So let's summarize week 8 now. The key topics we've covered, we start off thinking about traveling the galaxy. Taking our twin paradox analysis and applying it to the possibility of a trip to the center of our galaxy, approximately 30,000 light years away which would seem even at the speed of light, it would take 30,000 years to get there. So it seem totally impractical to do that. And yet, using our twin paradox analysis which of course is based on time dilation, length contractions, relative to the simultaneity, we showed that for the person on the rocket, if they could get up to a high enough speed, which was very close to the speed of light, then they could even reach the center of the galaxy in two years and then come back in roughly another two years. So that, that was good news in a sense. It would be very interesting to take a trip like that. The more sobering news, however, is that when we thought about the energy actually involved to do that, not to mention that the practical consequences of, having to take enough food along and so on and so forth. But we're talking about the famous equation, Einstein's famous equation, E equals mc squared, the more general formula of it being equals gamma mc squared. And we saw that when we reduce that for the low velocity limit 0.1c or below approximately, it turn E equals mc squared, the famous part of the equation, plus a kinetic energy part. And so we identify the mc squared part with the energy associated with mass. That somehow mass and energy are equivalent to each other and one can be turned into another. So we gave them the usual examples of nuclear fission and nuclear fusion. We talked a little bit about that in qualitative terms how that relates to E equals mc squared there. Then we turned our attention to the general theory of relativity. Just a few of the foundational concepts there. We're talking about Einstein's way, he said was his happiest thought, his most fortunate thought of his life. And that came in 1907 when he realized that essentially being in an accelerated frame of reference such as if you're in an elevator that's accelerating upward, that is equivalent to being in a gravitational field. Being on that elevator just on the surface of the Earth and feeling the force of gravity pulling you down. And using that what became known as the equivalence principle. So again, we have the equivalence of energy and mass from here, but this is a different equivalence principle. And that is, the equivalence of an accelerated frame of reference, being in an accelerated frame of reference. In other words, you're accelerating and a gravitational field. And what that allowed Einstein to do was to do thought experiments, especially in an accelerated frame of reference, like an elevator accelerating upwards, and derive certain consequences of that. And then say, well, because of the equivalence principle, it must be the same way for a gravitational field. And these were his initial forays into what became the general theory of relativity. And we pointed out two consequences of this equivalence principle, the qualitative derivations of, one is gravitational time dilation. In other words, that, in a gravitational field, because we showed this for an accelerated frame of reference, that a clock up here versus a clock down here okay? This clock, the bottom clock, closer to really feeling more gravity as it were, would run slower than this clock. Or clocks higher up on a gravitational field will run faster than clocks lower lower down. And so that was one result that came out of the equivalence principle. Second result, again using sort of an elevator analysis, was that light would bend in a gravitational field. So those two results came out of this idea of the equivalence principle. And then we went on to talk about just sort of final comments, one of those was the idea of crucial experiments. The popular idea of sciences, you get an experiment of result, yeah, if it's against a theory at the time then you might roll the theory out. But we pointed out that theory is built in to experiment as well, the theory laid in this of experiment. So in many cases you're not quite sure what's wrong. And, so, we cited the example of experiments by, of author Kaufmann of Germany. Right about the time of the miracle year, that essentially disproved Einstein's version of the special theory of relativity, and Lorentz's related version as well. And the data seemed to be good. The results seemed to be good. Max Planck pointed out a few things where it's not quite as clearcut as maybe Kaufmann made it out to be. But, still they were concerned that what are you going to do, Einstein, with this? And Einstein's response essentially was I'm going with my theory because my theory is based on very fundamental general principles. And these experiments and some of the alternative theories they seem to support were more ad hoc in nature and not as general and so on and so forth. And it turned out a few years later, about ten years later or so, that as they continued to do more experiments, that there were some flaws found in Kaufmann's earlier work on this. So that was one crucial experiment, where it was rejected, it was also rejected and turn out to be right. Another example were the 1919 Eclipse expedition results that seem to show, that did show, the bending of starlight around the sun in the gravitational field of the sun. The question was whether the bending actually matched up with Einstein's prediction or not. And again we cited it, to indicate the human element of some of these things, because you have to, in data analysis, you have the data there, and you're going to, do we throw out that data point, because we think it's bad? Something went wrong with that measurement? Or maybe we keep it in because we think it's a good measurement, it just doesn't fit, what we think it should be. And so on and so forth. So in the end they did announce in 1919 the British Expedition lead by Arthur Eddington that it was in favor of Einsteins theory. It was confirmation of Einsteins Theory and that really was the beginning of Einsteins rise to fame. Not only in the scientific world, which he had been fairly well known before that, but in the general public, in the realm of general public as well. So, crucial experiments, said a few words about the nature of genius. One of the things we mentioned was it's the combination of passion plus time, time spent on task, and going deep into a subject, plus talent. So it's not just talent. It's not going to say just time, it's not going to say just passion, but it's a combination of those. And just to an extent, being in the right place at the right time. Einstein came together with his, the various, his background, his upbringing, the things he was passionate about and thinking about, and the insight he was able to degenerate from all that in a way we would call genius. And it matched up with some of the key problems of the time and he was able to make progress there. Later on in his life, he wasn't quite as successful. Although, he did for the next 20 years or so after 1905, he seemed almost every couple of years he would come up with a new result, even things we did not get a chance to talk about in this course. Even modern day things like lasers go back to some of the theoretical work Einstein did in the early part of the 20th century. So, Einstein's fingerprints, as it were, are throughout modern science. Not only in relativity, but also in quantum mechanics, and in many other ways as well. So, that was a few words on the nature of genius. And then relativity versus relativism, we noted they're very different things, although people often confuse them. And in fact, a somewhat ironic misunderstanding, because relativity really is theory of invariance, the unchanging quantities. Well relativism as a philosophical theory says everything's relative and everything changes in a sense that way. You can't choose between the things as it were. And then finally, just asked the question is all of this really real? This time dilation stuff and length contraction, the twin paradox and everything else. And we said it actually is. It is part of the marvelous structure of reality. One of the original quotes we quoted from Einstein that it's the way the world works. We have good experimental evidence for it from, we cited the example of the muons, the GPS navigation system, particle acceleration, so on and so forth. None of that makes sense without the underlying foundation that Einstein provided with his special theory of relativity, and then later on his general theory of relativity.