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

1. The pole and barn paradox If v=0.866c,  = 2, l(pole) =10m In frame of barn, the pole should fit in the barn, l =20m 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 OR: the train, the tunnel and the swiss bandits. OR: brushed noses of passengers on symmetric rrains: no perpendicular contraction 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.

two doors close at different times. two doors close Simultaneously. Barn’s frame: Pole’ frame t0 = 0 t’0 = 0 Back door closes then opens t1 =38.49ns, x1=10 (10m/0.886c) Back door closes then opens t’ 1= 19.25ns (5m/0.866c) Front door closes and then open t2= 38.49ns, x2 = 0 Front door closes and then open: t’2 =76.98ns, (20m/0.886c) pole moves out t3 = 2*38.49=76.98ns pole moves out: t’3 = 19.25+76.98=96.23ns 10 m 5 m 20 m 10 m two doors close at different times. two doors close Simultaneously. By Lorentz Transformation: The back door closes: The front door closes: --The surprising result is that the back gate is seen to close earlier, before the back of the pole reaches the front of the barn. --The door closings are not simultaneous in Pole’s frame, and they permit the pole to pass through without hitting either door.

2. Lorentz transformation in Spacetime diagram: C t ’ x ’ S’ a) How to set the second Reference Frame in the space time diagram of the first frame? C t x --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. A B S --They have to point in this way shown in blue lines: C t ’ x ’ S’ We have derived L-Transformation, which relates two sets of spacetime Coordinates of two different RFs for the same one Event. How to express LT in spacetime diagram? We label the space and time axes for the first observer x, t, the worldline of a light Pulse propagation is drawn as the angle bisector of the first quadrant (half light cone), how to put the space time axes for the second observer? Let's label the space and time axes for the second observer by x´ and ct´, and choose their zeros to coincide with the point at which the pulse was emitted . In order for second observer to measure the speed of light to be c, the distance x´ covered in time t´ must be ct´. This means that the lengths A and B must be equal. The only way this can be true is if the primed axes are tilted at equal angles as shown above. b) How to measure with respect to axes which are not right angles to one another ? --The diagram on the right shows the set of all points (in purple) with some particular value of x´.

c) How is relativity of simultaneity described in space time diagram? Ct’’ x’’ C t ’ x ’ S’ Ct x C t ’ x ’ S’ Ct x Increasing speed d) what happens if the relative speed of the observers is larger? 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 x axis). But the two events don’t occur at the same time t´ as measured by the second observer. Let's label the axes for a third observer (going faster than the second one) by x´´ and ct´´. 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 one than the ct´ axis is, as shown in the above right figure. ´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.

Two such events are said to be causally connected. 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: Inspired by the discussion of the relativity of simultaneity, you may be wondering ……. A little bit more work will convince you that the answer is ‘no’. tB > tA, B occurs after A A B C t x Can we perform a Lorentz transformation such that tB < tA, B occurs before A?

x x Here’s a representative case: A B C t X’ C t ’ t'B > t’A, B occurs after A 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´): A B C t x C t ‘ t'B < t’A, B occur before A But it is not true for v < c.

Causality and prohibition of motion faster than light.

2. Spacetime (Minkowski) diagram: For u = 3/5 c However, axial stretching occurs: We have derived L-Transformation, which relates two sets of spacetime Coordinates of two different RFs for the same one Event. How to express LT in spacetime diagram? We label the space and time axes for the first observer x, t, the worldline of a light Pulse propagation is drawn as the angle bisector of the first quadrant (half light cone), how to put the space time axes for the second observer? Let's label the space and time axes for the second observer by x´ and ct´, and choose their zeros to coincide with the point at which the pulse was emitted . In order for second observer to measure the speed of light to be c, the distance x´ covered in time t´ must be ct´. This means that the lengths A and B must be equal. The only way this can be true is if the primed axes are tilted at equal angles as shown above.