An Illustrated Guide to Relativ Tatsu Takeuchi

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.Newton incorporated this into his theories by assumingthat the inertial and gravitational masses were the same [5].General Relativity provides a geometrical explanation why they are the same by associatinggravity with the structure of spacetime itself.2.6 What’s so “special” about Special Relativity?293Galilean relativity3.1 Basic questionsLet us recall the basic questions we asked about the motion of an object in section 2.1, namely:Q1: Is it moving or is it at rest?Q2+Q3: If it is moving, what is its velocity?As we have seen, in order to answer these questions we must first choose a frame, and the answers depended on our frame choice.Let us actually try this out.Consider the car moving along a straight horizontal road as shown in the figure.In the tree-frame, at every instant in time the clock fixed to the tree (though it’s not drawn on the tree, assume that it is) will give some reading t (in seconds) and at the same instant the car will be somewhere along the road at some position x (in meters).The figure shows the sequence of this position from time t = 0seconds to t = 4 seconds in 1-second intervals.As you can see, we can tell from the figure that:A1: the position of the car is changing with time so it is obviously moving, andA2+A3: the position of the car is changing at a rate of +1 meters every second so its velocity is +1 meters per second.(The plus sign indicates that the motion is toward the right, so this number tells us not only the speed of the car but also its direction of motion.)The figure also shows a ball which is also moving horizontally, and we can see that the ball is moving at a velocity of +0.5 meters per second.Of course, the tree is at rest in this frame, so we can assign it a velocity of 0 meters per second.3.1 Basic questions31Motion of tree, car, and ball observed in the tree-frame.0tx–1123451tx–1123452 tx–112345tx3–112345tx4–11234532Galilean relativityWhat will the exact same motion look like if observed from the car-frame?1 In the car-frame, time is specified by the reading of a clock fixed to the car.Let’s call this reading t (in seconds) to distinguish it from the reading of the clock fixed to the tree t.To keep things simple, we will assume that the car-clock is synchronized with the tree-clock so that their readings, t and t, agree at all times.2 The location of an object in space in the car-frame is specified by its distance from the car, which we will call x (in meters) to distinguish it from the location of an object in the tree-frame x which was the distance from the tree.Then, at each instant labeled by t in the car-frame, the relative locations of the objects will be the same as that observed in the tree-frame at t = t.At t = 1 second, for instance, the relative locations of the tree, car, and ball are the same as those in the tree-frame at t = 1 second, which was shown in the second from top figure on the previous page.The only difference is that we must now use the distance from the car to specify the location of objects.So the tree, which is at x = 0 meters in the tree-frame, is at x = −1 meter in the car-frame.The car, which is at x = +1 meter in the tree-frame, is at x = 0 meters in the car-frame.And the ball, which is at x = +3 meters in the tree-frame, is at x = +2meters in the car-frame.This is depicted in the second from top figure on the opposite page.We can figure out the locations of the objects for all other times in a similar manner, and we will obtain the sequence of figures shown here.The clock is actually fixed to the car though it is drawn on the margin.This sequence describes the motions of the tree, the car, and the ball in the car-frame.If we follow how the location of the tree is changing with time, we can see that, after each second, the tree moves to the left by one meter.So the velocity of the tree is −1 meters per second, where the minus sign indicates that the motion is toward the left.Similarly, the ball is moving at a velocity of −0.5 meters per second.The car is at rest so its velocity is 0 meters per second.3.1 Basic questions33Motion of tree, car, and ball observed in the car-frame [ Pobierz całość w formacie PDF ]

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