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Special Relativity 1

# Special Relativity 1 - Transformations1 In the simplest...

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Transformations 1 In the simplest terms, Special Relativity is all about transforming the description of events observed in one inertial frame of reference into valid descriptions in some other inertial frame of reference. Geometric Transformation x’ θ P y x y’ Consider the Fgure above. Point P is some point - it exists, therefore it is, and it is, independent of any system we may choose to describe it in. Now - having said that - suppose the position of P is described in the x ° , y ° coordinate frame by the 2-vector: ° r ° = ° x ° p y ° p ± What would ° r , the description of P in the x , y frame, look like? If the x , y and x ° , y ° coordinate frames share a common origin 2 , the description of P in x ° , y ° may be taken to a valid description of P in x , y by the transformation operation: ° r = M ( θ ) ° r ° ° x p y p ± = ° A ( θ ) B ( θ ) C ( θ ) D ( θ ) ±° x ° p y ° p ± x p = A ( θ ) x ° p + B ( θ ) y ° p y p = C ( θ ) x ° p + D ( θ ) y ° p It remains only to Fnd the elements of M ( θ ), the transformation operator. 1 This is the frst in a set oF notes written to accompany my lectures. Enjoy! //Corbin 2 While we’ll be able to avoid this complication, it’s a good exercise to consider what happens to the transFormation in the event the origins don’t coincide. 1

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Using Constraints to Defne the TransFormation Operator There are certain ways a transformation has to behave. These constraints may be used to determine the elements of the transformation operator. In our case: i ) The distance from P to the origin must be preserved. ii ) M (0) = ° 10 01 ± iii ) M ( π 2 )= ° ± The Frst constraint tells us. . . ° r · ° r = ° r ° · ° r ° So. . . A ( θ ) 2 + C ( θ ) 2 =1 B ( θ ) 2 + D ( θ ) 2 A ( θ ) B ( θ )+ C ( θ ) D ( θ )=0 which leads us to. . . A ( θ ± D ( θ ) B ( θ ± C ( θ ) and. . . M ( θ ° A ( θ ) B ( θ ) ± B ( θ ) ± A ( θ ) ± where A ( θ ) 2 + B ( θ ) 2 The last two constraints conspire to tell us. . . M ( θ ° A ( θ ) B ( θ ) B ( θ ) A ( θ ) ± where A ( θ ) 2 + B ( θ ) 2 A (0) = 1 A ² π / 2) = 0 B (0) = 0 B ² π / 2) = 1 2
Upon inspection, M ( θ ) reveals itself to be 3 M ( θ )= ° cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ± Transformations in Physics Stu f 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) coordinate 4 and as many as three spacial coordinates. r = ct x y z Frames of Reference: O Lab Rocket O’ EVENT! v = c β S S’ x y y’ x’ Following the usual convention, the “ Rocket Frame ”( S ° ) moves through the Lab Frame S ) with a velocity ° v = ° β c (measured in the lab frame). Measure- ments in the rocket frame will generally be primed, measurements in the lab frame will generally not be primed.

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Special Relativity 1 - Transformations1 In the simplest...

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