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Rutgers, The State University of New Jersey

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McMaster University

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welcome to the first section in our unit on relativity. In this section, we're going to start out by looking at how we've dealt with relative motion in the past and then see why there's something that's not quite right with it. So in the past we've used what are called Galilean transformations, meaning that if we have to reference frames A and B and we start be moving with a constant velocity V at Time T equals zero, assuming the and be start from the same point, we will move ahead with this velocity V relative to a Now, if we have an observer at both of these in both of these frames of reference and they're both trying to classify the position of some object to see, then how would they do that? Well, if we wanted to know what the position is with respect to reference frame A, we could find the position with respect to reference frame B and add to it the velocity times, the amount of time that has passed since t equals zero, or if we wanted to go with knowing that we what the position is with respect to a we could then transfer it into what it is with respect to be. Now we see that in the Y direction because there's no velocity. We already have the Y for a is equal to the Y for being. Now when we want to find the velocity transformations, all we have to do is take a time derivative of both sides. And what we end up with is that the velocity with respect to a can be measured as the velocity with respect to be plus ve. What we've assumed here is that we have a constant velocity be. This is very important because while these air fairly simple transformations, if we hadn't acceleration, life would get sticky. Quick as it is, we can look at the acceleration and say, Well, if we have a constant velocity and we take another time derivative, we find that the acceleration in the X is the same for both reference frames and the acceleration from the why is the same. For both reference frames, this is really convenient. It's not as it is what is known as an inertial reference frame. That is to say, it's a reference frame in which Newton's law of inertia that is F equals M A still works. Notice that if we had had reference frame be with an acceleration relative to a moving is going faster and faster and faster relative to a then objects Thio relative to be would seem to start accelerating all on their own. And that would violate Newton's second law because there's nothing exerting a force on those objects. Therefore, it would be violated. But as long as we stick with inertial reference frames for which the math is significantly easier, then we can expect that Newton's laws will work as they always have. And since mechanics is really based on those, all of the mechanics we've done in physics 101 in physics 102 should continue toe work. Now there's one problem here, and it's a big one, and that problem is light. You see, Light has a speed that is dictated by universal constants, one over the square root of the permitted Vitti of free space multiplied by the permeability of free space. If you work it all out, it comes out to be this large constant number, and the problem with light having a constant velocity is that When some light comes in to these two reference frames, both B and A will measure the same speed for or if light went past it in the opposite direction. The reason this is problem is because when we go to plug it into our Galilean transformations, we've run into a conundrum that you X, with respect to a would be equal to you have X With respect to B Plus V, we'll be is going to measure. The speed is being C. But if we had worked it out the other way, then we would find that the speed of light with respect to be would have to be equal to C minus V. So this is a problem. We're getting results that don't make sense. So what Ah, Young Einstein and others postulated was that perhaps all the laws of physics would be the same in an inertial reference frame, meaning that a and B measure the exact same speed of light, and thus something's not quite right about Galilean relativity. In order to enlighten ourselves on this, we can think about what's known as an event. An event is just simply something that happens at a particular position at a particular time, and we can classify that event as having happened at Position X y Z at Time T and recognize that an event has a unique set of coordinates for every reference frame. So it will have an X, say Y Z and even a time with respect to a and a separate X Y Z B and T B. Now, in order to measure velocity for these two, we would say that you X A is equal to some change, some displacement with respect to X and some change in time with respect to a and similarly for B. Now, when we look at the speed of light at light traveling, we know that these must turn out to be the same. But the Delta X is in the delta. Tease can't possibly be the same. So what is going on here? Something isn't quite right. Maybe it's with our measurement of displacement. Measurements are measurement of time. Something is going wrong, and we need to fix it in order to get a little closer idea of what's going on here. We can also look at a concept known as simultaneity. Simultaneity is saying we have to events. Event won, an event to which is some explosion, and we have to reference frames here will say we have a spaceship A and a spaceship be where B is moving relative to a Now the two lights go off. And when we look at them for a respect in terms of reference, frame A, we say Okay, well, a sees that has the same distance between start to and star one or explosion one an explosion, too. And that means since they both have the same speed of light and they have the same distance that will take the same amount of time for each of the light from each to get here. And if the person on a were to go out and measure this distance after the fact, they would realize, Oh, I was halfway in between meaning that these two events occurred simultaneously, they occurred at the same time. However, if we look at it in the perspective of B, who's moving forward, be experiences some displacement. Delta X, as the light from these two events attempts to travel to it, meaning the light from two must travel to here and the light from one travels to here. As such, the distance our two is greater than the distance are one, and the time t two is greater than the time to you one. And thus the person in reference frame be the observer and reference frame. Be says these events were not simultaneous, but the person in reference Frame A says that these events are simultaneous. This means that simultaneity is relative. It means that there could be this quality to the way that we perceive time that doesn't match up with someone who is moving relative to us, and it's because that light has this same constant velocity. Now we'll continue to dig more into this in future sections, but for now we're going to leave it there and look at a few problems.

University of North Carolina at Chapel Hill