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Lec29 - time Consider again the Feynman light clock In one...

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time. Consider again the Feynman light clock. In one reference frame, 1 2 c Δ t 2 = h 2 + 1 2 Δ x 2 In another frame 1 2 c Δ t 2 = h 2 + 1 2 Δ x 2 . Therefore, 1 2 c Δ t 2 - 1 2 Δ x 2 = h 2 = 1 2 c Δ t 2 - 1 2 Δ x 2 . We therefore find that the combination c 2 Δ t 2 - Δ x 2 is an invariant, even though separately Δ t and Δ x are not. We therefore define the spacetime interval Δ s 2 := c 2 Δ t 2 - Δ x 2 . Why does such an invariant exist? The reason is that we can measure Δ s 2 without invoking a reference frame at all: we can measure it with a clock in an inertial frame that is present at both events. (This clock is static with respect to the Feynman light clock, or, in fact, is the Feynman light clock itself.) Notice, that we tacitly assumed in this proof that the separation of the two events is timelike, because we gave light su ffi cient time to do the round trip. Problem : A firecracker explodes at the origin of an inertial reference frame. Then, 2 μs later, a second firecracker explodes 300 m away. Astronauts in a passing rocket measure the distance between the explosions to be 200 m. According to the astronauts, how much time elapses between the two explosions? Solution: The spacetime interval in the rest frame is Δ s 2 = c 2 Δ t 2 - Δ x 2 = (600 m) 2 - (300 m) 2 = 2 . 7 × 10 5 m 2 In the astronauts’ frame, Δ s 2 = 2 . 7 × 10 5 m 2 = c 2 Δ t 2 - Δ x 2 = c 2 Δ t 2 - (300 m) 2 whose solution is Δ t = 1 . 86 μ s. The Lorentz transformation As di ff erent observers disagree on measurements of space and time, we wish to connect these measurements. In Newtonian physics this is done with the galilean transformation equations, specifically (for relative motion in the x direction) Δ x = Δ x - v Δ t Δ y = Δ y Δ z = Δ z Δ t = Δ t 99
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The galilean transformation equations present us with a problem: when we use them for the Maxwell equations (more specifically, the electromagnetic wave equation derived from the Maxwell equations), we violate the Principle of Relativity. Our conclusion that the speed of light is c
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