Special Relativity 1

Which ones well ill give you a hint the origin of the

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Unformatted text preview: alilean constraints (space and time are the same for everyone in all inertial frames) lead directly to the result that velocities transform by simple addition. Lorentz Transformation To build a 2 × 2 transformation operator, we needed three constraints. The three Galilean constraints led uniquely to the Galilean transformation operator we played with in the last section. Now, we’d like to put together a transformation operator built on Einstein’s postulates: • The laws of physics are the same in all inertial frames of reference. • The speed of light in vacuum, c, is the same in all inertial frames of reference. Since we only get three constraints, we’re going to have to replace two of the Galilean constraints. Which ones? Well, I’ll give you a hint - the origin of the rocket frame still moves through the lab frame with a velocity β cx. That’s right ˆ - we can no longer guarantee that all observers will experience space and time the same way - more on that later. The new constraints are as follows: • i) Light (in vacuum) travels with the same speed, c, in all inertial frames of reference. • ii) The laws of physics are the same for all observers. • iii) The origin of the rocket frame (O￿ ) was co-located with the origin of the lab frame (O) at t = t￿ = 0 and moves through the lab frame with a velocity ￿ = β cx v ˆ Suppose a photon leaves the (mutual) origin at t = t￿ = 0, traveling in the +ˆ x direction. According to the first constraint, its position in the rocket at any time t￿ will be given by x￿ = ct￿ , and its position in the lab at any time t will be given by x = ct. The spacetime coordinates that describe the photon in each frame should look like: 7 r= ￿￿ ct ct r￿ = ￿ ct￿ ct￿ ￿ These two descriptions must be related by a Lorentz transformation: ￿￿ ￿ ￿ ￿ ￿￿ ct L 0 ,0 ( β ) L 0 ,1 ( β ) ct = ct L 1 ,0 ( β ) L 1 ,1 ( β ) ct￿ so. . . L 0 ,0 ( β ) + L 0 ,1 ( β ) = L 1 ,0 ( β ) + L 1 ,1 ( β ) If we send the photon in the −x direction instead, the spacetime coordinates ˆ look like: ￿ ￿ ￿ ￿￿ ct ct ￿ r= r= −ct￿ −ct and they are related by: ￿ ￿￿ ct L 0 ,0 ( β ) = −ct L 1 ,0 ( β ) L 0 ,1 (...
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This note was uploaded on 12/03/2013 for the course PHYSICS 105A taught by Professor Corbin during the Winter '13 term at UCLA.

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