Dirac_8 - all inertial frames of reference i.e same in O...

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PHY4604 R. D. Field Department of Physics Dirac_8.doc University of Florida Lorentz Transformation (4-Vector Notation) 4-vector “dot product”: Define the 4-vector dot product as follows: 2 3 2 2 2 1 2 0 2 0 ~ ~ x x x x r r x r r = r r where = 3 2 1 0 ~ x x x x r and = 3 2 1 x x x r r Lorentz Transformations: Any four quantities x 0 , x 1 , x 2 , x 3 that transform from one inertial frame to another according to the Lorentz transformation is a Lorentz 4-vector. r L r = ~ ~ r L r ~ ~ 1 = where = 1 0 0 0 0 1 0 0 0 0 0 0 γ βγ L and = 1 0 0 0 0 1 0 0 0 0 0 0 1 L with 2 1 / 1 / β = = c V . Lorentz Invariant: A “Lorentz invariant” is any quantity that is the same in
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Unformatted text preview: all inertial frames of reference ( i.e. same in O and O' frame). The square of a Lorentz 4-vector is a Lorentz invariant ( i.e. Lorentz scalar ). 2 2 ) ~ ( ~ ~ ~ ~ ~ r r r r r r ′ = ′ ⋅ ′ = ⋅ = Space-Time and Energy-Momentum 4-vectors: ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = z y x ct r ~ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = z y x cp cp cp E p ~ 2 2 ) ( ~ ~ mc p p = ⋅ Space-Time 4-Vector 4-vector 3-vector 3-vector dot product Energy-Momentum 4-Vector...
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