# Relativistic relative velocity

1. Oct 16, 2012

### center o bass

Hi. I'm reading some quantum field theory and I'm a bit rusty in my relativistic kinematics. I stumbled across the formula

$$E_1E_2 v_{rel} = ((p_1p_2)^2 - m_1^2m_2^2)^{1/2}$$

where 1 and 2 are two collinearly colliding paritcles with their respective masses and $v_{rel}$ are their relative velocity. My question is; how is this relation derived?

Last edited: Oct 16, 2012
2. Oct 16, 2012

### tom.stoer

what is the context? what do the indices 1 and 2 mean?

3. Oct 16, 2012

### center o bass

Hi! 1 and 2 are two collinearly colliding paritcles with their respective masses and $v_{rel}$ are their relative velocity.

Last edited: Oct 16, 2012
4. Oct 16, 2012

### ghwellsjr

5. Oct 16, 2012

### Fredrik

Staff Emeritus
You should post a better reference. Name the book and the page number, and if possible, link directly to the page at Google Books.

6. Oct 17, 2012

### center o bass

7. Oct 17, 2012

### ghwellsjr

Are you going to answer my question in post #4?

8. Oct 17, 2012

### center o bass

Yeah of course. I thought that was clear from the title and the statement

"where 1 and 2 are two collinearly colliding paritcles with their respective masses and $v_{rel}$ are their relative velocity."

but yes, it is their relative velocity :)

9. Oct 17, 2012

### Staff: Mentor

Last edited: Oct 17, 2012
10. Oct 17, 2012

### ghwellsjr

How are you defining and/or measuring their individual velocities?

11. Oct 17, 2012

### Bill_K

It's a pretty formula, but I don't believe it. For slow velocities, the right hand side becomes imaginary.

12. Oct 21, 2012

### robphy

I think $p_1p_2$ is the dot-product of the two 4-momenta.
...in terms of components: $E_1 E_2 - \vec p_1 \cdot \vec p_2$, where the spatial dot-product is used.
...in terms of rapidities ["angles" in spacetime]: $m_1 m_2 \cosh(\theta_1-\theta_2) = m_1 m_2 \gamma_{12}$, where $\gamma_{12}=\frac{1}{\sqrt{1-v_{12}^2}}$ is in terms of $v_{12}=\tanh(\theta_1-\theta_2)$, the velocity of object-1 according to object-2, what I would call the "relative velocity" (see below).

So, the quantity under the radical sign on the right-hand side ( $((p_1p_2)^2 - m_1^2m_2^2)^{1/2}$ ) is non-negative, even for small velocities.

However, I think the formula in Mandl is incorrect for another reason.
(2nd ed) http://books.google.com/books?id=Ef4zDW1V2LkC&pg=PA129#v=onepage&q&f=false (p. 129, eq 8.9)
(1st ed) http://archive.org/details/IntroductionToQuantumFieldTheory (p. 185, eq 23)

The (proposed) equation in Mandl [for spatially-parallel 3-momenta according to us... i.e. the 4-momenta of the two particles and us are coplanar in spacetime]
$$E_1 E_2 v_{rel} \stackrel{?}{=} ((p_1p_2)^2 - m_1^2m_2^2)^{1/2}$$
translates into rapidities as \begin{align} (m_1\cosh\theta_1) (m_2\cosh\theta_2) v_{rel} &\stackrel{?}{=} ((m_1m_2\cosh(\theta_1-\theta_2))^2 - m_1^2m_2^2)^{1/2} \\ \cosh\theta_1 \cosh\theta_2 v_{rel} &\stackrel{?}{=} ((\cosh(\theta_1-\theta_2))^2 - 1)^{1/2} \\ \cosh\theta_1 \cosh\theta_2 v_{rel} &\stackrel{?}{=} \sinh(\theta_1-\theta_2) \\ v_{rel} &\stackrel{?}{=} \frac{\sinh(\theta_1-\theta_2)}{\cosh\theta_1 \cosh\theta_2 } \end{align}
However, I would have expected
$$v_{rel} \stackrel{expected}{=} \tanh(\theta_1-\theta_2) = \frac{\sinh(\theta_1-\theta_2)}{\cosh(\theta_1-\theta_2)} = \frac{\sinh(\theta_1-\theta_2)}{\cosh\theta_1\cosh\theta_2 - \sinh\theta_1\sinh\theta_2}$$
so that Mandl's formula should probably read
\begin{align} (E_1 E_2 - \vec p_1 \cdot \vec p_2)v_{rel} \stackrel{expected}{=} ((p_1p_2)^2 - m_1^2m_2^2)^{1/2} \\ (p_1 p_2)v_{rel} \stackrel{expected}{=} ((p_1p_2)^2 - m_1^2m_2^2)^{1/2} \end{align} in its simplest form.

The further clue that something is wrong with Mandl's formula is that
eq. 8.10a on p. 129, 2ed and eq. 24 on p. 185, 1ed
appears to describe "relative velocity" in the Galilean way as the difference of two velocities.
If there are special cases or approximations being taken, they are not obvious to me.

Did I make a mistake somewhere? in interpretation?

13. Oct 21, 2012

### Bill_K

Very good! That looks right. (With the assumption included that v1 and v2 are collinear.)

14. Oct 21, 2012

### robphy

There must be more to this story because Weinberg discusses this in his Quantum Theory of Fields book: p.137 - p.139

p.139 ... it can take values as large as 2.

Aha!
I see what it is now. It's a terminology confusion.
Mandl's and Weinberg's "relative velocity" is what DaleSpam and others here at PF call "separation velocity"... literally v1-v2.

In terms of rapidities, Mandl's formula is:
\begin{align} E_1 E_2 v_{separation} &= ((p_1p_2)^2 - m_1^2m_2^2)^{1/2}\\ (m_1\cosh\theta_1) (m_2\cosh\theta_2) v_{separation} &= ((m_1m_2\cosh(\theta_1-\theta_2))^2 - m_1^2m_2^2)^{1/2} \\ \cosh\theta_1 \cosh\theta_2 v_{separation} &= ((\cosh(\theta_1-\theta_2))^2 - 1)^{1/2} \\ &= \sinh(\theta_1-\theta_2) \\ v_{separation} &= \frac{\sinh(\theta_1-\theta_2)}{\cosh\theta_1 \cosh\theta_2 } \\ &= \frac{\sinh\theta_1\cosh\theta_2-\sinh\theta_2\cosh\theta_1}{\cosh\theta_1 \cosh\theta_2 } \\ &= \tanh\theta_1-\tanh\theta_2 \end{align}

So, while the relative-velocity $v_{rel}=v_{12}=\tanh(\theta_1-\theta_2)$ is a scalar (a Lorentz-invariant quantity),
the separation-velocity $v_{sep}=v_{1}-v_{2}=\tanh\theta_1-\tanh\theta_2$ is not Lorentz invariant.
(As we know, of course, these two quantities are equal in the Galilean case, as well as Galilean-invariant.)

However, $\cosh\theta_1\cosh\theta_2 v_{sep}=\gamma_1\gamma_2(v_1-v_2)=\sinh(\theta_1-\theta_2)$ is a Lorentz-invariant, the "relative celerity". (See http://en.wikipedia.org/wiki/Proper_velocity )
Thus, $E_1 E_2 v_{sep}$ is a Lorentz-invariant... as Weinberg motivates.

Whew... hopefully this clears up the confusion, as well as answers the original poster.
[Ok, great... now back to grading.]

Last edited: Oct 21, 2012
15. Oct 21, 2012

### Fredrik

Staff Emeritus
This sort of error can often be fixed by changing the country part of the domain name (.no) to your own country's code, or to .com. It worked for me with this one. (The message means "you have either come to a page that can't be displayed, or reached the limit for what you can display of this book").

Last edited: Oct 22, 2012
16. Oct 22, 2012

### harrylin

Please leave me out: I use apparently the same definition as Mandl and Weinberg (and Einstein, Alonso&Finn, ..). :tongue:
(I was going to suggest that it's probably a definition issue, but evidently you figured it out already).