In the laboratory reference frame (LRF), a horizontally moving (with constant speed) flat capacitor would have a different size of plates therefore resulting in different capacity $C'$, namely $$ C' = \frac{1}{\gamma}C, $$ where C is a capacity in its own reference frame. The energy of a capacitor is $$ W' = \frac{q^2}{2C'} = \gamma W. $$ Here I assumed the value of a charges remain constant in different inertial reference frames (otherwise we could distinguish one reference frame from another). So if the capacitor is closed on a resistor in his reference frame then for me in LRF would be seen like there was more heat produced on a resistor since $Q=W'$ no matter what current was. So, wouldn't it be the way I distinguish one inertial reference frame from another?
4 Answers
At lowest order, there's no contradiction. Using the relativistic field transformation, the field in the capacitor goes up by a factor of $\gamma$, while the volume of the capacitor goes down by a factor of $1/\gamma$, so the total field energy goes up by a factor of $\gamma^2/\gamma = \gamma$. This is all dissipated in the resistor.
Meanwhile, let the four-momentum given to the resistor in the rest frame be $p^\mu = (U, 0, 0, 0)$. Then the four-momentum given to the resistor in the boosted frame is $p'^\mu = (\gamma U, \gamma v U, 0, 0)$.
That is, in both cases we expect the energy dissipated in the resistor to be $\gamma$ larger than in the rest frame, so there is no contradiction.
However, in the boosted frame there's also a magnetic field which contributes energy at order $v^2$, and I haven't been able to figure out where that energy goes. There's a possibly related subtlety, which is that something must hold the capacitor plates apart. This means there is a pressure somewhere in the system, which can be Lorentz boosted into an energy, and must be counted. This is absolutely essential to account for the case where the boost is perpendicular to the plates, and might fix up the magnetic field energy contribution in this case.
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Related: Is the inertia of light equal to the inertia of mass under $E=mc^2$?
The source of inertia of the capacitor is its stress-energy, just like the source of gravity of the capacitor is its stress-energy.
When we accelerate a box that contains light, the box must have reduced inertia, because the light has increased inertia - because the box as a whole should have normal inertia.
In the case of a capacitor we have charges under pressure and containers of charges under stress. The charges have increased inertia, the containers have decreased inertia.
When some charges move from the positive plate to the negative plate, stress in both plates decreases. Which means that the inertia defect decreases in both plates, in other words inertia of plates increases.
As the plates gain inertia while keeping the same velocity, the plates gain momentum and kinetic energy. The momentum and the kinetic energy come from the electric field.
Magnetic fields are just relativistic effects on electric fields, so magnetic fields have not been in any way left out or forgotten in the above analysis.
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So if capasitor is closed on a resistor in his reference frame then for me in LRF would be seen like there was more heat produced on a resistor since $Q=W'$ no matter what current was. So, wouldn't it be the way I distinguish one intertial reference frame from another?
In the frame where the capacitor with resistor are at rest, all EM energy in the capacitor eventually transforms into increase in internal energy and manifests as increased temperature and increased mass of the system.
In the frame where the capacitor with resistor move, EM energy available is higher, but here not all EM energy needs to transform into increase of internal energy; some of it may transform into increased kinetic energy.
So different frames imply different division of available energy. This does not mean, as I see it, any indication that some reference frames are preferred in the sense of ether theories.
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If we somehow contract the plates of a charged capacitor, the charges get closer to each other and the repulsive Coulomb forces between charges become stronger. The voltage of the capacitor increases. The capacitance of the capacitor decreases.
If we somehow contract the plates of a charged capacitor, and also the fields of the charges, the repulsive Coulomb forces between charges stay the same. The voltage of the capacitor stays the same. The capacitance of the capacitor stays the same. This is what happens when we accelerate or boost a capacitor.
Let's say we have a spherical fleet of spaceships, every ship is positive charged. Adding a positive ship to the fleet takes lot of energy. Now let's say the fleet accelerates to high speed keeping it's dimensions unchanged, like Bell's spaceships. Now it's easy to add a ship to the end of the formation, because the electric fields are contracted, and it's easy to add a ship to the side of the formation because magnetic attraction is helping. The sphere seems to have a larger capacitance when it moves.
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