Chapter1_10

# Chapter1_10 - = x t c x ct cosh sinh sinh cosh r L r ′ =...

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PHY3063 R. D. Field Department of Physics Chapter1_10.doc University of Florida Analogy with Rotations Consider two frames of reference the O-frame (label points according to x,y) and the O'- frame (label points according to x',y'). Let the origins of the two frames coincide and rotate the O'- frame about the z-axis by an angle θ . The two frames are related by the following transformation ( i.e. by a rotation). θ cos sin sin cos y x y y x x + = = cos sin sin cos y x y y x x + = + = Vector Notation: = y x y x cos sin sin cos r R r = r r = cos sin sin cos R = y x r r = y x r r Rotational Invariant: 2 2 2 2 2 2 r r r y x y x r r r = = + = + = = r r r r Lorentz Transformation: Let cosh θ = γ and sinh θ = βγ then
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Unformatted text preview: = x t c x ct cosh sinh sinh cosh r L r ′ = ~ ~ = cosh sinh sinh cosh L = x ct r ~ ′ ′ = ′ x t c r ~ Lorentz Invariant: 2 2 2 2 2 2 ~ ~ ~ ) ( ) ( ~ ~ ~ r r r x t c x ct r r r ′ = ′ ⋅ ′ = ′ − ′ = − = ⋅ = y x y' x' θ Point P: (x,y) P': (x',y') O O' r = r' sin 2 θ + cos 2 θ = 1 cosh 2 θ- sinh 2 θ = 1 Hyperbolic cosine Hyperbolic sine Length of vector invariant under rotations “Length” of vector invariant under “rotations”...
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