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I want to understand the derivation of the relativisticrelative velocity
$$v_{\text{rel}}=\frac{1}{1\vec{\beta}_1 \cdot \vec{\beta}_2} \vec{\beta}_1\vec{\beta}_2.$$
I was studying how to derive the crosssection formula in the CoM frame from Mandl & Shaw QFT's book, and they state the following formula for the relative velocity (I'm going to use Vanhees71's notation though)
$$\omega_1 \omega_2 v_{rel} = [(p_1 p_2)^2  m_1^2 m_2^2]^{1/2} \ \ \ \ (2)$$
Then the relative velocity in the CoM system follows:
$$v_{rel}=\frac{\vec p_1}{\omega_1}+\frac{\vec p_2}{\omega_2}=\vec p_1\frac{\omega_1+\omega_2}{\omega_1\omega_2} \ \ \ \ (3)$$
Then Vanhees71 pointed out that (2) is wrong.
At #9 Vanhees71 stated:
$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$
'Your Eq. (2) is wrong. It's an invariant and thus you have the Minkowski product p_1 p_2 in the first term under the square root and not ##\vec{p}_1 \cdot \vec{p}_2##.
Eq. (3) cannot be right either, because the relative speed is covariantly defined as
$$v_{\text{rel}}=\frac{I}{p_1 p_2}=\frac{P\omega}{\omega_1 \omega_2+P^2}.$$
Indeed the general formula for the relative velocity, as derived in
https://itp.unifrankfurt.de/~hees/pffaq/srt.pdf
Eq. (1.6.5), confirms the above formula since or collinear velocities ##\vec{\beta}_1=\vec{p}_1/E_1=\vec{p}/E_1## and ##\vec{\beta}_2=\vec{p}_2/E_2=\vec{p}/E_2## the formula simplifies to
$$v_{\text{rel}}=\frac{1}{1\vec{\beta}_1 \cdot \vec{\beta}_2} \vec{\beta}_1\vec{\beta}_2.$$
In your formula (3) the factor in front of the "naive" formula for the relative velocity. Unfortunately this is wrong in some textbooks. I cannot check Mandl and Shaw, whether it's correct in there, but I guess not, given your Eq. (3).'
$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$
I did not really understand why Mandl & Shaw formulas were wrong so I thought the best was going through Vanhees71's derivation first... And here we go! :)
The intention is to go through the whole derivation (Section 1.6, Relative Velocity), so let's start from the very beginning
Two particles have the following fourvelocities with respect to an arbitrary inertial frame
$$u_1^{\mu} = \frac{1}{\sqrt{1\vec \beta_1^2}} \ \begin{pmatrix}
1 \\
\vec \beta_1 \\
\end{pmatrix}, \ \ \ \
u_1^{\mu} = \frac{1}{\sqrt{1\vec \beta_2^2}} \ \begin{pmatrix}
1 \\
\vec \beta_2 \\
\end{pmatrix}
\ \ \ \ (*)$$
Then the rotationfree Lorentz boost to the rest frame of particle 1 is given by
$$(\Lambda^{\mu}_{ \ \ \nu}) = \hat B( \vec \beta_1)=
\begin{pmatrix}
\gamma_1 & \gamma_1 \vec \beta_1^T \\
\vec \beta_2 & 1_3 + (\gamma_11) \hat \beta_1 \hat \beta_1^T \\
\end{pmatrix} \ \ \ \ (**)$$
My first questions are:
1) I do not follow ##(*)##
I know what's the definition of proper velocity; it's simply the change of the spacetime coordinate ##x^{\mu}## per unit of proper time:
$$u^{\mu} = \frac{d x^{\mu}}{d \tau}$$
My point is that I'd expect to have the scalar factor multiplied by a ##4 \times 1## matrix instead of a ##2 \times 1## matrix.
2) I do not understand how to write down the rotationfree Lorentz boost as presented
I understand that the rotationfree Lorentz boost can be written in matrix form as follows
$$\begin{pmatrix}
\bar x^0 \\
\bar x^1 \\
\bar x^2 \\
\bar x^3 \\
\end{pmatrix}= \begin{pmatrix}
\gamma & \gamma \beta & 0 & 0 \\
\gamma \beta & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix}
x^0 \\
x^1 \\
x^2 \\
x^3 \\
\end{pmatrix}
\ \ \ \ (***)$$
But how can we go from Eq. (**) to Eq. (***)?
Sources: Introduction to Electrodynamics, Griffiths and Manuscript in Special Relativity Theory, Vanhees71
PS: please note I am not used to modern SRT notation so I may ask too many naive questions. I am also aware that Griffiths' Electrodynamics book is maybe not the best source to study SRT. If you have any book suggestions please feel free to share
Thank you
$$\omega_1 \omega_2 v_{rel} = [(p_1 p_2)^2  m_1^2 m_2^2]^{1/2} \ \ \ \ (2)$$
Then the relative velocity in the CoM system follows:
$$v_{rel}=\frac{\vec p_1}{\omega_1}+\frac{\vec p_2}{\omega_2}=\vec p_1\frac{\omega_1+\omega_2}{\omega_1\omega_2} \ \ \ \ (3)$$
Then Vanhees71 pointed out that (2) is wrong.
At #9 Vanhees71 stated:
$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$
'Your Eq. (2) is wrong. It's an invariant and thus you have the Minkowski product p_1 p_2 in the first term under the square root and not ##\vec{p}_1 \cdot \vec{p}_2##.
Eq. (3) cannot be right either, because the relative speed is covariantly defined as
$$v_{\text{rel}}=\frac{I}{p_1 p_2}=\frac{P\omega}{\omega_1 \omega_2+P^2}.$$
Indeed the general formula for the relative velocity, as derived in
https://itp.unifrankfurt.de/~hees/pffaq/srt.pdf
Eq. (1.6.5), confirms the above formula since or collinear velocities ##\vec{\beta}_1=\vec{p}_1/E_1=\vec{p}/E_1## and ##\vec{\beta}_2=\vec{p}_2/E_2=\vec{p}/E_2## the formula simplifies to
$$v_{\text{rel}}=\frac{1}{1\vec{\beta}_1 \cdot \vec{\beta}_2} \vec{\beta}_1\vec{\beta}_2.$$
In your formula (3) the factor in front of the "naive" formula for the relative velocity. Unfortunately this is wrong in some textbooks. I cannot check Mandl and Shaw, whether it's correct in there, but I guess not, given your Eq. (3).'
$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$
I did not really understand why Mandl & Shaw formulas were wrong so I thought the best was going through Vanhees71's derivation first... And here we go! :)
The intention is to go through the whole derivation (Section 1.6, Relative Velocity), so let's start from the very beginning
Two particles have the following fourvelocities with respect to an arbitrary inertial frame
$$u_1^{\mu} = \frac{1}{\sqrt{1\vec \beta_1^2}} \ \begin{pmatrix}
1 \\
\vec \beta_1 \\
\end{pmatrix}, \ \ \ \
u_1^{\mu} = \frac{1}{\sqrt{1\vec \beta_2^2}} \ \begin{pmatrix}
1 \\
\vec \beta_2 \\
\end{pmatrix}
\ \ \ \ (*)$$
Then the rotationfree Lorentz boost to the rest frame of particle 1 is given by
$$(\Lambda^{\mu}_{ \ \ \nu}) = \hat B( \vec \beta_1)=
\begin{pmatrix}
\gamma_1 & \gamma_1 \vec \beta_1^T \\
\vec \beta_2 & 1_3 + (\gamma_11) \hat \beta_1 \hat \beta_1^T \\
\end{pmatrix} \ \ \ \ (**)$$
My first questions are:
1) I do not follow ##(*)##
I know what's the definition of proper velocity; it's simply the change of the spacetime coordinate ##x^{\mu}## per unit of proper time:
$$u^{\mu} = \frac{d x^{\mu}}{d \tau}$$
My point is that I'd expect to have the scalar factor multiplied by a ##4 \times 1## matrix instead of a ##2 \times 1## matrix.
2) I do not understand how to write down the rotationfree Lorentz boost as presented
I understand that the rotationfree Lorentz boost can be written in matrix form as follows
$$\begin{pmatrix}
\bar x^0 \\
\bar x^1 \\
\bar x^2 \\
\bar x^3 \\
\end{pmatrix}= \begin{pmatrix}
\gamma & \gamma \beta & 0 & 0 \\
\gamma \beta & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix}
x^0 \\
x^1 \\
x^2 \\
x^3 \\
\end{pmatrix}
\ \ \ \ (***)$$
But how can we go from Eq. (**) to Eq. (***)?
Sources: Introduction to Electrodynamics, Griffiths and Manuscript in Special Relativity Theory, Vanhees71
PS: please note I am not used to modern SRT notation so I may ask too many naive questions. I am also aware that Griffiths' Electrodynamics book is maybe not the best source to study SRT. If you have any book suggestions please feel free to share
Thank you