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I still don't really understand what it means apart from cosht being the x coordinate of the intercept of a ray passing through the origin intercepts a hyperbola, where the rays angle is 2A

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- Thread starter SpartanG345
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- #1

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I still don't really understand what it means apart from cosht being the x coordinate of the intercept of a ray passing through the origin intercepts a hyperbola, where the rays angle is 2A

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Pengwuino

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Hurkyl

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If you rotate an orthonormal coordinate system for Euclidean space, the way the coordinates of any point change can be expressed with a matrix of circular functions. For example, a rotation about the

[tex]\begin{bmatrix}\cos{\theta} & \sin{\theta} & 0\\ -\sin{\theta} & \cos{\theta} & 0\\ 0 & 0 & 1\\\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}[/tex]

In special relativity, besides rotating an orthonormal coordinate system for spacetime in this way, you can make another kind of change of coordinates that also preserves (spacetime) distances between points. This switches to a coordinate system moving at some velocity relative to the one you started with, for example moving along the

[tex]\begin{bmatrix}\cosh{\phi} & -\sinh{\phi} & 0 & 0\\ -\sinh{\phi} & \cosh{\phi} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}t\\x\\ y\\ z\end{bmatrix}[/tex]

where [itex]\phi = \text{tanh}^{-1}\left({\frac{v}{c}}\right)[/itex], the inverse hyperbolic tangent, "artanh", of the speed of the new coordinate system as a fraction of the speed of light.

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hotvette

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