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Home , Light cone

Einstein's special has two standard of absolute rest. fundamental postulates: the This principle is familiar to anyone who has ever and the principle of the invariance of the speed of been in a train. When leaving a station, the train light. often starts and stops several , and the

engine vibrates so you can’t tell whether the train The principle of relativity is moving at any particular just by feeling the vibrations of the wheels. Sometimes you Consider two spaceships which are far enough think your train is at rest with respect to the from other masses that the effects of gravity can station, and you look out the window and see be neglected. Suppose that these spaceships are another train which appears to be in motion. moving with constant velocity. (An astronaut However, you later discover that the other train inside a spaceship can easily determine whether was at rest with respect to the station all along, the ship is moving with constant velocity by and its apparent motion was caused by the trying to float freely at a fixed position relative to motion of your own train.

the spaceship. If the ship’s velocity is not constant, then the astronaut will crash into one of Invariance of the the walls of the ship.) Suppose an astronaut in one of the ships looks at This principle says that the speed of light is the the other one and sees it moving. He is perfectly same for all observers moving with constant free to consider himself at rest, and to attribute velocity. In contrast to the principle of relativity, the relative motion entirely to the motion of the this is not part of everyday experience. It is not other ship. Similarly, an astronaut in the other true for a wave in water, for example. The speed ship can consider herself to be at rest if she of the wave relative to an observer depends on chooses. The principle of relativity says that no how fast the observer is moving with respect to experiment can be performed which will the water. Another example: a ball thrown at 100 determine which astronaut is “right”. There is no km/h in the forward direction by someone on a train moving at 200 km/h has speed 300 km/h sphere with a plane slicing through the middle of relative to an observer on the earth. But the sphere, you get an expanding circle. Now according to the principle of the invariance of the make a three-dimensional plot with time t on the speed of light, the light from a flashlight pointed vertical axis. At each point on the vertical axis, in the forward direction by someone on the train place the corresponding circle. The resulting moves at speed c (the speed of light) for both figure is called the light cone. If you put ct on observers. the vertical axis, then the cone has sides with Graphical representations slope equal to 1:

In order to continue, we have to get used to ct translating back and forth between actual 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 expanding outwards at the speed of light. The resources for this section contain a movie showing this expanding sphere. Here is a If you don't like drawing three-dimensional plots, snapshot of such a sphere at three successive there's an even easier way to represent the times: situation. If you take the intersection of the expanding 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 ct on the vertical axis and the locations x of these two points on the horizontal axis. 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 If you take the intersection of the expanding with when only one spatial dimension matters.

ct emitted. These axes cannot point in just any old direction because they must be oriented such that the second observer also measures speed c for the light pulse. They have to point this way:

ct ct«

x

Lorentz transformations x«

B Suppose there is another observer observing this same burst of light. This second observer is A travelling with a constant velocity with respect to the first observer. Let's orient our coordinate x system so that the second observer moves in the In order for second observer to measure the first observer's positive x-direction. The big speed of light to be c, the distance x« covered in question is: what does the second observer see? time t« must be ct«. This means that the lengths A The principle of relativity says that the second and B must be equal. The only way this can be observer also sees a sphere expanding out with speed c. We would like to know what this true is if the primed axes are tilted at equal as shown above. implies about the way in which the second observer measures space and time. We will be Aside: how to measure with respect to axes led to two of the main features of relativity, which are not at right angles to one another. namely and . You can understand how to interpret the primed Let's label the space and time axes for the second axes in the above figure by asking yourself what observer by x« and ct«, and choose their zeros to is the set of all points with some particular value coincide with the point at which the pulse was of x«. The answer is shown in magenta in the next figure: figure:

ct« ct ct« same t but different t «

x«

x«

set of all points with this value of x'

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

ct ct« ct«« ball is moving at three-quarters the speed of light relative to the train. This means that its is tilted three-quarters of the way towards the

x«« light cone with respect to the primed axes.

ct ct«

x« increasing speed

world line of ball

x«

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

to think a bit about how two events are related, if ct ct« one is the cause of the other. B The answer is that the two events can only be cause and effect if they can be connected to one 10 10 10 another by something moving at speed less than or equal to the speed of light. A signal of some sort must connect them. Two such events are said x« to be causally connected.

Diagrammatically, B is causally connected A x to event A if B lies within or on the light cone In contrast, let’s suppose that it were possible to centered at A: go into a frame moving faster than light. Then

ct the ct« axis would tilt past the light cone, and the

B order of events could be reversed (B could occur at a negative value of t«:

ct x«

light cone B ct« centered at A

A x Obviously, the value of t at B is greater than the

value of t at A. We say B occurs after A. The question is, is it possible to perform a Lorentz A x

transformation such that B occurs before A in the 1111111100000000 1111111100000000 1111111100000000 1111111100000000 1111111100000000 1111111100000000 new frame? It’s also easy to show that if a body were able to travel faster than the speed of light, some A little bit of doodling will convince you that the observers would observe causality violations.

Effects would precede their causes. ct ct « Let’s say that a person is born at event A and travels faster than the speed of light to event B, where he dies. His world line is shown in red:

ct

x«

B

A x 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000

B To the second observer, the death occurs at a

value of t« which is less than that of the birth, as

A x shown by the dashed line in the last figure. The person dies before he is born, violating causality. According to the observer in the inertial frame shown, all is well. The death occurs at a value of Such causality violations are not observed t which is greater than that of the birth. experimentally. This is evidence that bodies cannot travel faster than the speed of light. Now consider another observer in motion with respect to the inertial frame shown above, such that the second observer’s axes are oriented as Length contraction shown: The form of the Lorentz transformation discussed above follows directly 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?

One result is the famous length contraction. ct ct« world line of end of

Consider two rods which have the same length 10 10 10 10 when they are at rest with respect to one another. rod at rest in S Suppose the rods are now carried by observers who are moving with constant velocity relative to

one another. The special theory of relativity x« turns out to predict that each observer will

measure the other's rod to be shorter than his end of rod as seen own. at t«=0 in S«

We will choose the two coordinate systems so 1111111100000000 1111111100000000 1111111100000000 1111111100000000 x

that their origins coincide, as usual. We will 1111111111 0000000000 10 10 10 10 1111111111 0000000000 place the rods in both systems with one end at the Now the big question is: how long does the first origin of the spatial coordinate and the other end rod appear to the second observer, compared to at a positive value of this coordinate. his own rod? That is, what are the units along the x« axis? To answer this, we have to plot the The next figure shows the world sheet of the rod world line of the end of the second observer's at rest in one observer's own frame (we will call rod.

this frame S ; the axes of S and the rod at rest in S are shown in red). It also shows the axes of the ct ct«

other observer's frame S« in blue, and the position of the end of the same rod as viewed by the world line of end of second observer at t«=0. Note that length rod at rest in S« measurements are, by definition, performed simultaneously in any given frame. Because of x« the relativity of simultaneity these need not be

simultaneous in another frame. That is why the end of rod as seen position of the end of the rod at t«=0 in S« is not at t=0 in S the same as its position at t=0 in S. x 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 Now here's the point: the principle of relativity implies that both observers must measure the length contraction, also called Lorentz same effect. Each must measure the moving rod contraction. to be shorter or longer than his own, and in the In contrast, the following situation is not allowed same proportion. by the principle of relativity because one The only way this can happen is shown in the observer (the one in S) measures the moving rod following figure (in which only the world lines of as contracted while the other (in S«) measures it the ends of the rods are shown, for clarity): as expanded:

ct ct ct« ct«

x« x«

x x (If you were going to remember only one of the There is absolutely no contradiction involved in many diagrams in this section, this would be it.) both observers measuring the same effect. This happens in everyday life too; consider four kids (Note: the units of length along the x and x« axes of equal height standing in a row. From the first are different! You can’t directly compare a kid's point of view, the second kid is taller than length measured along one axis with a length the third (because the second kid is closer, of measured along the other.) course). But from the point of view of the fourth The only way each can measure the same effect kid, the second kid is shorter than the third. It all is if each measures the moving rod as contracted depends on your point of view. The only way compared with his own. This is the famous you can tell who is taller is to view them both from the same distance. In the next section, we will consider the mathe- How does the length contraction depend on the matical form of the Lorentz transformation. relative speed of the observers? If the speed is We’ll find out by how much lengths are predicted large, the effect is larger, as shown in the to contract, and we’ll also derive the famous time following figure: dilation effect.

ct ct«

x«

x Conversely, if the speed is small, the effect is small. This is why we don’t ordinarily see the effect; common speeds we can observe are much smaller than the speed of light. Here is a movie showing the changing orien- tation of the primed axes and the magnitude of the Lorentz contraction effect, 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 Ð0.8c, and then returns to zero. The cycle repeats indefinitely. What’s next?