it goes tick tock less frequently, less time elapses. That is an astonishing conclusion. Before Einstein’s realization of this effect called time dilation, everybody thought that time is universal. Isaac Newton. Imagine that there was this cosmic clock out there in the universe relentlessly ticking forward second after second after second independently of who’s doing observing, independently of who might be in motion. Yet Einstein realizes that that universal cosmic clock is a fiction. It’s a fantasy. The rate at which time elapses is very much in the eye of the beholder. It depends on who is in motion and who is doing the observing. That is a stunning realization. It cracks the universality of the Newtonian notion of time leaving us with an Einsteinian version in which time elapses differently for different individuals if they are moving relative to one and other.

We can now go a little bit further and let’s get to the equation which will tell us the factor by which time slows down on a moving clock. We can work that out. It’s a pretty straightforward calculation. Let’s do it together. I’m going to do it right here on the iPad. That’ll come up on your screen. What am I going to do? Well, here is my stationary light clock. In this case, the ball of light is going up and down to go tick tock. Let’s call it distance L that it has to travel to go up and down. Now let’s compare that to say the moving light clock and the moving light clock, the ball of light is going on this double diagonal to go up and down. In fact, I’m really only going to focus on the tick. The tock is the same. The ball of light goes like that. Let’s call this distance L prime. If you want to figure out the ratio of time elapsed on the stationary clock compared to time on the moving clock, well, again, more time elapse on the stationary compared to the moving clock. That therefore has the ratio of L prime over L.

Imagine just for argument’s sake that the L prime, the diagonal trajectory for the moving clock, imagine it’s whatever, three times larger than the straight up and down trajectory on the stationary clock. Then the moving clock will tick slower by a factor of three, less time will elapse on it, therefore by a factor of three. Therefore, the ratio of time on the stationary clock compared to time on the moving clock will be three. Three times more time will elapse in the stationary compared to the moving clock if L prime over L was equal to three. One concrete example, but what we want to do is we want to imagine that this clock here has a given velocity called V and we want to work out L prime over L in terms of V itself. How do we do that? Well, it’s actually a straightforward calculation, makes use of nothing more profound than the Pythagorean theorem. It’s really quite beautiful. Let’s imagine that you and I are watching the moving clock in motion.

What do we say? We say from our perspective, imagine that it took a time delta S for the ball of light to go from the bottom of the moving clock to the top mirror of the moving clock. If that was a time delta S, then we would say that the clock move from here to here. How far will it go? Well, it’s velocity times time, so this would be V times delta T S. We say that this up and down distance is equal to L. Now you see that we have a beautiful little right triangle and that will allow us using the Pythagorean theorem to relate the various times at hand. Right? How do we do that? Well, we know, I’ll just write over here that L squared plus V squared delta T S squared, that’s A squared plus B squared in high school language, is equal to L prime squared. Now L prime squared, I can also be a little bit more cunning about. This distance will be equal to delta T S, the amount of time that it took the ball of light to go from the bottom mirror to the top mirror times the speed of light, the constant speed of light C.

I can now set L prime squared equal to delta T S squared times C squared. Okay, that’s great because now I can write L squared is equal to delta T S squared times C squared minus V squared, and therefore L is equal to Delta T S times the square root of C squared minus V squared. Okay, now I can take that and plug it upstairs into our little ratio over here. L prime itself is equal to C times delta T S. Then if I put down the L downstairs that we just calculated, that is the square root of C squared minus V squared delta T S. Cancel out the delta T S and divide top and bottom by C just to simplify, and to get one over the square root of one minus V squared over C squared. That little expression is the beautiful answer that we get to. It has a name, maybe I should give it another color just to highlight it. This guy has a name, it’s called the gamma factor.

It tells us as we just derived that the ratio of elapsed time on the stationary clock, this guy over here compared to elapsed time on the moving clock is given by one over the square root of one minus V squared over C squared. Just to summarize that, we just found that delta T S is equal to one over the square root of one minus V squared over C squared times delta T moving. We are the stationary observer watching that clock in motion. This factor over here, which I gave the name gamma. Notice the gamma is always greater than one because V is always between zero and C and therefore the denominator that we have in that expression will always be less than one and the reciprocal of a number less than one is always a number that will be bigger than one. There it is. There is this beautiful time dilation formula and you notice that for every day velocities, V is so much smaller than C that this gamma factor is so close to one that we don’t notice it.

That is why for thousands of years we have been fooled into thinking that time is universal, that time is independent of our motion. It’s not independent of our motion. If we grew up in a world in which we routinely travel at velocities V that were near the speed of light, we would all know that in our bones, but we don’t know it in our bones because we don’t live in that world. But Einstein revealed that if speed is close to the speed of light, this gamma factor gets bigger and bigger and that means that the time dilation gets larger and larger. In fact, I can even show this to you in the little demonstration that I’ll bring up over here. Great. Here are two clocks side by side. In this demonstration I’m going to pick the speed of the moving clock and show you how time elapses on that clock compared to the stationary clock.

Imagine that that moving clock stays in a rocket ship or picking the speed of the rocket ship. Okay, let me put an ordinary number. I’ll put in, whatever, thousand miles an hour. Let me put this in. Okay. Thousand miles an hour and I hit, and there you see that there’s virtually no difference between the ticking of the two clocks because the slowdown of the moving clock is barely noticeable. Now I’m going to put in a bigger number. Let me put in, say 667 million miles an hour, now look at what happens. Look at the hand on the clock on the rocket ship, it is ticking off time so much slower than time on our watch, time on a stationary clock. This is really how time itself behaves. Time slows down for a clock that is in motion. Indeed, if you imagine therefore going on that rocket ship, going out into space, turning around and coming back on your moving clock from our perspective, we will see that you’ve only aged one year.

These durations that clocks are measuring are real time. It’s like biological time. It’s psychological time. We will see that you have aged only one year say on that roundtrip journey, but when you step off the ship, our clocks on earth, they’ve been spinning around really quickly. Therefore it might be a thousand years into the future, 10,000 years into future, a million years into future, all dependent on how closely your speed V gets to the maximum speed, the speed of light. It’s a wonderful, profound insight. Many puzzles if you think about this, things that go into the name of the twin paradox, which I’m happy to answer questions about this if you want to ask about them in the comment section, you should think about it too. The basic question is, from the perspective of the person on the rocket ship, could you flip the argument around and say, the clock on earth is taking off time more slowly than their clock? Think about that one.

There’s an answer for why that argument fails and why truly is the case that the person stepping off the rocket ship would be actually younger than the person who stayed on earth. I’ll let you think about that one. That’s our equation for today, the time dilation equation that comes out of Einstein’s special theory of relativity. Again, love to hear your suggestions for other equations. I think tomorrow I will likely look at the so-called length contraction formula, which shows the effect of motion on space. Today we did time. From there, I don’t know exactly where we’ll go. I’ll take your suggestions, maybe a little bit more of relativity and so forth. Join us and I look forward to seeing you at the next episode tomorrow. Okay. Take care.