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Passionflower
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Does anybody have a reference or can write out the general (so not just a boost in only one direction) Lorentz matrix in terms of rapidities?
Passionflower said:Does anybody have a reference or can write out the general (so not just a boost in only one direction) Lorentz matrix in terms of rapidities?
Thaakisfox said:Yep, here it is:
[tex]\Lambda = \left( \begin{smallmatrix} \pm 1& 0 \\ 0& \pm \textbf{I} \end{smallmatrix} \right)\left( \begin{smallmatrix} \cosh\eta & -\textbf{n}\sinh\eta\\ -\textbf{n}\sinh\eta& \textbf{I}+\textbf{n}\circ\textbf{n}(\cosh\eta -1) \end{smallmatrix} \right)\left( \begin{smallmatrix} 1&0\\ 0&\textbf{R} \end{smallmatrix} \right)[/tex]
Mentz114 said:That looks useful. I think I can work out what n is. What is R ?
George Jones said:An arbitrary 3x3 (spatial) rotation matrix.
George Jones said:Do you mean a boost in an arbitrary direction, or do you mean an arbitrary (restricted) Lorentz transformation (which is not necessarily a boost)? If you mean the former, look at page 541 of the second edition of Jackson.
JDoolin said:He defines
[tex]\begin{matrix}
L = \omega \cdot S - \zeta \cdot K
\\
A=e^L
\end{matrix}[/tex]
so that if [itex]\omega=(w,0,0)[/itex] you get a rotation matrix, and if [itex]\zeta = (z,0,0)[/itex] except for one little issue. There's a mysterious lack of imaginary numbers anywhere, but he's still getting cosines and sines when he takes the e^{L} for the [itex]\omega[/itex] part.
It takes a little getting used to, that taking a constant to the power of a matrix gives you a matrix.
Should L be, instead
[tex]L = i \omega \cdot S - \zeta \cdot K[/tex]
or is this hidden in the notation somewhere?
Ben Niehoff said:The cosines and sines come from taking the matrix exponential. Try computing
[tex]\exp \begin{pmatrix}0 & -\theta \\ \theta & 0\end{pmatrix}[/tex]
and see what happens.
S1 = {{0,-Theta}, {Theta, 0}}
A = IdentityMatrix[2] (*When I tried to do the k=0 term, I got a 0^0 error.*)
For[n = 1, n < 6, n = n + 1, (*Technically this should go to n->infinity*)
A = A + S1^n/n!; (*This is apparently the key to doing the matrix exponential*)
Print[MatrixForm[A]]]
Clear["Global`*"]
S = {{0, -T}, {T, 0}};
S0 = {{1, 0}, {0, 1}};
S1 = S
S2 = S.S
S3 = S.S.S
S4 = S.S.S.S
MatrixForm[Expand[S0 + S1 + S2/2 + S3/6 + S4/24]]
TeXForm[MatrixForm[Expand[S0 + S1 + S2/2 + S3/6 + S4/24]]]
JDoolin said:The software I am using is not doing Matrix multiplication correctly when I take S1^{n}. It is just multiplying term by term.
A Lorentz Matrix is a mathematical representation of the transformation between two frames of reference in special relativity, where the relative velocity between the frames is constant. Rapidities, denoted by the Greek letter "eta" (η), are a measure of the relative velocity between the frames and are used in the Lorentz Matrix to calculate the transformation equations.
The Lorentz Matrix is expressed as a 4x4 matrix, with the rapidities ηx, ηy, ηz representing the relative velocities in the x, y, and z directions respectively. The diagonal elements of the matrix are cosh(η) and the off-diagonal elements are sinh(η), where cosh and sinh are hyperbolic functions.
Rapidities play a crucial role in the Lorentz transformation as they are used to calculate the transformation equations for time, length, and momentum between two frames of reference in special relativity. They are also important because they are additive, meaning that the rapidity of an object moving with a certain velocity relative to one frame will be equal to the sum of the rapidities of that object and the frame relative to another frame.
Rapidities and relativistic velocities are directly related through the formula v = c*tanh(η), where v is the velocity, c is the speed of light, and tanh is the hyperbolic tangent function. This formula allows for the conversion between rapidities and relativistic velocities, which are both measures of the relative velocity between two frames of reference in special relativity.
Yes, rapidities can be negative. A negative rapidity indicates a relative velocity in the opposite direction of a positive rapidity, and the magnitude of the negative rapidity will be equal to the magnitude of the positive rapidity. This is because rapidities are a measure of the magnitude of the relative velocity, rather than the direction.