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.(adsbygoogle = window.adsbygoogle || []).push({});

1st: I know the T_{00}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:

n_{E}= [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex]E^{2}

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)/r^{2}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= r_{0}/[itex]\gamma[/itex]

r_{0}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-(v^{2}/c^{2})}[/itex]

Having said this, if r = r_{0}/[itex]\gamma[/itex] then

r^{2}= (r_{0})^{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) / ((r_{0})^{2}/ [itex]\gamma[/itex]^{2}) = (KQ[itex]\gamma[/itex]^{2})/ (r_{0})^{2}

This would mean that E^{2}would be (K^{2}Q^{2}[itex]\gamma[/itex]^{4})/ (r_{0})^{4}

Finally going back to the energy density expression n_{E}= [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex]E^{2}, this would change to:

n_{E}= [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex] * (K^{2}Q^{2}[itex]\gamma[/itex]^{4})/ (r_{0})^{4}

Therefore my derived quantity for the energy density T_{00}= [itex]\frac{1}{2}[/itex][itex]\epsilon[/itex] * (K^{2}Q^{2}[itex]\gamma[/itex]^{4})/ (r_{0})^{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?

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# 1st stress-energy tensor component for an electric field

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