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2 (1 + 1 2 )2 1 1
2 1 2
(1 + 1 2 )2
2 2 2 2 2 2 1 + 2 1 2 + 1 2 1 1 2 + 1 2
(1 + 1 2 )2
This simpli es to
2 3 = 2 2 ( 1 + B2 )2 2 1 2 + 1 + 2 = (1 + 1 2 )2 (1 + 1 2 )2
Taking square roots of both sides, we obtain 3 = 1 + B2 1 + 1 2
Special Relativity
Now we use = v/c to write v 1 /c + v 2 /c v3 = c 1 + v 1 v 2 /c2 Multiplication of both sides by c gives the velocity composition law v3 = v1 + v2 1 + v 1 v 2 /c2 (1.26)
Light Cones and Spacetime Diagrams
It is often helpful to visualize spacetime by considering a ash of light emitted at the origin. As we discussed earlier, such a ash of light is described by a spherical wavefront. However, our minds cannot visualize four dimensions and it s not possible to draw it on paper. So we do the next best thing and suppress one or more of the spatial dimensions. Let s start with the simplest of all cases, where we suppress two spatial dimensions. Doing so gives us a simple spacetime diagram (see Fig. 1- for the basic idea). In a spacetime diagram, the vertical axis represents time while one or two horizontal axes represent space. It is convenient to work in units where c = 1. The upper half plane where t > 0 represents events to the future of the origin. Past events are found in the lower half plane where t < 0. The motion of light in
t Future
x Light moves on lines t = x Past
Fig. 1-3. The division of spacetime into future and past regions. Light rays move on the lines t = x and t = x. These lines de ne the light cone while the origin is some event E in spacetime. The inside of the light cone in the lower half plane is the past of E, where we nd all events in the past that could affect E. The future of E, which contains all events that can be causally affected by E, is inside the light cone de ned in the upper half plane. Regions outside the light cone are called spacelike.
World line of a stationary particle t
Special Relativity
Future (t > 0) x Light moves on lines t = x
Past (t < 0)
Fig. 1-4. The worldline of a stationary particle is a straight line.
such a diagram is then described by lines that make a 45 angle with the x-axis, i.e., lines that satisfy t2 = x2 In the rst quadrant, the paths of light rays are described by the lines t = x. The motion of a particle through spacetime as depicted in a spacetime diagram (Fig. 1-4) is called a worldline. The simplest of all particle motion is a particle just sitting somewhere. To indicate the worldline of a stationary particle on a spacetime diagram, we simply draw a straight vertical line that passes through the x-axis at the spatial location of the particle. This makes sense because the particle is located at some position x that does not change, but time keeps marching forward.
t timelike
spacelike
Fig. 1-5. A light cone with two spatial dimensions.
Special Relativity
Special relativity tells us that a particle cannot move faster than the speed of light. On a spacetime diagram, this is indicated by the fact that particle motion is restricted to occur only inside the light cone. The region inside the light cone is called timelike. Regions outside the light cone, which are casually unrelated to the event E, are called spacelike. More insight is gained when we draw a spacetime diagram showing two spatial dimensions. This is shown in Fig. 1-5.
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