Causality in special relativity --The ladder and barn paradox --Spacetime diagram: --Causality

1. Causality in special relativity --The ladder and barn paradox --Spacetime diagram: --Causality -- Nothing travels faster than C

2. 1. The pole and barn paradox In frame of barn, the pole should fit in the barn, If v=0.866c,  = 2, l(pole) =10m In frame of pole, the barn is shorter,  = 2, l(barn) =5 m, the situation is even worse, the pole can’t fit into the barn at all l =20m They cannot both be right!! --Let’s assume the doors of the barn are kept open in usual state, and are designed in such a way that it can be triggered by passing of the pole to close and then open again immediately after, so that the pole can keep a constant motion passing through the barn. --The back door of the barn: will close when the front of pole just approaches the back door, then open again immediately and the front door will close just when the rear end of pole passes it and open again immediately.

3. 10 m two doors close at different times. two doors close Simultaneously. 10 m 5 m 10 m 20 m 5 m 20 m 10 m By Lorentz Transformation: The back door closes: 5 m 20 m The front door closes: 5 m 10 m 20 m 10 m --The surprising result is that the back gate is seen to close earlier, before the front of the pole reaches it. --The door closings are not simultaneous in Pole’s frame, and they permit the pole to pass through without hitting either door.

4. 2. Lorentz transformation in Spacetime diagram: C t ’ C t ’ a) How to set the second RF in the space time diagram of the first frame? S’ S’ C t --The Ct’ and x’ cannot point in just any odd direction because they must be oriented such that the second observer (S’) also measures speed c for the light pulse. x ’ x ’ S B --They have to point in this way shown in blue lines: A b) How to measure with respect to axes which are not right angles to one another ? x --The diagram on the right shows the set of all points (in purple) with some particular value of x´.

5. c) How is relativity of simultaneity described in space time diagram? Ct’’ C t ’ C t ’ x’’ S’ S’ Increasing speed x ’ x ’ Ct Ct d) what happens if the relative speed of the observers is larger? ´The point with x´´=0 covers even more ground in a given time interval than did the point with x´=0. Thus, the ct´´ axis (the set of all points with x´´=0) is inclined even more towards the light cone than the ct´ axis is, as shown in the above right figure. x x

6. 3. Causality Causality means that cause precedes effect : an ordering in time which every observer agrees upon. Q: Whether it is possible to change the order of cause and effect just by viewing two events from a different frame. A: two events can only be cause and effect if they can be connected to one another by something moving at speed less than or equal to the speed of light. Two such events are said to be causally connected. Diagrammatically, event B is causally connected to event A if B lies within or on the light cone centered at A: tB > tA, B occurs after A C t B Can we perform a Lorentz transformation such that tB < tA, B occurs before A? x A

7. Here’s a representative case: C t ’ t'B > t’A, B occurs after A X’ In contrast, let’s suppose that it were possible to go into a frame moving faster than light. X ‘ Then the ct´ axis would tilt past the light cone, and the order of events could be reversed (B could occur at a negative value of t´): C t ‘ C t C t B B t'B < t’A, B occur before A But it is not true for v < c. x x A A

8. Causality and prohibition of motion faster than light.