B How do we derive p = γmv ?

1. Mar 1, 2017

Cozma Alex

I thought this mental experiment: consider an inertial frame of reference solidal to a particle moving with velocity v respect to another inertial frame of reference, i gotta find the momentum in the second frame of reference (in the first is 0 since is solidal with the particle)

p= m* dx/dt

And then i plug instead of x and t the lorentz tranformation thwt connects them with t' and x'...

But it doesn't work, where am i wrong? Is that method correct? If not why?

2. Mar 1, 2017

haushofer

That doesn't work, because the (spatial component of the!) relativistic momentum is p = ymv to start with. You should transform the 4-momentum. This 4-momentum is defined by the relativistic action of a point particle.

3. Mar 1, 2017

SiennaTheGr8

The idea is to find an expression that:
• is conserved in closed systems (and additive) even when $v \rightarrow c$,
• increases in magnitude without bound as $v \rightarrow c$ (because otherwise it would have a maximum limit like $v$ does, and this would contradict the first criterion), and
• reduces to $\mathbf{p} \approx m \mathbf{v}$ in the classical limit.
The famous Tolman/Lewis thought experiment demonstrates that if such a quantity exists, then it must be $\mathbf{p} = \gamma m \mathbf{v}$, which obviously meets the second and third criteria. See here: https://books.google.com/books?id=FrgVDAAAQBAJ&pg=PA76.

As for the first criterion, experiment verifies that this vector is indeed conserved at all possible values of $v$. (Think particle accelerators.)