- #1
space-time
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- 4
I have recently gone over the derivation of the stress energy momentum tensor elements for the special case of dust. This case just used a swarm of particles. Now that I understand that case, I am now trying to see if I can derive the components for an electric field. I just want you guys to please tell me if you agree with what I came up with thus far or if I'm off.
1st: I know the T00 component to be the relativistic energy density. The classical energy density of an electric field (which can be used to find the amount of energy stored in a capacitor) is:
nE = [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex]E2
where: [itex]\epsilon[/itex] is the relative permeability of the material and E is the magnitude of the electric field.
Well, I noted that E=(KQ)/r2 where K is Coulomb's constant, Q is charge and r is the distance from the charge that is responsible for the electric field.
K and Q are invariants, but r is not invariant because r is a length and length can be contracted due to lorentz contraction.
r= r0/[itex]\gamma[/itex]
r0 is the length that is seen in the electric field's rest frame of reference.
[itex]\gamma[/itex] is the typical 1/[itex]\sqrt{1-(v2/c2)}[/itex]
Having said this, if r = r0/[itex]\gamma[/itex] then
r2 = (r0)2 / [itex]\gamma[/itex]2
Now going back to the formula for the magnitude of the electric field, the formula would change to:
E= (KQ) / ((r0)2 / [itex]\gamma[/itex]2) = (KQ[itex]\gamma[/itex]2)/ (r0)2
This would mean that E2 would be (K2Q2[itex]\gamma[/itex]4)/ (r0)4
Finally going back to the energy density expression nE = [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex]E2 , this would change to:
nE = [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex] * (K2Q2[itex]\gamma[/itex]4)/ (r0)4
Therefore my derived quantity for the energy density T00 = [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex] * (K2Q2[itex]\gamma[/itex]4)/ (r0)4
What do you guys think? Am I right, a little off or way off base? If it is either of the latter two, then can you please explain where I went wrong?
1st: I know the T00 component to be the relativistic energy density. The classical energy density of an electric field (which can be used to find the amount of energy stored in a capacitor) is:
nE = [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex]E2
where: [itex]\epsilon[/itex] is the relative permeability of the material and E is the magnitude of the electric field.
Well, I noted that E=(KQ)/r2 where K is Coulomb's constant, Q is charge and r is the distance from the charge that is responsible for the electric field.
K and Q are invariants, but r is not invariant because r is a length and length can be contracted due to lorentz contraction.
r= r0/[itex]\gamma[/itex]
r0 is the length that is seen in the electric field's rest frame of reference.
[itex]\gamma[/itex] is the typical 1/[itex]\sqrt{1-(v2/c2)}[/itex]
Having said this, if r = r0/[itex]\gamma[/itex] then
r2 = (r0)2 / [itex]\gamma[/itex]2
Now going back to the formula for the magnitude of the electric field, the formula would change to:
E= (KQ) / ((r0)2 / [itex]\gamma[/itex]2) = (KQ[itex]\gamma[/itex]2)/ (r0)2
This would mean that E2 would be (K2Q2[itex]\gamma[/itex]4)/ (r0)4
Finally going back to the energy density expression nE = [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex]E2 , this would change to:
nE = [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex] * (K2Q2[itex]\gamma[/itex]4)/ (r0)4
Therefore my derived quantity for the energy density T00 = [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex] * (K2Q2[itex]\gamma[/itex]4)/ (r0)4
What do you guys think? Am I right, a little off or way off base? If it is either of the latter two, then can you please explain where I went wrong?