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05-lorentz

# 05-lorentz - The Lorentz Boost Math21b Oliver Knill This...

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The Lorentz Boost Math21b, Oliver Knill This background information is not part of the course. The relation with special relativity might be fun to know about. We will use the functions cosh( x ) = ( e x + e - x ) 2 , sinh( x ) = e x - e - x 2 on this page. LORENTZ BOOST . The linear transformation of the plane given by the matrix A = cosh( φ ) sinh( φ ) sinh( φ ) cosh( φ ) is called the Lorentz boost . The angle φ depends on the velocity v . The corresponding transformation x y 7→ A x y with y = ct is important in relativistic physics. PHYSICAL INTERPRETATION . In classical mechanics, when a particle travels with velocity v on the line, its new position satisfies ˜ x = x + tv . The Galileo transformation is x ct 7→ x + tv ct . According to special relativity, this is only an approximation. In reality, the motion is described by a linear transformation x ct 7→ A x ct , where A is the above matrix and where the angle φ is related to v by the formula tanh( φ ) = v/c . Trigonometric identities give sinh( φ ) = ( v/c ) /γ, cosh( φ ) = 1 , where γ = p 1 - v 2 /c 2 . The linear transformation is A ( x, ct ) = (( x + vt ) /γ, t + ( v/c 2 ) /γx ). For small velocities v , where the value of
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