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

[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.

 

[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.]

 

[PAllen]

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

 

[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.

 

[fluidistic]

I see PAllen, thanks for the confirmations.
However I don't really understand your last statement

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.

. If it is coordinate independent, then how come it is not true with general coordinates?

 

[PAllen]

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?

trace=0 is true in all coordinates.

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

 

[fluidistic]

trace=0 is true in all coordinates.

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

Ah I see! Thanks a lot. (I had to reread).