Conservation of Four-Momentum in Inertial Frames

[brainpushups]

Homework Statement

a) Suppose that the total three-momentum of an isolated system is conserved in all inertial frames. Show that if this is true (which it is), then the fourth component of the total four-momentum has to be conserved as well. b) Using the zero-component theorem you can prove the following stronger result very quickly: If anyone component of the total four-momentum is conserved in all frames, then all four components are conserved.

Homework Equations

Zero component theorem: If q is a four vector and one component of q is found to be zero in all inertial frames then all four components of q are zero in all frames.

The Attempt at a Solution

a) The total three momentum is $P=\gamma_{1}m_{1}v_{1}+\gamma_{2}m_{2}v_{2}+...$

The fourth component of the total four-momentum is

$P_{4}=c(\gamma_{1}m_{1}+\gamma_{2}m_{2}+...)$

My idea was to show that if the time rate of change of the three momentum is zero that this must somehow guarantee that the quantity in parentheses for the fourth component also has a time rate of change of zero.

The time rate of change of the total three momentum is (notice the dots above gamma and v they are hard to make out).

$P^{.}=\gamma^{.}_{1}m_{1}v_{1}+gamma_{1}m_{1}v^{.}_{1}+\gamma^{.}_{2}m_{2}v_{2}+gamma_{2}m_{2}v^{.}_{2}+...=0$

The time rate of change of the fourth component is

$P^{.}_{4}=c(\gamma^{.}_{1}m_{1}+\gamma^{.}_{2}m_{2}+...)$

I'm not seeing anything that allows me to proceed from this point.
As an alternative I thought I could use the invariant scalar product in some way, but it wasn't clear how this possible alternative approach would work out either.

b)
I guess I’m not really sure where to start with part b) except to show that the time rate of change of at least one of the components of the total four-momentum is zero in all frames. I’m not sure if the author is implying that this is given information because of the phrasing of the question. If we assume $P^{.}$ to be zero then this seems very trivial because simply stating the zero component theorem is about enough to prove it!

 

[dauto]

The derivative of a 4-vector with respect to proper time (a scalar) is also a 4-vector

 

[brainpushups]

I understand that, but I'm not clear on how that helps me.

 

[dauto]

If one component of a 4-vector is conserved, its derivative with respect to proper time is zero. By the zero component theorem, all the other components of the derivative must vanish as well which means the other components of the 4-vector are also conserved.

 

[brainpushups]

Ok that makes sense for part b. I thought you were referring to part a. Any thoughts about part a and how to show this without invoking this theorem? Thanks for your thoughts.