SR: magnitude of a particle's 3-momentum

[lomidrevo]

Homework Statement

We observe an experimentalist moving by with 4-velocity $u^\mu$ and a particle zipping by with 4-momentum $p^\mu$. Show that magnitude of the particle's 3-momentum as seen by the experimentalist is given by
$$ |\vec p| = \left [ (p \cdot u)^2 + (p \cdot p)^2 \right ] ^{1/2}$$

Homework Equations

Scalar dot product between 4-vectors:
$p \cdot q = -p^0q^0 + p^1q^1 + p^2q^2 + p^3q^3 = -p^0q^0 + \vec p \cdot \vec q$

The Attempt at a Solution

Let's denote our reference frame as unprimed, and the experimentalist's frame of reference as primed. As the scalar dot product between two 4-vectors is invariant under Lorentz's transformation we can write:
$p \cdot p = p' \cdot p' = -p'^0p'^0 + \vec p' \cdot \vec p'$
$p \cdot u = p' \cdot u' = -p'^0u'^0 + \vec p' \cdot \vec u'$
but in the experimentalist frame of reference $u'^\mu = (1, \vec 0)$, therefore the second equation can be written as:
$p \cdot u = p' \cdot u' = -p'^0$
and combining with the first equation, we get:
$\vec p' \cdot \vec p' = (p \cdot u)^2 + (p \cdot p)$
expressing for the magnitude of the 3-momentum as seen in the experimentalist's reference frame:
$|\vec p'| = \sqrt{(p \cdot u)^2 + (p \cdot p)}$

In the problem statement the $(p \cdot p)^2$ comes with power of 2 under the square root, in my solution it comes only with power of 1: $(p \cdot p)$. Could you pls have a look at my solution and to advice whether I am wrong at some point? Or possibly, is there a typo in the textbook from where I got this problem?

Note: I believe it is obvious, but better to emphasize it: author uses such units, that $c=1$ and therefore it is omitted in the equations.

 

[Orodruin]

The square in the $p\cdot p$ term leads to a dimensional inconsistency and is therefore quite clearly wrong.

 

[PeroK]

Or possibly, is there a typo in the textbook from where I got this problem?

As pointed out above, the book solution is wrong. I suspect the author got caught between writing $p.p$ and $p^2$.

 

[lomidrevo]

@Orodruin, @PeroK, thank you! I've suspected that the equation in the textbook might be wrong, but wasn't sure about that.