36. Special Relativity

Consequences of Relativity

# Time Dilation

Patrick Ford

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Hey, guys, In this video, we're gonna talk about the specific consequences of a special relativity, starting with time dilation. All right, let's get to it. Now, Time dilation is always introduced with the same thought experiment, and it's pretty much the same thought experiment that Einstein used in 1905 to introduce the concept of special relativity. The only difference is they didn't have lasers back then. And we have lasers now. Besides that, it was done in the train. Okay, imagine that we have a rest frame s, which is shown immediately above me here. My finger gets cut off, so I can't quite point to it. And then above that, we have a moving frame s prime. So as prime is moving at the speed of the train, so the train appears to be at rest and s prime and s is the lab frame. So the train is just moving at whatever speed it's moving. Okay, remember, that s prime. We're going to consider the proper frame because we are interested in what's happening inside the train in this instance. And the lab frame, as always, is just the one that is at rest with respect to the earth. Okay, so let's say that there was an observer inside the train. So this is an observer inside s prime, and they're watching as a ray of light leaves. Some source bounces off a mirror at the top of the train and then comes back down and is detected at the bottom. Okay. And the distance from the floor to the mirror at the ceiling is some distance. H Okay. How much time is that observer going to measure passes. Okay, well, the distance traveled, right was to H just twice the distance that it had to go from the bottom to the top. And so the time measured is gonna be the distance divided by the speed. Right? Which is to h oversee, because light is traveling at sea the speed of light. Now, what about the same experiment? Measured from an outside observer, An observer in the lab frame. Well, that observer isn't going to see, like going up and down that observers going to see light starting over here and traveling at an angle, bouncing off the mirror and coming back down. Right? Because the train itself is moving, so it's gonna see light start here, travel up and then travel down in this triangular direction. Okay, let me minimize myself here really quickly. This height is the same. It's still just a church. Okay, but the light actually has to travel a further total distance, right? As we can see in this triangle right here that it's not traveling to times H. It's traveling this high pot news, which by definition has to be longer than H that high pot news doubled. So it's traveling to L not to h right into l is bigger than two h. Now I do the actual math here just to show you what is involved in it. But you don't need to worry about this per se. It's just that this is clearly longer than H because it's h plus some number. Okay, so far. Like I said, here, there's nothing strange about this. This is just geometry. Now where the strangeness comes in is the next bit, which is the fact that if we want to measure the time that the light took to travel that triangle, we're going to need the speed of the light right. However, the speed of that light is the same in the lab frame as it is in the moving frame. Remember that We got that. The time measured in s prime was to h oversee. The time measured in s is now going to be to l over See that same speed of light so you can see right away that if the distances are different, the amount of time that you measure to have passed has to be different. And it has to be different because that freaking speed of light is the same in both frames. This is why stuff gets so weird and special Relativity. It's because the speed of light has to be the same in both frames. If you were just throwing a ball right inside the train, the dude in the trains throwing the ball up and down, up and down that guy would measure some time. It takes for the ball to go up and down on outside observer, would see the ball go up and then down, and that would be a larger distance. But ah, ball is not the same as light. Ah, balls. Speed is not the same in both frames. The speed is going to be different in both frames. So if you did the same analysis, what you would arrive at is the time measured in both frames is the same. Because the ball's speed is justifiably sorry. Not justifiably is different enough to compensate for that change in distance. Right, that this distance right here is longer. So this speed is going to be faster, so the time measured is the same. But that's not true for light. Now, let me minimize myself everything here is all of the math required to come to the conclusion this right here to come to the equation That compares the speed of light measured in the moving frame to the speed of light measured in the rest frame. Okay, just in case you need to know this for your class, Okay? But the important thing to take away is this sort of simplified equation that the time measured in s the lab frame is going to be something called Gamma times. The time measured in the moving frame as prime where gamma something called the Lawrence Factor. And it's this denominator right here. One over the square root of one minus. You squared over C squared. Don't forget that you is the speed of the frame relative of the moving frame relative to the lab frame. Okay, Now, once again, I had been using from the beginning the concept of a proper frame and a lab frame in this case, the time and s Sorry. Let me do this. Yeah. Okay. So I did write it out like this. So the time and s is actually known as the dilated time. I had wanted to talk about the proper time first, but this is where we are. It's the dilated time, okay? And it's always going to be greater than the proper time. Let me minimize myself here the proper time. Because the event remember that we were talking about was the laser going up and down inside the train. That was the event we were interested in. So when we're measuring the time taken for that light to travel from the laser at the floor of the train to the top and back, that is the proper time. So the time in s prime is called the proper time, and the time and s is called the dilated time. And if you look at Gamma right here. As you gets larger, the denominator gets smaller and as the denominator gets smaller, gamma gets larger. Okay, so as you gets larger, gamma gets larger. So you are taking the proper time and multiplying it by a number greater than one. So this dilated time is always going to be larger than the proper time. Okay, now, typically, this is where the notation could get a little bit weird. And we're going to continue using this notation from now on. Just because this is how people do it. Don't ask me why the dilated time is typically given by Delta T Prime. Now, the reason why this can be weird is because the dilated time is actually the time an s not the time and s prime this notation Delta t Prime doesn't have anything to do with reference frame. It doesn't have to do with s or s prime. This is just the dilated time, Okay? Given our particular choice off s and s prime, the time measured in s happens to be the dilated time, okay? And the proper time or the time measured at rest with respect to the event, right? That is t not right? Delta t not. Okay, let's do one quick example here. It just says, um, spaceships have to travel faster than 11.2 kilometers per second in order to escape the earth. Gravity. This is called the escape Velocity of earth. Okay, we want to know. Can astronauts measure any noticeable amount of time dilation on a spaceship traveling at 11 kilometers per second? Basically so the dilated time is going to be a gamma times the proper time. Now if we look at mineralized myself really quickly if we look at the spaceship right here, right? Okay. Traveling fast away from the Earth's surface than the lab frame s is going to be the frame that an observer watching the spaceship leave the earth at rest on the earth is measuring right? And then the astronauts inside the spaceship are going to be in a frame s prime. Okay, that's moving at this velocity. You relative toe s. Now, if we want to know how a clock inside the spaceship is ticking, that's going to be the proper time. Okay? And if we wanted to know how a spaceship Sorry a clock on the earth is taking relative to the clock in the ship. That's going to be the dilated time. Okay, Often times it's easy to remember that the moving clock measures time or slowly. The moving clock will be the proper clock. The stationary clock will be the dilated clock. Okay, so let's just look at Gamma. Basically one minus. You squared over C squared times Delta t not right. This is going to be the square root of one minus. Let's just call it 10 kilometers per seconds. Let's 10 times 10 to the three over three times. 10 to the eight. We could just call it one times 10 to 8 because this number is actually going to be zero anyway and then squared. So what you're getting right here? This number on the interior is going to be. This is 10 to the four in the numerator. Tend to the eight in the denominator. That's 10 to the negative four. But you still have to square it. So this whole number is going to end up being 10 to the negative eight. So look at what you're doing. You're doing one minus 10 to the negative eight, and then you're squaring you're taking the square root of that. If you plug that into your calculator, it's going to tell you it's one okay. Or maybe 0.999999 something. All right, but it's most likely just gonna tell you that it's one. This means that astronauts traveling at this speed 11.2 kilometers per second do not notice any difference in time. Measured by their clocks relative to clocks on Earth, there is no noticeable time dilation in a foreign astronauts on this ship leaving Earth at the regular escape velocity just 11 kilometers per second. OK, so this wraps up our sort of introduction into time dilation, and now we're gonna follow this by some specific practice problems to get more comfortable with making these calculations. All right, thanks so much for watching guys.

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