Unit 4 - Physics 225 Relativity and Math Applications...

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Physics 225 Relativity and Math Applications Spring 2010 Unit 4 4-vectors and the Expanding Universe N.C.R. Makins University of Illinois at Urbana-Champaign © 2010
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Physics 225 4.2 4.2
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Physics 225 4.3 4.3 Exercise 4.1: Lorentz 4-Vectors Here is our beloved Lorentz transformation ( LT ) yet again: ct = γ ( ct β x ) x = ( x ct ) y = y z = z To get rid of all the Δ ’s, we use the now-familiar tactic of having observers S and S agree to synchronize the space-time origins of their coordinate systems. 1 The Lorentz transformation shows how relativity insists that we treat space and time on an equal footing. A famous quote from Hermann Minkowski expresses this beautifully: Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a union of the two will preserve an independent reality. We really should introduce some notation that removes the separation between space and time … and so we shall! We introduce the space-time 4-vector = the usual 3-vector of position ( x , y , z ) plus a “zeroth” component denoting time: x μ ! ( ct , x , y , z ) where the index = 0,1,2,3 The 4-vector x provides a complete description of both where and when an event occurred. To put space and time on a truly equal footing, we set the time-component x 0 of our 4-vector to the combination ct instead of t in this way, all components of x have the same units = length. With 4-vector notation in hand, it’s time to introduce the elegance of matrix notation . A matrix is nothing more than a superb way to represent systems of linear equations . That’s it. The Lorentz equations are perfect candidates. Have you learned how to do matrix multiplication ? The rule is dead-easy: “Row dot Column” (where “dot” means the familiar dot-product operation between vectors of equal length). Instead of explaining any further, just stare at the two versions of the Lorentz transform below … keep staring … it’ll be obvious in about 60 seconds how this matrix notation/multiplication works: Lorentz Transformation Equations: Matrix version of the exact same thing: ct ' = + ! ct " !# x x ' = " ct + x y ' = y z ' = z c ! t ! x ! y ! z " # $ $ $ $ % & = ( ) (* 0 0 ) 0 0 0 0 1 0 0 0 0 1 " # $ $ $ $ % & ct x y z " # $ $ $ $ % & . 1 If S and S use some common space-time event to set their zero of position and zero of time, they will be measuring all other events relative to that common origin . We can get rid of the Δ ’s because all their measurements are Δ ’s!
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Physics 225 4.4 4.4 The LT matrix usually goes by the symbol Λ . If indices are included it is Λ μ ν where is the row index and is the column index. In this index notation, the matrix multiplication on the previous page is written as follows: ! x = " # x = 0 3 $ . Finally, since the y (= x 2 ) and z (= x 3 ) coordinates are never affected by a boost in the direction x (= x 1 ), the most common way to see the Lorentz transformation is with only x 0 and x 1 included: The transformation is super-easy to remember in this 2x2 form!
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This note was uploaded on 10/06/2011 for the course PHYS 225 taught by Professor Makins during the Spring '10 term at University of Illinois, Urbana Champaign.

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Unit 4 - Physics 225 Relativity and Math Applications...

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