Imagine an LC circuit in which the capacitor consists of two metal discs of area $A$ and spacing $D_0$ which is connected to a solenoid coil which for convenience has length $D_0$, cross-sectional area $A$ and contains $N$ turns of wire. The capacitance of the capacitor at rest is designated $C_0$ and the inductance of the coil is $L_0$ where

$$C_0 = \frac{e_0A}{D_0}$$ and $$L_0 = \frac{u_0N^2A}{D_0}$$

Where $e_0$ is the permittivity of free space and $u_0$ is the permeability of free space.

We provide a means of momentarily asserting a current through the coil or a voltage on the capacitor in order to kick it into oscillation. The circuit contains no resistance and so its oscillations do not die away. We also provide an ammeter in series with the circuit that has a needle that swings back and forth to allow nearby observers to measure the period of the oscillations. The circuit’s resonant frequency at rest is

$$f_0 =\frac{1}{\sqrt{L_0C_0}}$$

We orient the capacitor and the inductor so their axies of symmetry are pointing in one direction, check the circuit’s resonant frequency by watching the ammeter needle, and then accelerate the circuit in that direction. As the circuit’s speed $v$ becomes relativistic, both the capacitor and the inductor experience foreshortening, as seen by a stationary observer nearby: the gap $D_0$ between the plates appears to have shrunk to $D(v)$, and the length $D_0$ of the coil also appears to have shrunk to $D(v)$ where we use the length contraction factor $\gamma = \sqrt{1-\frac{v^2}{c^2}}$ to calculate $D(v)$:

$$D(v) = D_0\sqrt{1-\frac{v^2}{c^2}}$$

That observer then calculates the capacitance $C(v)$ of the foreshortened capacitor and the inductance $L(v)$ of the foreshortened coil as

$$C(v) =\frac{e_0A}{D(v)}$$ which is greater than $C_0$, and

$$L(v) = \frac{u_0N^2A}{D(v)}$$ which is greater than $L_0$.

Because the foreshortened capacitor’s capacitance went up and the foreshortened coil’s inductance went up, we would expect the resonant frequency to go down to some value $f(v)$ where

$$f(v) = \frac{1}{\sqrt{L(v)C(v)}}$$

Substituting in the values for $L(v)$ and $C(v)$, we get

$$f(v) =f_0 \sqrt{1-\frac{v^2}{c^2}} $$

So we used the length contraction factor $\sqrt{1-\frac{v^2}{c^2}}$ to shrink the width of the capacitor gap and the length of the coil, and then obtained an equation for a shrinkage in the resonant frequency of the circuit.

Now if we had instead applied the gamma factor of relativistic time dilation directly to the time between peaks of the moving circuit’s oscillation, we would get exactly the same reduction in the resonant frequency of the circuit as we did by relativistically foreshortening the physical dimensions of the oscillator’s components.

Is this seeming coincidence worth understanding more deeply, or is it simply a trivial result due to some circularity in my reasoning?

  • $\begingroup$ Are you sure electromagnetic laws leading to the frequency formula still hold true for components (capacitor, solenoid) moving at fast speed wrt lab? $\endgroup$ – Elio Fabri Oct 8 '18 at 19:07

Interesting question, unfortunately circuit theory is non relativistic. Circuits that satisfy the assumptions of circuit theory in their rest frame do not satisfy them in other frames. For example, one of the assumptions of circuit theory is that all of the components of a circuit have no net charge (Nilsson and Riedel, Electric Circuits). But a current density in one frame is a charge density in another frame, so some components gain net charge (Panofsky and Phillips, Classical Electricity and Magnetism). Also, the geometry of a circuit does not matter in circuit theory, but it does matter in relativity.

So, to answer the question, I would say that it is probably not merely a coincidence, but I am not sure that there is much that can be done to pursue it more deeply. The circuit theory is fundamentally a non relativistic approximation, so it would require quite a bit of theoretical work to get it to the point where it could be proven that it is more than a coincidence

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Basic idealized analogue circuit theory is based on Maxwell's equations. Such ideas have without a doubt a relativistic version. One must be careful with the formulas to know which ones change and which ones remain valid.

Capacitance and Inductance are geometrically defined. For example in a circular loop, if the object has relativistic motion along the axis perpendicular to the plane of the loop there is no change. However at any other angle, the circular loop stops being circular and becomes oblate. One must now recalculate the relationship that current flowing through an oblate circuit has to the inductance.

The short answer is Einstein's answer, namely, the laws of physics hold true in all inertial frames. If a contradiction is noted then an error was made (seldom is it with the theory unless one is doing cutting edge experiments).

In the example, it may or may not be true that the length of the coil D(0) changes to D(v). It depends on how the coiled object is oriented with respect to the disk, and the relative motion. The coil length D(0) changes to D(v) (the changed disk spacing) only when the coil is unwound and its line segment is parallel to the relativistic motion also presumed to be the axis perpendicular to the capacitor disks and their spacing.

Diagrams and symbols are always easier than words. But if you have your coil loop parallel to your capacitor disks, such that the disk spacing is affect by relativistic motion, then the coil length will not be changed.

Applying Lorentz-Fitzgerald contraction in all the right ways and right places will always result in the frequency measurements by a non stationary observer to match the direct calculation of the non stationary observer's computation of the component pieces.

To get one's bearings in situations like this it is best to start with what are the invariants. Charge, Q, is an invariant. So the formula Q = V C, where V is the voltage and C is the capacitance shows us that it is the electric field itself that transforms (thus changing the voltage V) and the capacitance transforms due to changes in the geometry. Now this is the key, and they both do so in such a manner as to leave Q unchanged.

In this specific case, I cannot tell for certain but it seems that there may be two coincidental errors or omissions that led to an agreement but would not otherwise. I don't see the justification for the coil length changing. It doesn't help that they were given the same name. I always give each physically distinct quantity, a mathematically distinct symbol to ensure I avoid confusion. Then at the end of a correct calculation I can check for degeneracy or coincidence.

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