Unit 7 - Physics 225 Relativity and Math Applications...

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Physics 225 Relativity and Math Applications Spring 2010 Unit 7 The 4-vectors of Dynamics N.C.R. Makins University of Illinois at Urbana-Champaign © 2010
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Physics 225 7.2 7.2
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Physics 225 7.3 7.3 Unit 7: The 4-vectors of Dynamics Back in Unit 4, we introduced the notion of a 4-vector . Here is our boxed definition: A Lorentz 4-vector is any 4-component quantity which transforms from a fixed frame to a moving frame via the Lorentz transformation matrix Λ μ ν . We introduced and worked with both the Lorentz transformation (LT) matrix Λ and the space-time 4-vector. Here they are for reference: x μ ! ( ct , x , y , z ) where the index μ = 0,1,2,3 Remember how to use these objects? To boost a 4-vector into another frame, you write it in column form (making it a 1-column matrix), then hit it with the LT matrix Λ . The result is the 4- vector in the new frame, also in column form. All is explained by staring at the following: 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 " # $ $ $ $ % & . The space-time 4-vector x is not the only 4-vector in nature … far from it. We’re in the dynamics (mechanics) business now, where energy, momentum, velocity, and mass are the main quantities of interest. How do we transform those quantities between frames ? Simple: we figure out the 4-vector form of those quantities. The ones we are after today are the 4-velocity η and the 4-momentum p . By construction, those 4-vectors will transform in exactly the same way as x . (That’s what 4-vector means.) In matrix notation: boost position and time: x ' = ! " x boost momentum and energy: p ' = ! p boost velocity: ' = " # Today we’ll make sure we understand the origin of these dynamical 4-vectors, and learn how to use them with some examples taken straight from particle physics. ! # $ % $& 0 0 % 0 0 0 0 1 0 0 0 0 1 ( ) ) ) ) * + , , , ,
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Physics 225 7.4 7.4 Exercise 7.1: Building new 4-vectors The main strategy for building new 4-vectors is to combine a known Lorentz 4-vector with a known Lorentz scalar in some way. A Lorentz scalar , a.k.a. a Lorentz invariant , is a quantity that is unaffected by a Lorentz boost. Thus, when you combine a Lorentz 4-vector (boosts with the Λ matrix) with a Lorentz scalar (doesn’t boost at all), the result will be a new 4-vector (also boosts with the Λ matrix). Before we continue, we need to appreciate exactly what kind of system we are working with: Relativistic dynamics is the physics of rapidly moving objects. The problem we’re trying to address is this: A particle of rest-mass m 0 travels at velocity ! u relative to some “stationary” observer S. A second observer S is moving at velocity ! v relative to S. Following our standard conventions, we pick the + x direction to be parallel to ! v . Our goal: boost the particle’s velocity !
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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 7 - Physics 225 Relativity and Math Applications...

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