# Momentum for my exam tomorrow

1. Apr 15, 2008

### stunner5000pt

1. The problem statement, all variables and given/known data
Let P be the 4 momentum
u be the 4 velocity
a) Evaluate the Lorentz invariant $P^2$
b)Differentiate $P^2=P_{\mu}P_{\mu}$ and show that
$$\vec{u}\cdot\frac{d}{dt}\left(\frac{m_{0}\vec{u}}{\sqrt{1-u^2/c^2}}\right)=m_{0}c^2\frac{d}{dt}\left(\frac{1}{\sqrt{1-u^2/c^2}}\right)$$

2. The attempt at a solution

The first part yields an answer of $$E^2 -p^2 c^2=m^2 c^4$$

Now for part b. Does the P^2 have anything to do with the equality that needs to be proven? Do i need to differentiate P^2 with respect to time? Do i hav to differentiate
$$E^2 -p^2 c^2=m^2 c^4$$ with respect to time?

Thanks

2. Apr 15, 2008

### nrqed

Yes, differentiate with respect to time. The derivative of P^2 obviously gives zero since m^2 c^4 is constant.

Now, write P^2 = E^2 - p dot p c^2

(my small p is the three momentum and dot is the dot product

so d/dt(P^2) = 2 E dE/dt - 2 c^2 p dot dp/dt

This must be zero. Now write E= gamma mc^2 and p = gamma m u where u is the ordinary three velocity