For our final video of the course, let's try to recap some of the key concepts and principles we've covered, sort of like our road map or our journey through the course. Obviously it won't cover everything, challenged us to get it all on board but I think we got it all on here without squeezing it too much. So why don't we start off just talking about Einstein in context which meant looking at the scientific background of his time, some of the key scientific discoveries and developments throughout the 1800s, Einstein being born in 1879 of course. Also the technology of the time because remember his family was very involved in electro-technology and, of course, later on he worked as a clerk at the patent office and so saw a lot of the things that were going on. Especially things like synchronization of clocks and other things going on in electrical technology of the time. We can't draw a direct connection between those things to some of his work, especially in the special theory of relativity and the relativity of simultaneity and clock synchronization as we talked about, but it's certainly part of the environment there and it may have in some way inspired his thinking on some of these matters. So, both the science and the technology at the time was important. We talked about 1905, his miracle year, some of his early struggles as well if you if you've read the young Einstein article, so it talked about the key papers in the miracle year, with a focus of course on the June 1905 paper on the electro-dynamics of moving bodies which we know as special relativity paper. We also noted that what really bothered him were certain theoretical asymmetries in terms of what was going on. We talked about some of the experimental results. Especially the Michelson Moorely experiment and stellar aberration. They did not seem to have as big an influence on Einstein's thinking as just some of the, as he termed it, the asymmetries involved. We talked about the bar magnet and the coiled wire, and how depending on which was considered to be moving, physicists had one explanation for one scenario and another explanation for another scenario, and things like that really bothered Einstein. To the point that he described it as, after years of thinking about this by this time being 26 years old, it was almost a state of psychic tension for him. And then out of that, through some conversations with his friend Michele Besso, you may remember his flash of insight that time was suspect as he later termed it. So, what he then did is he took two key principles that were well established in physics, one, the principal of relativity, so we spent some time talking about Galilean relativity and just the basic principal of motion of objects and how you can convert from one frame of reference to another frame of reference if they're moving at a constant velocity with respect to each other. We talked about that a little bit and the other principle was when he turned the principle of light constancy that the movement of a source of light or a source of waves means that it's not going to actually affect the velocity of the waves coming out of that. So we spent some time just talking about how waves work to get some background on that, and understand this principle of light constancy. And then also the conflict between them, because the principle of light constancy implied that it either existed and it was just assumed, that that had to have some sort of medium for, flight was a way, it had to have some sort of medium through which it traveled, and the idea was that it was this thing called the ether. So they spent a lot of time creating models of the ether, wondering about how it might work, be put together, and so on and so forth. And the problem with that though is that it went against the principle of relativity because if you have it ether you can assume the ether was the absolute reference frame and so you can have absolute motion against measured against ether or just the state absolute rest would be ether the underlying foundation of the universe, as it were. And so Einstein's law is conflict here and he said his insight was actually they are both true here. And out of those two principles then, we showed that you get the invariance of this speed of light that combine these two things together. Its not just that if you have a moving source, the light coming from that moving source has a constant invariant velocity. But, it's any motion at all. One observer to light or to another observer moving with respect to light. Speed of light will always be the same to all observers. And that came from the combination of these two seemingly incompatible principles. That was Einsteins key insight that set him off on this path. And then from that, we looked at okay, what do we get out of that? Well, one key thing was the so-called relativity of simultaneity. That clocks synchronized in one reference frame will not be synchronized in a moving reference frame. We talked about the concept of a lattice of clocks for each reference frame that would be synchronized. And so the relativity of simultaneity, or the relativity of clock synchronization, that was a big news for the time, because the standard assumption was, well of course everybody measures the same time. Time is absolute, doesn't matter how you're moving with respect to somebody else's clock, you'll see the same time as they will. But Einstein showed, and we sort of followed in his footsteps here, that put these to principles together, except both of these well accepted principles and even though they seem to be in conflict, you get, if you accept them as true you get the relativity of simultaneity. And from that then other results tumbled out. We talked about time dilation using the light clock example, again, coming from the invariance of the speed of light. So that a moving clock, an observer who observes a moving clock will observe it to tick more slowly than an identical clock that he or she has next to them. So time dilation. And from that we also got the Lorenz factor gamma, the scaling factor for time dilation really. And then from that we get length contraction as we analyzed it further that moving objects contract in the direction of motion to an observer who's observing a moving object. If you're riding along with a moving object of course everything in your frame of reference, everything is normal, has this normal length because you're in it's so called rest frame. So it's rest length or proper length. So time dilation, the Lorentz factor gamma, length contraction, I won't write down the formulas here because you should know them by now of course, you can remind yourself, review them. The invariant interval, I actually did write down this formula just to remind us that also came out of analysis of a light clock thought experiment and we show that for all observers this combination of things from one event, two events and one frame versus another frame c squared t squared minus x squared will always be constant and will allow use to do a few calculations with that and then the phrase leading clocks lag. We talked about the relativity of simultaneity, relativity of synchronization. But that as you observe two clocks go by and they're moving with respect to you, the leading clock will lag behind the trailing clock. And that was the key result, we did both qualitatively and then later quantitatively once we figured out the Lorentz transformation. And that was probably the most complicated analysis we did, mathematical analysis we did, the derivation there but we got that such that we could transform between inertial frames of reference very easily. If again Bob analysis, if Bob measured a certain event in his frame of reference at a certain position and a certain time if we knew that and we knew then the velocity between his frame of reference and Alice's frame of reference we can easily find out where would that event occur in space time. In Alice's frame of reference, so that was why the Lorentz transformation became handy to do things like that. And of course, we talked about space times diagrams. We actually introduced them back here when we talked about relativity, but we made use of them in terms of Lorentz transformation, getting both frames of reference, as it were, onto the same plot, and did a few things we that. And then just looked at some of the famous implications of all this. Not that time dilation, length contraction aren't famous in it of themselves. But we consider the situation can you have faster than light travel? We did one example where we had the spaceship going from San Francisco to St. Louis to New York and showed that actually in that case it's an optical illusion. That you get faster than light travel the spaceship never goes faster than the speed of light but it seems like to an observer in New York say that things are traveling faster than light. We looked at cause and effect situation where if you do have faster than light travel you get a reversal of cause and effect, which is a strong argument against the possibility of faster than light travel because we just don't see things like that happening. The universe doesn't work that way. There seems to be a definite change of cause and effect. Although of course, in quantum mechanics it gets a little more complicated than that. But even in quantum mechanics, you don't have time running in reverse essentially, at least in experimental situations. In theoretical situations you can actually imagine time running in reverse, but that's a whole other topic, which we didn't get into. So we definitely show the cause and effect that if you have faster than light travel you can have something happening before it's caused, and we did the example of the spaceship traveling away from a planet and then the bad guy is inventing a faster than light spaceship and trailing after them. And actually destroying them and the good guys frame of reference they get destroyed before the fast and light spaceship was even invented. So we say there was a contradiction there and therefore can't happen. So, more implications of course. Paradox, we talked about the pole in the barn paradox, we talked about the spaceships on a rope paradox, and of course we talked about the twin paradox and that led us on. In the final week we talked a little bit about the galactic travel. Is it possible, especially given the results of the twin paradox? In principle, it is, that you can travel to the center of the galaxy in a reasonable amount of time, if you are on the spaceship. Go back on Earth thousands of years will pass by. But then we ran into problem when we consider the famous equation, e equals m c squared, or the general form of it, e equals gamma m c squared. We discovered that the amount of energy to get up to the speeds needed, very close to the speed of light, to make galactic travel possible even in. And just theory is far more than we certainly know how to get at, at this point in time. And then finally, just a few words on general relativity, the equivalence principle between accelerating frames of reference and gravity. And therefore that led on to showing that actually a gravitational field time dilation occurs. We mentioned this is a factor in the GPS navigation system that has to be taken into account that satellites are up here and we're down here and therefore satellite clocks will actually run faster than clocks here if they're identical down on the surface of the Earth so that has to do, that time, it's not really a time delay, it's a time dilation factor. The delay of speed of light signals moving from one and the other but in addition to that we have just simple time dilation. The clocks on the surface of the Earth will run more slowly than identical clocks on the satellite so that has to be factored into the system, and the bending of light in a gravitational field as well. So a lot of things we covered, amazing actually we can get it all on one board at least the key concepts here, so you can see where we've come hopefully that helps you get a big picture of you because again sometimes we get so caught up in the details we lose sight of where we've been and where we might be going with it. But I hope as we start off the course one of the key quotes we talked about not only the mystery of the marvelous structure of reality, we mentioned that several times, But also Einstein's quote to the point that just the struggle to understand, and yes it really is a struggle to understand a lot of these things within the details or just even the big picture concepts as well. But that the struggle to understand can be an ennobling experience, and even an enriching experience, so I hope you've found some of that as we've journeyed through this course together.