Special Relativity. The principle of relativity. Invariance of the speed of light

Transcription

1 Special Relativity Einstein's special theory of relativity has two fundamental postulates: the principle of relativity and the principle of the invariance of the speed of light. The principle of relativity Consider two spaceships which are far enough from other masses that the effes of gravity can be negleed. Suppose that these spaceships are moving with constant velocity. (n astronaut inside a spaceship can easily determine whether the ship is moving with constant velocity by trying to float freely at a fied position relative to the spaceship. If the ship s velocity is not constant, then the astronaut will crash into one of the walls of the ship.) Suppose an astronaut in one of the ships looks at the other one and sees it moving. He is perfely free to consider himself at rest, and to attribute the relative motion entirely to the motion of the other ship. Similarly, an astronaut in the other ship can consider herself to be at rest if she chooses. The principle of relativity says that no eperiment can be performed which will determine which astronaut is right. There is no standard of absolute rest. This principle is familiar to anyone who has ever been in a train. When leaving a station, the train often starts and stops several times, and the engine vibrates so you can t tell whether the train is moving at any particular time just by feeling the vibrations of the wheels. Sometimes you think your train is at rest with respe to the station, and you look out the window and see another train which appears to be in motion. However, you later discover that the other train was at rest with respe to the station all along, and its apparent motion was caused by the motion of your own train. Invariance of the speed of light This principle says that the speed of light is the same for all observers moving with constant velocity. In contrast to the principle of relativity, this is not part of everyday eperience. It is not true for a wave in water, for eample. The speed of the wave relative to an observer depends on how fast the observer is moving with respe to the water. nother eample: a ball thrown at 100 km/h in the forward direion by someone on a

2 train moving at 200 km/h has speed 300 km/h relative to an observer on the earth. But according to the principle of the invariance of the speed of light, the light from a flashlight pointed in the forward direion by someone on the train moves at speed c (the speed of light) for both observers. Graphical representations In order to continue, we have to get used to translating back and forth between aual motion in space and the representation of that motion on a space-time diagram. Consider a flash of light emitted at a particular point in space, at a particular time. If you could slow things down enough, the flash would look like a sphere of light epanding outwards at the speed of light. The resources for this seion contain a movie showing this epanding sphere. Here is a snapshot of such a sphere at three successive times: If you take the interseion of the epanding sphere with a plane slicing through the middle of the sphere, you get an epanding circle. Now make a three-dimensional plot with time t on the vertical ais. t each point on the vertical ais, place the corresponding circle. The resulting figure is called the light cone. If you put on the vertical ais, then the cone has sides with slope equal to 1: If you don't like drawing three-dimensional plots, there's an even easier way to represent the situation. If you take the interseion of the epanding sphere with a measuring stick through the middle of the sphere, you get two points moving away from one another. Just make a two-dimensional plot with on the vertical ais and the locations of these two points on the horizontal ais. You get a V-shaped figure which is just a slice taken out of the light cone. This is the easiest type of representation to work with when only one spatial dimension matters.

3 emitted. These aes cannot point in just any old direion because they must be oriented such that the second observer also measures speed c for the light pulse. They have to point this way: Lorentz transformations Suppose there is another observer observing this same burst of light. This second observer is travelling with a constant velocity with respe to the first observer. Let's orient our coordinate system so that the second observer moves in the first observer's positive -direion. The big question is: what does the second observer see? The principle of relativity says that the second observer also sees a sphere epanding out with speed c. We would like to know what this implies about the way in which the second observer measures space and time. We will be led to two of the main features of relativity, namely length contraion and time dilation. Let's label the space and time aes for the second observer by and, and choose their zeros to coincide with the point at which the pulse was B In order for second observer to measure the speed of light to be c, the distance covered in time t must be. This means that the lengths and B must be equal. The only way this can be true is if the primed aes are tilted at equal angles as shown above. side: how to measure with respe to aes which are not at right angles to one another. You can understand how to interpret the primed aes in the above figure by asking yourself what is the set of all points with some particular value of. The answer is shown in magenta in the

4 net figure: figure: same t but different t set of all points with this value of ' It is just a line parallel to the ais, which is itself the set of all points with =0. Similarly, the set of all points with some particular value of is a line parallel to the ais, which is itself the set of all points with =0. End of aside. This transformation between the two sets of coordinates is called a Lorentz transformation. n important consequence of the form of this transformation is that two events which occur simultaneously at different locations for one observer will not be simultaneous for another observer. This is called the relativity of simultaneity, and is illustrated in the following The two events shown occur at the same time t as measured by the first observer (because they lie on a line parallel to the ais). But the two events don t occur at the same time t as measured by the second observer. One last point before we continue: what happens if the relative speed of the observers is larger? Let's label the aes for a third observer (going faster than the second one) by and. The point with =0 covers even more ground in a given time interval than did the point with =0. Thus, the ais (the set of all points with =0) is inclined even more towards the light cone than the ais is, as shown in the following figure:

5 ball is moving at three-quarters the speed of light relative to the train. This means that its world line is tilted three-quarters of the way towards the light cone with respe to the primed aes. increasing speed world line of ball Can you beat the speed of light? nother question is this: suppose you are riding on a train moving at three-quarters the speed of light. You are a tremendous pitcher, and can throw a ball at three-quarters the speed of light in the direion of motion of the train. Does an observer on the earth see the ball travelling at 3/4+3/4=1.5 times the speed of light? The answer is no. The rule which allows you to simply add the velocities does not work when such large velocities are involved. This is easily seen on the following space-time diagram. Let's go into the frame in which the observer on the earth is at rest, with coordinates and. You are on the train, with coordinates and. The primed aes are tilted towards the light cone. The Its world line does not tilt past the light cone, it just gets closer to it. So the ball does not travel faster than light. Causality in relativity Causality means that cause precedes effe. This implies an ordering in time which every observer agrees upon. Inspired by the above discussion of the relativity of simultaneity, you may be wondering whether it is possible to change the order of cause and effe just by viewing the two events from a

6 different frame. In order to answer this, we need to think a bit about how two events are related, if one is the cause of the other. The answer is that the two events can only be cause and effe if they can be conneed to one another by something moving at speed less than or equal to the speed of light. signal of some sort must conne them. Two such events are said to be causally conneed. Diagrammatically, event B is causally conneed to event if B lies within or on the light cone centered at : B light cone centered at Obviously, the value of t at B is greater than the value of t at. We say B occurs after. The question is, is it possible to perform a Lorentz transformation such that B occurs before in the new frame? little bit of doodling will convince you that the answer is no. Here s a representative case: In contrast, let s suppose that it were possible to go into a frame moving faster than light. Then the ais would tilt past the light cone, and the order of events could be reversed (B could occur at a negative value of t : B B It s also easy to show that if a body were able to travel faster than the speed of light, some observers would observe causality violations.

7 Effes would precede their causes. Let s say that a person is born at event and travels faster than the speed of light to event B, where he dies. His world line is shown in red: B ccording to the observer in the inertial frame shown, all is well. The death occurs at a value of t which is greater than that of the birth. Now consider another observer in motion with respe to the inertial frame shown above, such that the second observer s aes are oriented as shown: B To the second observer, the death occurs at a value of t which is less than that of the birth, as shown by the dashed line in the last figure. The person dies before he is born, violating causality. Such causality violations are not observed eperimentally. This is evidence that bodies cannot travel faster than the speed of light. Length contraion The form of the Lorentz transformation discussed above follows direly from one of the two main postulates of special relativity, the principle of the constancy of the speed of light. What happens when this is combined with the other main postulate, the principle of relativity?

8 One result is the famous length contraion. Consider two rods which have the same length when they are at rest with respe to one another. Suppose the rods are now carried by observers who are moving with constant velocity relative to one another. The special theory of relativity turns out to predi that each observer will measure the other's rod to be shorter than his own. We will choose the two coordinate systems so that their origins coincide, as usual. We will place the rods in both systems with one end at the origin of the spatial coordinate and the other end at a positive value of this coordinate. The net figure shows the world sheet of the rod at rest in one observer's own frame (we will call this frame S ; the aes of S and the rod at rest in S are shown in red). It also shows the aes of the other observer's frame S in blue, and the position of the end of the same rod as viewed by the second observer at t =0. Note that length measurements are, by definition, performed simultaneously in any given frame. Because of the relativity of simultaneity these need not be simultaneous in another frame. That is why the position of the end of the rod at t =0 in S is not the same as its position at t=0 in S world line of end of rod at rest in S end of rod as seen at t =0 in S Now the big question is: how long does the first rod appear to the second observer, compared to his own rod? That is, what are the units along the ais? To answer this, we have to plot the world line of the end of the second observer's rod world line of end of rod at rest in S end of rod as seen at t=0 in S Now here's the point: the principle of relativity

9 implies that both observers must measure the same effe. Each must measure the moving rod to be shorter or longer than his own, and in the same proportion. The only way this can happen is shown in the following figure (in which only the world lines of the ends of the rods are shown, for clarity): length contraion, also called Lorentz contraion. In contrast, the following situation is not allowed by the principle of relativity because one observer (the one in S) measures the moving rod as contraed while the other (in S ) measures it as epanded: (If you were going to remember only one of the many diagrams in this seion, this would be it.) (Note: the units of length along the and aes are different! You can t direly compare a length measured along one ais with a length measured along the other.) The only way each can measure the same effe is if each measures the moving rod as contraed compared with his own. This is the famous There is absolutely no contradiion involved in both observers measuring the same effe. This happens in everyday life too; consider four kids of equal height standing in a row. From the first kid's point of view, the second kid is taller than the third (because the second kid is closer, of course). But from the point of view of the fourth kid, the second kid is shorter than the third. It all depends on your point of view. The only way you can tell who is taller is to view them both

10 from the same distance. How does the length contraion depend on the relative speed of the observers? If the speed is large, the effe is larger, as shown in the following figure: In the net seion, we will consider the mathematical form of the Lorentz transformation. We ll find out by how much lengths are predied to contra, and we ll also derive the famous time dilation effe. Conversely, if the speed is small, the effe is small. This is why we don t ordinarily see the effe; common speeds we can observe are much smaller than the speed of light. Here is a movie showing the changing orientation of the primed aes and the magnitude of the Lorentz contraion effe, as the relative velocity of the two frames varies. The velocity starts out small and positive, grows to 0.8c, decreases all the way back to 0.8c, and then returns to zero. The cycle repeats indefinitely. What s net?