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JD_PM

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- I want to understand the derivation of the relativistic-relative 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 cross-section 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.uni-frankfurt.de/~hees/pf-faq/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 four-velocities 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 rotation-free 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_1-1) \hat \beta_1 \hat \beta_1^T \\

\end{pmatrix} \ \ \ \ (**)$$

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.

I understand that the rotation-free 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:

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.uni-frankfurt.de/~hees/pf-faq/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 four-velocities 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 rotation-free 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_1-1) \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 rotation-free Lorentz boost as presentedI understand that the rotation-free 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*, Vanhees71PS: 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