In the last video lecture, we considered the case where we had the spaceship moving from San Francisco to St. Louis to New York. And the observer in New York, where they were actually saw the spaceship seemingly coming toward them at four times the speed of light or even nine times the speed of light. In other words, is a case of apparent faster than light motion. Even though the way we setup, clearly the ship was not moving faster than light is moving at 0.6 or 0.8 or 0.90, or whatever it chose. But putting the numbers together it seemed like, just because of the way the flashes of light worked as it traveled across and. The observer in New York would see the flashes of light coming right after each other, such that the effect was, seemed that the ship was moving faster than light. What we want to do in this video clip is actually see what happens when you have a ship that can move faster than light, or just something in general moving faster than light. Just like the last example, actually this is from Taylor and Wheeler's book on space time physics, although it's a classic example, you can find it in many other places. I've modified it, just a little bit as well, to get at the salient points, that we want to make here. And so the theme is cause and effect, or vice versa. And so, it gives the ending away just slightly but let's set up the situation here. We've got a diagram, but here are the events that we're going to imagine occurring. We've got the good guys and bad guys. And they don't like each other very much. But, at a certain point, they sign a peace treaty. So in year zero, there's a peace treaty signed and the negotiating team for the good guys then leaves at, leaves the home planet of the bad guys. They actually went to the home planet of the bad guys and negotiated the peace treaty and got it all set and everything signed and then they left in their ship to go back home at a velocity of 0.6 times the speed of light. And turns out that in the mean time, after they have left, over the next four years. Okay, so that was, we're calling that year zero. Over the next four years the bad guys are able to invent a faster than light space ship. And being bad guys, they say, we're going to take off in pursuit of are the good guys, our enemies. Because now we have a chance to really put it to them. And so, they use their faster light ship and take off at a velocity of three times the speed of light. And that happens in year four. And then in year five, they're able to catch up to the good guys, and they launch a sneak attack against them. So diagrammatically, on the space time diagram, this is what it looks like, okay? So this is the frame of reference of the bad guys, for b here, for bad guys. It's in light-years and years, so the velocity of light is one, light-year per year here. And the green line represents the negotiating team of the good guys, taking off from the home planet of the bad guys, which is right here, and traveling back towards their home wherever it might be. Now, again, remember, they are actually not traveling at an angle. They are traveling along the x axis here, going in this direction, but the space-time diagram shows their world line progress through time as well. So, space that way and then moving through time this way. And if the speed of light, again, is at a 45 degree angle here this roughly is 0.6c as we've drawn it. In other words, in one year they go about 0.6 of a light year. In two years they go about 1.2 of a light year and so on and so forth. Not precise, but pretty close there in terms of the units involved, and the scale involved. So they're traveling along here, going back home. Really, traveling along here, of course, this is their world line at 0.6 c, and then in year four, the bad guys invent their faster than light spaceship. At that point they take off from their home planet. So, they've just been sitting on their home planet here as time goes on. One year, two years, three years, four years, trying to make this faster than light spaceship work and they finally got to work in year four. So, they take off at a speed of three times the speed of light because the other ship was going 0.6 times the speed of light you work it out. They were able to catch up with them in one year. They go, essentially, in one year from here to here. They're able to go three light years because it's three times the speed of light. So, three light years per year is what their speed is. Over five years, the other ship goes five times 0.6c which again, is three light years. So, they both end up three light years away from the bad guys planet there. And that is where the sneak attack occurs. And from the bad guys perspective that sneak attack occurs at x of b equals three. Three light years away and a time of five years. Okay, taking off from zero here. And what we'd like to figure out is, what is the time and location of this attack in the good guys frame of reference? Okay, where they assume their on their ship, they actually see their home planet receding away from them, they consider themselves at rest or can consider that. And let's just do the Lorentz transformation, see what we get here. So that's what we're dong here, where the difference in the two frames of reference is 0.6c. And also note that from the good guy's perspective, the bad guy's planet is moving to the left, in the negative x direction, so we use the minus sign version of the Lorentz transformation. So, the X location of the sneak attack here is going to be gamma X sub B, minus vtB, where, we've got x sub B and t sub B here, three and five. Gamma, for 0.6c turns out to be 1.25. So we just plug in the numbers, 1.25 for gamma, 3 for X sub B, the location of the bad guy frame where the sneak attack occurred and the time where the sneak attack occurred in the bad guy frame and we're transforming it now into the good guys' coordinates. And know what happens here, X sub B is three, V is 0.6c really, but we're using unit c as one. So, we can just put 0.6 there, t sub B is five and therefore, 5 times 0.6 is 3. We get 3-3, we get 0 there. You say, well, does that make sense? Well, think about it a minute. From the perspective of the good guy ship, the sneak attack occurs right at their ship. That's their zero point, that's the origin in their frame of reference. So yes, it does make sense that they get X sub G equals zero. It occurs right at their ship, which again is their origin point for their frame of reference. Now, what about the time, though, where they see that occur? So again we plug in the numbers, equation gamma t sub B minus v over c squared x sub B, of course again with the minus sign because from the good guys, the planet, the bad guy frame of reference is moving to the left, negative x direction. And plug in the numbers, you get a 5-0.6 over c squared, again c is one. So, that just becomes 1 times 3. So, this is 3 times 0.6 is 1.8. 5 minus 1.8 is, what, 3.2. Multiply that by 1.25 and you get four years. Okay? So that in the good guys frame of reference which we're not showing on this plot, we could put it on here but we're not going to get it too complicated in terms of a plot that's going on. But we see their clocks read four years at that point. And really, that makes sense because you just have a time dilation effect going on here, between the bad guys' clocks, which took off five years, but in from the bad guys' perspective, the good guys' clocks are running slow. And so they'll tick off four years, assuming they've both had their zero point for their clocks when the good guys left the planet there. Okay, so nothing too amazing there or confusing, just in case of how confusing it might be in terms of time dilation and things like that. But nothing that is overly surprising there. Now however, let's ask the question what is the time of the invention of the faster than light ship? And the launch of that ship in the good guy frame. So, that's this point right here. In the bad guy frame, they launch their ship, put a black mark there to indicate that's the launch of the ship the bad guys launch. The invention of the ship, and the launch of the fast and light ship, what time does that occur in the good guy's frame? We know the sneak attack occurred four years in their frame. Let's see what this does. Same equation again. Well actually, just do it over here. So we have in this case, t sub G equals gamma times t sub B minus v over c squared, x sub B. And for the invention and launch right here, notice that, well we could put in gamma in first. So we'll do that, so it's 1.25. What time does that occur in the bad guy's frame? Well it occurs at t = four, right? That's the time that occurs, so it's four. And then, minus 0.6 over c squared, and where does that launch occur in the bad guys frame? The x over b coordinate, is just at zero, it's on their home planet. So, that's the zero point in their coordinate system. So actually this whole thing, this second thing here, is just times zero. That whole second term disappears and we're just left with four in the middle here, it's 4- 0 and 4 times 1.25 is 5, and our units are years. So, look what we've got here? In the good guy frame, the launch, the invention and launch, of the spaceship occurs at five years according to their clocks, and the sneak attack occurs at four years. In other words, the sneak attack occurred in their frame before the ship was even invented. So, how can you have that? In other words, it reverses cause and effect. Normally cause happens and then you get the effect. Here you have the effect happening before the cause. And since as far as we know that never happens. If things like that did happen, science would always be impossible because you never know when you would have a cause and effect and the order of events was and so on so forth. So, this is another indication that things cannot be faster than light. Because if a material object or else even a signal baring information, if we could faster than light, then you have situations like this where the effects happens before the cause. And again, what we see here is a sneak attack by agreement transformation calculation occurred at four years according to the good guy's clock. And at that point, according to their clocks the ship that just attacked them hadn't even been invented and hadn't been launched yet. Because that's only going to occur in year five in their system of clocks. And so, you get a very deep contradiction here. And so, given this, we'd say nuh you can't have anything moving faster than the speed of light. At least when you talk about material objects. As we've mentioned before, there are somewhat esoteric theories, where people have tried to imagine, what if you could have these things called tachyons? Which would be particles that could move faster than the speed of light. People worked with those for a while, it didn't seem to really get anywhere, they weren't able to do anything with it that was consistent, and so had pretty much given up on that. Doesn't mean it's not impossible, necessarily. Might come up with a new theory or just new experimental evidence, but certainly in terms of cause and effect we've never seen that sort of thing violated. Some of you may think here well what about quantum mechanics, quantum physics, all the weird stuff that's going on there. If you analyze it carefully, you don't get violations of the special theory of relativity. There are other weird things going on there, but the special theory of relativity seems to hold up. Again, it doesn't mean people aren't thinking about it in ways that might not be satisfied or we can go beyond it and occasionally every once in a while experimental evidence does come up that seems to suggest maybe something is going faster than the speed of light. Although so far in every case, further analysis has shown, the analysis was just, wasn't quite correct, or there's some mitigating factors, confounding factors, that made the special theory of relativity still hold true.