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Infinitesimal Lorentz transformations

  • Thread starter Amentia
  • Start date
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1. Homework Statement
Show that an infinitesimal boost by v^j along the x^j-axis is given by the Lorentz transformation
[tex] \Lambda^{\mu}_{\nu} =
\begin{pmatrix}
1 & v^1 & v^2 & v^3\\
v^1 & 1 & 0 & 0\\
v^2 & 0 & 1 & 0\\
v^3 & 0 & 0 & 1
\end{pmatrix}
[/tex]
Show that an infinitesimal rotation by theta^j along the x^j-axis is given by
[tex] \Lambda^{\mu}_{\nu} =
\begin{pmatrix}
1 & 0 & 0 & 0\\
0& 1 & \theta^3 & -\theta^2\\
0 & -\theta^3 & 1 & \theta^1\\
0 & \theta^2 & -\theta^1 & 1
\end{pmatrix}
[/tex]
Hence show that a general infinitesimal Lorentz transformation can be written [tex]x'^{\mu} = \Lambda^{\mu}_{\nu} x^{\nu}[/tex] where [tex]\Lambda = 1 + \omega[/tex] with
[tex] \omega^{\mu}_{\nu} =
\begin{pmatrix}
0 & v^1 & v^2 & v^3\\
v^1& 0 & \theta^3 & -\theta^2\\
v^2 & -\theta^3 & 0 & \theta^1\\
v^3 & \theta^2 & -\theta^1 & 0
\end{pmatrix}
[/tex]

2. Homework Equations
No equations but in another exercises I have computed generators called K for the boost and J for the rotation.
3. The Attempt at a Solution
I don't understand the question. Why do we have velocities instead of gamma and beta or equivalently the "rapidity" defined in the book from gamma and beta? Does that come from the fact that v and theta are small compared to c and 1?

I am not sure how to start, although I can see that it looks like a dot product between a vector v and the generator K by identification... And a vector theta with the generator J for the second matrix. And the third matrix just looks like the sum of the first ones.

But what is the reasoning to obtain that? Is "j" a random direction? and we want to write the most general transformation possible for this little boost and little rotation going in all space directions?
Something like: [tex] \vec{\Lambda}\cdot\vec{e_{j}}[/tex] or [tex]\vec{\theta}\cdot\vec{e_{j}}[/tex]
?

Thanks for any help to clarify more my mind.
 
83
4
Sorry wrong title, I had started another thread that I cancelled and my browser mixed up the titles... But I don't know how to edit it. It should be something like "Problem about quantum mechanical transformations" if a moderator is able to change that.
 

nrqed

Science Advisor
Homework Helper
Gold Member
3,540
181
1. Homework Statement
Show that an infinitesimal boost by v^j along the x^j-axis is given by the Lorentz transformation
[tex] \Lambda^{\mu}_{\nu} =
\begin{pmatrix}
1 & v^1 & v^2 & v^3\\
v^1 & 1 & 0 & 0\\
v^2 & 0 & 1 & 0\\
v^3 & 0 & 0 & 1
\end{pmatrix}
[/tex]
Well, there is something wrong in this expression, there should be a division by "c" (unless you set c=1 in your class).
I don't understand the question. Why do we have velocities instead of gamma and beta or equivalently the "rapidity" defined in the book from gamma and beta? Does that come from the fact that v and theta are small compared to c and 1?
They are working in first order of ##v_i/c##, indeed. So you should use the transformation you know in terms of beta and gamma and Taylor expand them to lowest order.
 

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