A few very basics questions regarding 4 and 3 momenta and the trace of

1. Sep 17, 2013

fluidistic

Hi guys! I've got 2 extremely simple questions, hence a single thread.
First, I want to know whether the conservation of the 4-momentum in a closed system implies the conservation of the energy and of the 3-momentum.
Let's assume we consider 2 different times, $t_i$ and $t_f$. Then $P_i=(E_i/c,\vec p_i)=P_f=(E_f/c, \vec p_f)$. Where the E's are the energy at the 2 different times and the lower capital p's are the 3-momenta at those 2 different times.
Now each component of $P_i$ must be equal to each component of $P_f$ right? If so, it follows that $E_i=E_f$ and that $\vec p _i = \vec p_f$. Is this correct?

Second question. I've "heard" that the trace of Faraday tensor is 0 and thus its diagonal entries are all 0. However the trace is definied as the sum of all the entries on the diagonal... so the fact that the trace is 0 does not imply that all the diagonal entries are worth 0. Is this correct?
Thanks.

Last edited: Sep 17, 2013
2. Sep 17, 2013

PAllen

Yes to your first question - all reasoning and conclusion is fine.

[edit: removed answer to second question, since you changed it while I was writing answer.]

3. Sep 17, 2013

PAllen

trace=0 does not, by itself, require all diagonal entries to be zero.

4. Sep 17, 2013

PAllen

As for the Faraday tensor, trace=0 is a coordinate independent statement. All diagonal elements zero is true in Minkowski coordinates, but not in general coordinates.

5. Sep 17, 2013

fluidistic

I see PAllen, thanks for the confirmations.
However I don't really understand your last statement
. If it is coordinate independent, then how come it is not true with general coordinates?

6. Sep 17, 2013

PAllen

trace=0 is true in all coordinates.

All diagonal elements zero is not true in general coordinates. However, the trace will still be zero.

7. Sep 17, 2013