Special Relativity 1

Events happen events happen independent of any system

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Unformatted text preview: (θ ) = − B ( θ ) A( θ ) where A( θ ) 2 + B ( θ ) 2 = 1 ￿ A π /2) = 0 ￿ B π /2) = 1 A(0) = 1 B (0) = 0 2 Upon inspection, M (θ) reveals itself to be3 ￿ cos (θ) M (θ ) = − sin (θ) sin (θ) cos (θ) ￿ Transformations in Physics Stuff happens. Events happen. Events happen independent of any system we may choose to describe them in. An event happens some-where and some-when once we place it in a frame of reference. A valid description of an event in some frame will consist of a temporal (time) coordinate4 and as many as three spacial coordinates. ct x r= y z Frames of Reference: EVENT! S y S’ y’ O O’ v =β c x’ x Lab Rocket Following the usual convention, the “Rocket Frame” (S ￿ ) moves through the “Lab Frame” (S ) with a velocity ￿ = β c (measured in the lab frame). Measurev￿ ments in the rocket frame will generally be primed, measurements in the lab frame will generally not be primed. As was the case in geometric transformations, and for essentially the same reason, it is incredibly beneficial to have the two frames share a common origin (in space and time!) - this is satisfied if we require the two spacial coordinate frames (x,y ,z and x￿ ,y ￿ ,z ￿ ) coincide at t = t￿ = 0. Setting up the Math Provided the frames share a common origin in space and time, it is relatively straightforward to transform the description of events from one inertial frame to another. . . 3 How does this differ from the rotation operator you memorized in your introductory calculus class? Why does it differ? 4 The factor of c in the temporal element is there provisionally to give that element the same dimensionality as the spacial elements. We will revisit this choice later. 3 r = L( β ) r ￿ ct L 0 ,0 ( β ) x L 1 , 0 ( β ) = y L 2 , 0 ( β ) z L 3 ,0 ( β ) L 0 ,1 ( β ) L 1 ,1 ( β ) L 2 ,1 ( β ) L 3 ,1 ( β ) L 0 ,2 ( β ) L 1 ,2 ( β ) L 2 ,2 ( β ) L 3 ,2 ( β ) L 0 ,3 ( β ) L 1 ,3 ( β ) L 2 ,3 ( β ) L 3 ,3 ( β ) ct￿ x￿ ￿ y z￿ Leading to. . . ct = L0,0 (β ) ct￿ + L0,1 (β ) x￿ + L0,2 (β ) y ￿ + L0,3 (β...
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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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