Infinitesimal Lorentz transformations

You will get the same result.In summary, the conversation discusses the infinitesimal boost and rotation along the x-axis, which are given by the Lorentz transformation matrices. The general infinitesimal Lorentz transformation can be written as x'^{\mu} = \Lambda^{\mu}_{\nu} x^{\nu} with \Lambda = 1 + \omega, where \omega is a matrix that combines the generators K and J for the boost and rotation, respectively. The use of velocities instead of beta and gamma is due to working in first order of v/c.
  • #1
Amentia
110
5

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]

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: [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.
 
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  • #2
Sorry wrong title, I had started another thread that I canceled 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.
 
  • #3
Amentia said:

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.
 

1. What are infinitesimal Lorentz transformations?

Infinitesimal Lorentz transformations refer to small changes in the coordinates and time of an event in the special theory of relativity. They are used to describe the effects of motion and time dilation in a local region.

2. How are infinitesimal Lorentz transformations represented mathematically?

Infinitesimal Lorentz transformations are represented by a 4x4 matrix, known as the Lorentz transformation matrix, which relates the coordinates and time of an event in one inertial reference frame to those in another inertial reference frame.

3. What is the significance of infinitesimal Lorentz transformations in physics?

Infinitesimal Lorentz transformations are essential in understanding the effects of special relativity, particularly in the context of high-speed motion and time dilation. They also play a crucial role in the formulation of Maxwell's equations in the theory of electromagnetism.

4. Can infinitesimal Lorentz transformations be applied to non-inertial reference frames?

No, infinitesimal Lorentz transformations can only be applied to inertial reference frames, which are frames of reference that are not accelerating or rotating. In non-inertial frames, more complex transformations, such as the general theory of relativity, are needed to describe the effects of motion and gravity.

5. How are infinitesimal Lorentz transformations related to the Lorentz factor?

The Lorentz factor is a key component in infinitesimal Lorentz transformations, as it represents the ratio of the time and length measurements between two inertial frames. It also accounts for time dilation and length contraction in special relativity.

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