Lorentz transformation of infinitesimal boost and rotation?

[jag]

Homework Statement

1. Show that the infinitesimal boost by v_j along the x_j axis is given by the Lorentz transformation (see attempted solution)
2. Show that infinitesimal rotation by theta_j by x_j is given by (see attempted solution)

Relevant Equations

Explained in attempted solution

1. Show that the infinitesimal boost by $v^j$ along the $x^j$-axis is given by the Lorentz transformation

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & v^1 & v^2 & v^3 \\
v^1 & 1 & 0 & 0 \\
v^2 & 0 & 1 & 0 \\
v^3 & 0 & 0 & 1 \\
\end{pmatrix}$$

Attempted solution

I know that for x-axis

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
\gamma^1 & \beta^1\gamma^1 & 0 & 0 \\
\beta^1\gamma^1 & \gamma^1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}$$

Replacing $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ and $\beta = \frac{v}{c}$ and setting $c = 1$ with $v \ll c$, I can get the following for the x-axis

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & v^1 & 0 & 0 \\
v^1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}$$

Similarly, I am constructing the y-axis Lorentz transformation

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & 0 & v^2 & 0 \\
0 & 1 & 0 & 0 \\
v^2 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}$$

and z-axis Lorentz transformation

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & 0 & 0 & v_3 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
v^3 & 0 & 0 & 1 \\
\end{pmatrix}$$

Then, I'm thinking of adding together the matrices but it doesn't yield the final answer, so I'm stuck here. Any pointers will be helpful.

2. Show that infinitesimal rotation by $\theta^j$ about $x^j$ is given by

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & \theta^3 & -\theta^2 \\
0 & -\theta^3 & 1 & \theta^1 \\
0 & \theta^2 & -\theta^1 & 1 \\
\end{pmatrix}$$

Attempted solution

I'm reading through https://en.wikipedia.org/wiki/Rotation_matrix but as far I can understand rotation matrix are presented in $\cos$ and $\sin$, so I'm not sure how to proceed here.

Looking forward to any assistance.

 

[anuttarasammyak]

Then, I'm thinking of adding together the matrices but it doesn't yield the final answer, so I'm stuck here. Any pointers will be helpful.

adding? not multiplying?

 

[jag]

@anuttarasammyak Sorry, I meant to write multiplying in the post.

 

[PhDeezNutz]

I haven't worked out the details myself so I apologize if I'm wrong (I'm addressing question number 1)

but usually a transformation matrix is constructed by projecting the old vectors onto the new vectors and arranging it in a matrix

that can be done via the dot product in usually euclidean space

but in minkowski space

you need to use "the metric" in order to do that. Use the appropriate signature.

 

[jag]

For question number (1), I multiplied the Lorentz transformation matrix for each axis and I get the result

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & v^1 & v^2 & v^3 \\
v^1 & 1 & v^1v^2 & v^1v^3 \\
v^2 & 0 & 1 & v^2v^3 \\
v^3 & 0 & 0 & 1 \\
\end{pmatrix}$$

My assumption is $v_iv_j = 0$ and hence, yielding the answer. I'm not sure whether this assumption is right.

 

[vela]

Did you use the fact that it's an infinitesimal boost?

 

[anuttarasammyak]

I'm reading through https://en.wikipedia.org/wiki/Rotation_matrix but as far I can understand rotation matrix are presented in cos and sin, so I'm not sure how to proceed here.

Apply usual rotation matrices and substitute $\cos\theta\approx 1$, $\sin\theta\approx \theta$ neglecting second and higher order infinitesimals.

 

[vanhees71]

Homework Statement: 1. Show that the infinitesimal boost by v_j along the x_j axis is given by the Lorentz transformation (see attempted solution)
2. Show that infinitesimal rotation by theta_j by x_j is given by (see attempted solution)
Relevant Equations: Explained in attempted solution

1. Show that the infinitesimal boost by $v^j$ along the $x^j$-axis is given by the Lorentz transformation

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & v^1 & v^2 & v^3 \\
v^1 & 1 & 0 & 0 \\
v^2 & 0 & 1 & 0 \\
v^3 & 0 & 0 & 1 \\
\end{pmatrix}$$

Attempted solution

I know that for x-axis

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
\gamma^1 & \beta^1\gamma^1 & 0 & 0 \\
\beta^1\gamma^1 & \gamma^1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}$$

Replacing $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ and $\beta = \frac{v}{c}$ and setting $c = 1$ with $v \ll c$, I can get the following for the x-axis

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & v^1 & 0 & 0 \\
v^1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}$$

Similarly, I am constructing the y-axis Lorentz transformation

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & 0 & v^2 & 0 \\
0 & 1 & 0 & 0 \\
v^2 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}$$

and z-axis Lorentz transformation

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & 0 & 0 & v_3 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
v^3 & 0 & 0 & 1 \\
\end{pmatrix}$$

Then, I'm thinking of adding together the matrices but it doesn't yield the final answer, so I'm stuck here. Any pointers will be helpful.

2. Show that infinitesimal rotation by $\theta^j$ about $x^j$ is given by

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & \theta^3 & -\theta^2 \\
0 & -\theta^3 & 1 & \theta^1 \\
0 & \theta^2 & -\theta^1 & 1 \\
\end{pmatrix}$$

Attempted solution

I'm reading through https://en.wikipedia.org/wiki/Rotation_matrix but as far I can understand rotation matrix are presented in $\cos$ and $\sin$, so I'm not sure how to proceed here.

Looking forward to any assistance.

To get from a Lie-group element, e.g., a rotation around a fixed axis to a Lie-algebra element you have to expand the Lie-group element in powers of the parameter (here the rotation angle) up to first order in the parameter. Just write down a rotation matrix around, e.g., the 3-axis and expand the cos and sin functions appearing there. Then write the result in a manifestly covariant way, and you get it for an arbitrary direction of the rotation axis or use the same arguments as for the Lorentz boosts.

 

[jag]

Hi All, it is super clear for me now. Thank you very much for your help! As a self-learner of physics, this forum has been really helpful for me!