- #1
Amentia
- 108
- 5
Homework Statement
Show that an infinitesimal boost by v^j along the x^j-axis is given by the Lorentz transformation
\Lambda^{\mu}_{\nu} =<br /> \begin{pmatrix}<br /> 1 & v^1 & v^2 & v^3\\<br /> v^1 & 1 & 0 & 0\\<br /> v^2 & 0 & 1 & 0\\<br /> v^3 & 0 & 0 & 1<br /> \end{pmatrix}<br />
Show that an infinitesimal rotation by theta^j along the x^j-axis is given by
\Lambda^{\mu}_{\nu} =<br /> \begin{pmatrix}<br /> 1 & 0 & 0 & 0\\<br /> 0& 1 & \theta^3 & -\theta^2\\<br /> 0 & -\theta^3 & 1 & \theta^1\\<br /> 0 & \theta^2 & -\theta^1 & 1<br /> \end{pmatrix}<br />
Hence show that a general infinitesimal Lorentz transformation can be written x'^{\mu} = \Lambda^{\mu}_{\nu} x^{\nu} where \Lambda = 1 + \omega with
\omega^{\mu}_{\nu} =<br /> \begin{pmatrix}<br /> 0 & v^1 & v^2 & v^3\\<br /> v^1& 0 & \theta^3 & -\theta^2\\<br /> v^2 & -\theta^3 & 0 & \theta^1\\<br /> v^3 & \theta^2 & -\theta^1 & 0<br /> \end{pmatrix}<br />
Homework Equations
No equations but in another exercises I have computed generators called K for the boost and J for the rotation.
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: \vec{\Lambda}\cdot\vec{e_{j}} or \vec{\theta}\cdot\vec{e_{j}}
?
Thanks for any help to clarify more my mind.