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

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  • #1
fluidistic
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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.
 
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  • #2
PAllen
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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
PAllen
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trace=0 does not, by itself, require all diagonal entries to be zero.
 
  • #4
PAllen
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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
fluidistic
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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?
 
  • #6
PAllen
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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.
 
  • #7
fluidistic
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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).
 

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