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Mathematics
Linear and Abstract Algebra
How to keep the components of a metric tensor constant?
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[QUOTE="jambaugh, post: 6039186, member: 76054"] Your use of Euclidean rotations will not properly reflect a Lorentz rotation because, whether you incorporate an ##i## factor for the x-coordinate or not, the quadratic form conserved by your simple rotations will be the Euclidean distance: ##r = \sqrt{ |ix|^2 + |ct|^2}##. Note you can rotate over 45degrees and convert a time like displacement to a space-like displacement. You should rather fully generalize to pseudo-Euclidean pseudo-rotations by utilizing hyperbolic trigonometric functions. [tex] \left[\begin{array}{c} x' \\ ct' \end{array}\right] = \left( \begin{array}{cc} \cosh(\beta) & \sinh(\beta) \\ \sinh(\beta) & \cosh(\beta) \end{array}\right) \left[ \begin{array}{c} x \\ ct \end{array}\right][/tex] [edit: Note the absence of a minus sign on the off-diagional sinh.] The pseudo-angle ##\beta## is the boost parameter which we can relate to the relative frame velocity by ## v/c = \tanh(\beta)##. This is a nice way to do things because the whole complicated addition of boost velocities problems has an elegant solution in that it is the boost parameters that add. ##\beta= \beta_1 + \beta_2##. Now there is a situation where it is valid to complexify the coordinates and perform a Wick rotation but that has to do with path integration which become path independent in the complex extension provide one accounts properly for poles. This is a matter of analytically extending the domain of the formal path integral including any pseudo-metric dependency, to the complex extension of space-time. Then using the mathematical result about path independence of the path integral (provided no poles are crossed) one can move the integral to a value-equivalent path on a real-Euclidean subspace of the complexification of the prior real pseudo-Euclidean space. [/QUOTE]
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How to keep the components of a metric tensor constant?
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