Understanding Lorentz Invariants and Momentum for My Exam Tomorrow

[stunner5000pt]

Homework Statement

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?

Please help!

Thanks

 

[nrqed]

Homework Statement

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?

Please help!

Thanks

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