9 - 5 - The Happiest Thought (16-53, High-Def) (1)

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  [SOUND]. In 1907, two years after the miracle year, Einstein had been asked to write a review article on the special theory of relativity. For a a science yearbook. It was really designed not for the general public, necessarily. But for other, other scientists. Just summarizing some of the key results of previous years. And he wrote later, reminisced later, that he had what he called The Happiest Thought or the most fortunate thought of his life. In fact, here's how he, he described the situation. he, he had been bothered as he thought more about the special theory of relativity, he was bothered by a couple things. And one thing was there's only constant velocity in motion and ,therefore it wasn't general enough. And also didn't take into account any gravitational effects. So he had been mulling those things over and later he reminisce that I was sitting in a chair in the patent office in Bern, Switzerland, when all of a sudden, a thought occured to me. If a person falls freely, he will not feel his own weight. And he says this, this revelation startled him. And as he thought more about it over the next eight years the end-result was the development of his General Theory of Relativity. So even though in one sense it's, it's beyond the course, we want to get at least a little insight into what that is all about. And see, actually in a couple of instances, well one instance at least, is the one we're going to talk about in this video, it gives us a result that is similar to the Special Theory of Relativity but different in a couple of key respects and that is a time dilation effect due to gravity. Oh, he talked about that if, if a person, as he, as he says. if a person falls freely he will not feel his, his own weight. And so, the classic example of doing that, and Einstein himself used this would be in an elevator. So let's imagine [UNKNOWN] it's a thought an elevator out there in the middle of  space. If it's falling freely or even just a real elevator if you cut the cable and there are no other safety mechanisms, as you fall freely, you will not feel your weight. if you drop a ball at that point, it will just stay there because [UNKNOWN] you're all falling down towards the center of the earth or down towards the surface of, of the earth. So, it will be like gravity doesn't exist anymore. And Einstein started analyzing that situation a little bit more and noted also that if you, say for your elevator again, you, you attach a cable to it, right? And maybe there's a crane or something, getting out in the middle of space, somewhat fanciful here. But the idea is you have something where now you can lift the, the elevator up. In fact, you can accelerate it in an upward direction at some acceleration a. Again, just like you, we would do in a real elevator. As the elevator goes up you feel that momentary push downward. You feel heavier. It's like gravity has actually increased. And so, an acceleration upwards, if you're in the elevator is, therefore, equivalent to a gravitational field. And this was essentially Einstein's Happiest Thought. And it came to be known as the Equivalence Principle. [SOUND] The Equivalence Principle. That, essentially, a gravitational field is equivalent to an accelerated frame of reference. Right? That this person inside here if, if they're walled off from everything. They could be in null space for all they know if the crane or the elevator whatever the cable pulls up on them at an acceleration that's equivalent to the acceleration due to gravity, then it feels like gravity to them. They couldn't tell whether they were in the elevator, just on the surface of the earth, feeling gravity pulling them down, or they're actually out in the middle of space, and some imaginary crane is actually pulling them at an acceleration such that they press down into the floor, as it were, and you feel that normal gravitational pull.  So that's the Equivalence Principle. Gravitational field is equivalent to an accelerated frame of reference. And so you see that, that opened up some things for Einstein because he is interested in going beyond the Special Theory of Relativity, which is only constant velocity and inertial frames of, of reference. And here was something that enabled him to perhaps, if he pursued this, to, to bring in accelerated frames of reference and also tied it into gravity as well. And so again over the next eight years in a number stops and starts and, and dead ends, and so on and so forth, plus he's working with other things along the way, he was able to come up with the general theory of relativity. And the, the key thing for analysis is that what it enables, the Equivalence Principle enables you to do is, if you could do an analysis involving acceleration, then the conclusions from that analysis also apply to a gravitational field. The situation where you just had a gravitational field occurring. And we want to do two examples of that. In this video we're going to do it, the time dilation example, and in the next video we'll do the bending of light example. And so, let's imagine a situation like this. So we're going to have an elevator here. And we'll have our, our two observers [UNKNOWN] two clocks. And so we'll put a clock at the top of the elevator, clock U for upper. And down here, we'll have clock L for lower. And these clocks emit little pulses of light. So, here's one [INAUDIBLE] get some nice red pulses of light coming up here in that direction and then appear another little laser and pulses of light going down and we'll have a, a detector here and there. Okay? And so the, the clock, each clock and these are, are synchronized clocks, the identical clocks and they, they emit these little pulses of, of light. That maybe they emit you know, ten pulses of light per one tick of each clock. Okay? Now another factor here is that we're going to assume any velocities involved  are much less than the speed of light. So, this example will have nothing at all to do with the special theory of relativity. We'll assume the velocities are low enough that we can ignore all relativistic effects. They're very, very small compared to everything else going on here. so we've got the clock. So that means the, the two clocks are in sync with each other. Right? We don't have to worry about leading clocks lag and all that because any velocities involved are much less than c. And we're going to now put this into acceleration in upward direction. So we're going to accelerate it in upward direction, but again, at very, very small velocities. You know, just sort of ordinary, everyday velocities so we can ignore any special relativistic affects there. And so lets think about what happens then to say clock L down here, the observer at clock L. they will be observing these pulses of light coming down to their detector here. And meanwhile, their pulses go up to detector for clock, clock U. But because, as these pulses are emitted here, of course they're traveling along now. But while they're traveling, the elevator moves up slightly. It's accelerated up a little bit here, such that, these, let's say there are you know again, 10 pulses per 10 pulses per tick of each clock. Okay? So clock L detected it though because its accelerated upward a little while these pulses are in motion. Will collect more pulses per tick of its clock. Or collect more of these pulses coming down, because it's, you know, sort of moving up a little bit and sort of sweeping through them a bit faster because of the acceleration involved here. And so maybe instead of receiving 10 pulses, which it normally would do per tick of it's clock, it receives 12 of these pulses of light coming down from the upper clock. So let's write that down. For, for clock L here, let's say it receives 12 pulses during one tick.
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