So we come to the conclusion now of the Michelson Morley Experiment. We need to take the analysis we've been doing with our airplane example and apply to the actual experiment that Michelson and Morley did in the later 1880s. And so here's the setup they used. They were using light, of course. So they had a light beam coming in. So in this case, the light beam is taking the place of our airplane, previously and we've got the ether wind. We're just imagining it's coming down here. This whole apparatus actually is on a they put out in a big sandstone slab. And so we're looking down at the top of the apparatus as it were. And so, they're assuming there's an aether wind coming from this direction here. This light beam comes in, and there's a half-silvered mirror here. What the half-silvered mirror does, means that some of the light if it hits, it will bounce off that way. And some of it will actually travel through the mirror. And so the light that bounces off goes up to this mirror, bounces off that, comes back down to the half-silvered mirror. And then some of that will actually bounce back the other way to the detector Down here so that's path A. Off the half-silver mirror to this full mirror back down again through the half-silver mirror and to the detector. And then path B which is going crossways either wind that's assumed to be there, goes through the half-silver mirror bounces off this full mirror back to the half-silvered mirror. Some of it does go through, just straight through, but some of it will bounce off the half-silvered mirror and at the detector. So essentially, what we're doing here is we're taking a light beam, we're splitting it into two, one half of the beam takes path A, and then back to the detector and the other half of the beam takes path B to detector. And in this orientation, path A has the head wind and then tail wind, and path b has the cross wind. And other than that, the analysis is exactly the same in terms of how we do, that with the airplane example with the tail wind, head wind, and then the cross wind. So the path A time which is the tail wind, head wind, tail wind Example, remember it was 2D / v sub plane. But here, this light is taking the place of the plane, so we have c, the speed of light, times ( 1 / 1- v w squared, the velocity of the aether wind. And we'll figure out what that might be in a second, / c squared again. And then the path B time which is the cross wind time 2D over C, 1 over square root of 1-VW squared over C squared where D here is the distance to the meters, from the half silver mirror to that mirror and that one as well. What would be the velocity either wind what it turns out that if we're assuming that we're doing this. And the earth is rotating around the sun obviously there's an affect of really the rotation 24 hour rotation to the earth. And then the evolving the revolution of the earth around the sun. The major effect of that is actually the Earth going around the sun because it's much faster. At the surface of the Earth, we're travelling, well the number I know is about 1,000 miles an hour as the Earth rotates. But, in terms of travelling through the solar system around the sun as it were it's about 30,000 meters per second so, much faster. So that traveling around the sun is the main velocity effect that's going on so the idea is as travel around the sun. As we travel around the sun we'll be experiencing either wind, we're traveling through the either just like riding a bicycle through a still air, you feel the breeze in your face. So, this number here for V sub W, about 30,000 meters per second and just for comparison, speed of light though is still much much greater than that. It's 300 million meters per second which is about 10,000 times faster than the speed of the supposed ether wind here. So path A time, path B time. The tailwind, headwind and the crosswind time. And what we'd like to do is just figure out, as you can see here. What's the difference in travel times? What's the effect of the ether wind on the light? And again this goes back to our some of our analysis. Qualitative analysis of just how waves work. That if a wave is in a medium and assumption here is that light waves, a medium of light waves is the Ether, and is that is moving then that affects the speed of the light the we would measure. So that we should see a difference here, so that one path should be a difference in travel time compared to the other path, when it gets to the detector here. And we'd like to know exactly what that difference might be for representative numbers here, with the speed of light and the Speed of the Ether wind there. Now we could just plug in some number here and we get an answer but we would like to get it in a little nicer form than that. And we'll use something we mentioned in the first video clip on Michelson-Marley effect here. So let's just see if we can simplify it slightly so we'll say. And so, also good example what physicist often do in situations like this so, we'll say we still have 2D over C here. And what I did here of course is just this minus this, they have the 2D over C factor. In each case, I pulled that out just that then, minus that doesn't really matter, which way you subtract them. 2D over C. I just want to rewrite this in here for a second, just slightly different rotation that we talked about before in terms of the exponential notation. This 1 over something is that something to the minus 1 power. So, let's write that so this becomes 1 minus VW squared over C squared to the minus 1 power. So we haven't really done any math there, just changed our notation. And then minus the second term. 1 minus vw squared over c squared to the one half power and the denominator is, means same thing to the minus one half power if we write it like this. 1 minus vw squared over c squared To the -1/2 power, just change the notation there. And now, again, to get room, we're going to erase this. So hopefully, you've been taking notes or pause it. It's related to the same equations we had before. We'll leave the diagram up here for reference because we'll be using that again in a little bit. So, Difference in travel times now is, so bring this up here is going to be equal to 2D/C of that still. Now here is where we use a mathematical technique that physicist love to use when they can, and that is remember, we talked about the binomial expansion. And just to remind you what that was, this was from the first video clip in the Michaelson Morley experiment. When we have something like this, say 1 + x to the n, if x is much smaller than 1 Then we can write this as approximately equals to, 1 plus nx. Okay, and note that this is why we wrote this here, this is the form we have, I have something, essentially, x in this case is Vw squared over C squared. And if I have a minus sign here, this just becomes the minus there, so either way works. So, I've got one minus something To the negative 1 power. So in this case n is negative 1, x is the vw squared over c squared and then just a minus sign. So I can do this, 1- x to the n is approximately equal to 1- nx. Where it can be positive or negative. So, well is this thing here much less than one? This is only if, just to remind, if x is much less than 1, there. Well, think about this a minute. We mentioned that the supposed velocity of the Ether wind, 30,000 meters per second, velocity of light 300 million meters per second. That's a 10,000 fold difference between the two, plus they're squared here, so really the difference is 10,000 squared, which is about 100 million times, if I did that right in my head. 10,000, 10 to the fourth, for those of you who do things like that, squared, you got 10 to the eighth, which is about 100 million times, so, this is a very small number when you do the vw square over c squared. And therefore, yes you can use the binomial expansion as an approximation note, works in this case. So, what do I have here for this one, again using binomial expansion, I've got, we'll write it out here, so I've got 1 -, okay, then what's n, n is negative one here. So, it's negative one times vw squared over c squared. Okay so, that is this part here. Just applying the binomial expansion there, the first couple terms of it actually. The first two terms, okay? And I have to erase that to get a little bit of room here. And then we're going to apply it to this term as well. So I took this term, that's what this is and then I've got a minus here. So I just to be clear we'll get a lot of parenthesis going on here. Minus then 1- (-1/2) because that's n value, the exponent value, times vw squared over c squared. So, one more parenthesis there to fill everything out, okay? It looks a little messy, but look it get very simple, very quickly. You still have the 2D over C on the outside. That's fine, now I have got 1- (-1), that is just 1 + that. So this becomes, 1 + vw squared / c squared. Over here I've got -, and we'll do it step by step here, 1 + one-half. That was- x a -, positive, one-half vw squared / c squared. And, we're almost there. = 2D over C. So what have we got here, I've got 1 + vw squared over c squared, minus 1,- 1/2 vw C squared, right? If I've got minus something in parenthesis it just becomes something like this. So this then becomes- 1- that, so we'll write in there. And I've got a 1- 1, 0, and I've got a v w squared / c squared- 1/2 vw squared / c squared. So this just becomes 2D / C, the one disappear. I'm left with 1/2 vw squared over c squared. The 2 is cancelled here. And finally I'm left with D over c x vw squared over c squared. And the reason we went through it partly is just to show you that physicists love to do these types of things to get it down to a much simpler form so that, assuming the approximation is correct, that we can use a binomial expansion in this case and it is. So well, what is this number? Remember, this is the difference in time between the two paths here, path A and path B in terms of the recombined lightnings they end up back in detector here, one it takes a little bit longer than the other, one in general. And this is the difference in time between them. Well what would you if you actually out in some numbers here what would you get? In the most famous version to Michelson and Morley the experiment D was 11 meters, okay? 11 meters, that's about more than 30 feet, 33, 35 feet, something like that. Just to give you an idea if you think in terms of feet rather meters. But 11 meters, so that's pretty big and you may ask did they really have a table that big? And actual fact, in the experiment, they had a big sandstone slab that was set on oil. So they have very nice solid foundation, free of vibration, things like that. There's an oil bath set in. And they actually have some other mirrors here. So they bounce the beam back and forth a few times in each way, so that they get a longer path length without having to have a huge table. So these 11 meters, C, speed of light. 300 millimeters per second. We mentioned Vw 30,000 meters per second. Again speed of light, the same thing as before. You work all this out and you get an answer that is approximately equal to say, wait, we won't use four, we'll use something closer to 3 point 67 x 10 to the minus 16. I got my numbers right there. That was right, seconds. Yeah, 367 times 10 minus 16 seconds, that clearly is a very short time variable. And there's no way Michelson and Morley could measure a time difference of the two light beams arriving at the detector that precisely. And you might ask, for those of you who are really thinking through this, you might say, well, also what about the assumption that when we're using D here 11 meters. How do we know that's exact the same in each path? And the actual answer is we don't. But they use it, ingenious method. What the did is they would do one measurement and of course you assume that Ether wind was coming from a certain direction. You don't quite know perhaps. But you can figure out in terms of the rotation of the Earth at any given time, where they are, what time of day it is, and so on so. And the direction of the Earth around the sun, so you can get pretty good idea where Ether wind should be coming from. So they do one measurement like this and then they just rotate their whole table the table 90 degrees, this big slab with all the equipment on it 90 degrees. And then do the measurement again. And it turns out, we won't go into an analysis of that, turns out if you do that then this time difference is doubled. So, instead of 3.67 we'll just say it's about 7 x 10 to the -16 seconds. Again, very small. But that's how you get rid of any dependence on the actual distance there. And so you might say well great, so what did they do? If you can't measure something like that. Well, Michelson was a brilliant inventor and experimental physicist and later won the Nobel Prize for it. And he developed techniques using interference techniques. Using interference phenomena of light waves. And that's why, a few video clips ago, we talked about constructive and destructive interference between two waves. And without revisiting all that, here's the basic idea of what happens. The two light beams take their different paths. They recombine back here at the detector. If there's been no path difference at all, no time difference, they both arrive back exactly at the same instant, then you can imagine that the light beams come in here. Let's use orange. Here come the light beams back into the detector here. And here comes the other one in. And you have a case of constructive interference. So you get a certain pattern of light patches, bright patches where the peaks are, and then dark patches where the troughs are of the wave. So you get a distinctive pattern there. Well, it turns out that the light they were using was sodium light. And for sodium light, let's erase some of our analysis here now. We don't need this. We've got our final answer down here, so here's our final answer for delta T, the time difference there and something like 16 seconds. So they use, as I mentioned, sodium light, Which is a very nice bright source of light that they could get into a beam and send into their apparatus like that. And I'll just give you some numbers, you don't have to memorize these or anything. But sodium light turns out, remember we used the symbol lambda for wavelength? Sodium light has a wavelength of 589 nanometers, billionths of a meter, so again a very short distance there. And another equation we mentioned when were talking about light. We have, let me use frequency, we used f for frequency here. So f, the frequency of light, remember is C over lambda, so given 589 nanometers, billionths of a meter and the speed of light, we could figure out what the frequency that sodium light represents. But also we know the period, which is the time between the peaks of our waves here, is one over the frequency which is lambda over C. And that's really what we're most interested here in. If you do this calculation, you will find the period for sodium is about 2 x 10 to the -15 seconds, again, still a very small amount, difference between the peaks in this bright sodium light, sort of yellowish light. And so 2 x 10 to the -15th seconds, how is that significant for us? Because look at this, we say the experimental analysis says that the time difference, if there's this ether wind blowing here, here's a cross wind and then a tailwind head wind case. The time difference is 7 x 10 to -16 seconds, really, really small, but compared to the period, not so small. In other words, if you do this, if you say, okay, I've got my time difference of 7 x 10 to the -16th seconds, very small. But in terms of the one period, the time between two peaks, so let's divide it by that, 2 x 10 to the -15th seconds. And if you get out your calculators, just do the math there. You find this is about 0.37. What does that mean? Well, what it means is that what will happen is, with that time difference, the two paths of the light here, one of them will be shifted by about a third of a period, okay. So instead of coming in exactly in line with each other again, one of them, this peak will get shifted over by a third, so all these peaks will get shifted over by a third. They will no longer be aligned with each other. You'll get a different interference effect at the detector. So that if the ether wind is blowing or really if Earth is moving through the ether, then you should see this interference effect. There should be a change in the so-called fringe pattern, as it is. We talked about the constructive and destructive interference fringes that they'll see. And it should be relatively in terms of the apparatus they had, not too difficult to see. One-third of a period meant that actually the light and dark areas would shift slightly by about a third. And again, with the apparatus they're using, should in principle be pretty easy to see. So what did they find? They found when they did this that the observed shift, they did it a number of times, different orientations, different times of the year, and so on and so forth. The observed shift, I think it's about 0.005, in terms of the experimental uncertainties they were dealing with, they did that the observed shift could not be greater than this. It might even be less than that. And so this is what's known as the famous null result, The famous null result of the Michelson-Morley experiment because they didn't see the effect of the ether wind. And physicists, they just didn't know what to do. They said, how can this be? Where is ether wind? Light is a wave, we know that. That's well confirmed, theoretically and experimentally. And we know that a wave has to have a medium that it travels through, a disturbance through some medium. We assume it's the ether. We give the name ether to that medium. We're not quite sure exactly how it's put together, what it consists of, and how it works and so on and so far. But it should be there, and yet we're not seeing any effect of it. So one possible explanation would be, remember, the whole idea here is that the Earth is revolving around the sun and moving through the ether that way. Well, what if at the surface of the Earth, maybe the ether gets sort of dragged along, just by the matter of the Earth or who knows, maybe the atmosphere is involved as well. So that you're sort of on the surface of the Earth, we're in a pocket. We're protected from the ether wind, so in fact, on the surface of the Earth, there is no ether wind blowing. And that could be a plausible explanation. Obviously, it would require some experimentation, some investigation to see if that actually made sense or not. And as it turns out, the reason they didn't immediately turn to that is there were other experimental results that showed that you could not have this so called ether drag effect, that the Earth did not drag the ether along with it. And in fact that's what we'll be working on in our next video clip. It's the so called stellar aberration effect. So the upshot of it is that going into all the details we did on Michelson-Morley experiment in terms of actually working through some of the math here. But conceptually it's fairly straightforward. It's really sort of that airplane effect, cross wind versus tail wind, head wind. Michelson-Morley had to use, as I said, some sort of path breaking experimental techniques to actually measure something that's in principle, this small, a time difference that small but by measuring using interference effects involved, they actually could in principle see the result and yet, they saw essentially nothing. And later on, we'll talk about what impact did this have on Einstein himself? Did he use this as a starting point for his theory of special relativity? Or maybe some other things were equally is important or maybe even more important to his thinking. Before we get there, next video clip we'll talk about the stellar aberration effect, just the idea behind it. So again it sets up a huge conundrum, a huge puzzle for physicists. They couldn't see the ether. They couldn't see the fact that the ether in the Michelson-Morley Experiment, the stellar aberration said the ether drag can't work either, so what do you do with the ether? How do you solve a problem like the ether? And so that's what in the second half or actually the later decades of the 19th century they were really puzzling over.