Do Kirchhoff's circuit laws work at relativistic speeds?

Question

To a pair of related questions, see this and this, regarding the applicability of Kirchhoff's laws in a relativistic regime @Dale rejected the possibility based on the following arguments:

1. all lumped elements have no net charge but a current density in one frame is a charge density in another frame, so some components gain net charge
2. there is no inductive coupling between lumped elements (a mutual inductance is considered a single lumped element with four terminals)
3. the circuit is small enough that c can be ignored and all effects are assumed to happen instantaneously the circuit is small enough that c can be ignored and all effects are assumed to happen instantaneously

And then he states that "Only with all three of these assumptions can circuit theory be derived from Maxwell’s equations. So without them any attempt to use circuit theory will inevitably lead to contradictions."

But I am skeptical of all three.

First #3. It is routine EE practice to add transmission line models if the network gets too large and its connections too long, and transmission line theory is nothing but Kirchhoff's laws over an infinite ladder. If you do not like "infinite" number L-s and C-s then model the delay by a finite set of lumped element circuits, such as in a Thomson-Bessel filters, etc.

Regarding #2 either there is inductive coupling between geometrically separated lumped elements or there is not. If there is such a coupling then, as a first approximation, it can be modeled by an ideal four-terminal inductive transformer. If that is not good enough an approximation, one can make it more complicated by adding more inductive transformers and/or capacitors. Again this does not limit the applicability of Kirchhoff's laws.

Finally #1. I do not understand how/where in Kirchhoff's laws would need that be excluded? If I place a bunch of electrons and freeze them in a dielectric between two plates of a capacitor why would that preempt me to model it as a capacitor in the network? If instead dynamically we get bunching but now equally as many positive as negative charges accumulate then if they are close enough to interact then it is a capacitor to Kirchhoff's and we have to include that capacitive coupling as it is routinely done on RF boards by engineers fighting the "strays". If they are separated so their interaction is negligible then it is negligible and just does not matter.

All of Maxwell's equations can be translated to Kirchhoff's KCL/KVL equations appropriately modified to include the transmission line models (see Schelkunoff, Dicke, Purcell, etc.) Whether it is practical to do so in any given problem is a separate question but for the simplest and most common distributed systems it is the most practical way, and this includes almost all waveguide circuits, antennas and their matching circuits, etc. Even most antennas can be analyzed this way and it is essentially the only way to find how they behave as a load to a driver or a source to a receiver.

To my surprise I could not find any relevant literature besides discussing such simpler cases as how a capacitance might change with velocity. So here I am asking for a revisit of these issues:

1. Is @Dale right when he rejects Kirchhoff's laws in the relativistic domain based on his three requirements? Do Kirchhoff's laws really need those assumptions?
2. Is it possible to reformulate KCL/KVL in a relativistic language despite relativistic kinematics' dependence on geometry while Kirchhoff's laws are dependent on topology? Is it enough to add transmission line models to move from topology to geometry?

Answer 1

To my surprise I could not find any relevant literature

I think that this is important to keep in mind. I also do not know of any literature describing a complete relativistic circuit theory. So my base assumption is that it is not possible, simply from the literature.

While I have specific arguments for some of @hyportnex’s objections, I do not have them for all. But I assume the missing arguments do not affect the outcome. Otherwise some clever theoretician in the last century would have developed a relativistic circuit theory.

Is it possible to reformulate KCL/KVL in a relativistic language despite relativistic kinematics' dependence on geometry while Kirchhoff's laws are dependent on topology?

This is, I think, the biggest issue. Relativity is a geometrical theory and circuit theory is topological. How could velocity even be introduced to a topological theory? Would a theory still deserve the name of “circuit theory” if you have to identify the path of each wire and the position of each circuit element?

Frankly, I think not, so I think relativistic circuit theory is a self contradiction from the beginning. If a theory requires the specification of the geometry, then I would not call it circuit theory. And I don’t see how a theory can be relativistic without the geometry.

I think this objection is not fixable. In any case, it is strong enough for me to require an actual working relativistic circuit theory as the counter proof. While the absence of such a working theory in the scientific literature is not a proof of its impossibility, it is certainly strong supporting evidence.

Do Kirchhoff's laws really need those assumptions?

Circuit theory textbooks state then quite clearly, so I would be hesitant to dismiss any of them. But let’s examine the impact of each assumption.

all lumped elements have no net charge …

I do not understand how/where in Kirchhoff's laws would need that be excluded?

This is the one for which I do not have a complete answer. If the total charge on any circuit element varies then Kirchoff’s current law (KCL) is violated as the current leaving the element is not the same as the current entering it.

So it is clear that the net charge on each element must be constant. While requiring it to be zero does imply that it is constant, zero is obviously not the only constant. So I cannot argue that it needs to be zero, merely that it needs to be constant.

The fact that the textbooks state it as zero and not just constant makes me suspect that I am missing some argument. But let’s investigate both the zero-charge and the constant-charge assumptions.

Now, in relativity a current density in one frame is a charge density and a current density in other frames. So the zero-charge assumption excludes relativistic inertial circuits. The constant-charge assumption would not exclude relativistic inertial circuits, but it would exclude relativistic accelerating circuits. KCL would fail as the accelerating circuit elements accumulate net charge.

there is no inductive coupling between lumped elements …

If there is such a coupling then, as a first approximation, it can be modeled by an ideal four-terminal inductive transformer.

Violation of this assumption makes it so that the voltages around a loop no longer sun to zero. This directly violates Kirchoff’s voltage law (KVL).

Your proposed approach for addressing such violations is clever, but you are probably overly optimistic about how it would work. First, you may need many more than four terminals in general. Second, and more importantly, in general you will not be able to treat the resulting multi-terminal element as a transformer.

To determine the current voltage relationship for this element will require a derivation or some good experiments. Such a derivation, in turn, could not use circuit theory (KVL is violated), but would typically require Maxwell’s equations. So what would be the point?

the circuit is small enough that c can be ignored and all effects are assumed to happen instantaneously …

It is routine EE practice to add transmission line models if the network gets too large and its connections too long, and transmission line theory is nothing but Kirchhoff's laws over an infinite ladder

This assumption is necessary for the electro quasi static and magneto quasi static approximations of Maxwell’s equations from which KVL and KCL are derived. In other words, the previous two assumptions are necessary for deriving KVL and KCL, but not sufficient.

While it is true that transmission line models are used by EE’s, so are Maxwell’s equations. The mere use of a technique by EE’s does not make that technique part of circuit theory. Transmission line models are somewhat of a bridge between circuit theory and the full Maxwell’s equations. But in themselves they do violate Kirchoff’s laws, so they are not part of circuit theory.

Transmission line theory has different assumptions than circuit theory: the small cross section assumption and the no mutual interference assumption. Additional assumptions can allow the use of circuit theory, but it is simply untrue that all transmission line models are part of circuit theory.

Obviously, this assumption in itself is not compatible with relativity. So since it is required for both KVL and KCL it is clear that a relativistic circuit theory could not be based on KCL and KVL. It is unclear what relativistic laws could replace them, and even less clear if the resulting theory would merit the label “circuit theory”.

So, in summary, I am not confident about the reason for the zero-charge assumption, but the weaker constant-charge assumption is required for KCL. The no-coupling assumption is required for KVL. The instantaneous assumption is required for both KVL and KCL. However, the strongest argument is the topological vs geometrical issue.

Answer 2

Do Kirchhoff's laws really need those assumptions?

Not all of those, not really.

KVL is a useful rule for formulation of circuit equations, based on the fact that electric potential is a single-valued function of position in space. This holds in the relativistic scenario as well. Values of potential and of parameters $R,L,C,\mathscr{E}$ of circuit components may have different values in the frame where the circuit moves, and it may be difficult to express them (e.g. due to external induced emfs) but the fact remains valid.

KCL states that for any region of space where there are no components but only perfect conductors connected to their terminals, sum of currents entering the region equals sum of currents leaving the region, or in other words, integral of current density over closed boundary of that region is zero. Mathematically, everywhere outside the components, $$ \nabla'\cdot \mathbf j' = 0, $$ which means that in those regions (assuming the Maxwell equations) $$ \partial_t' \rho' = 0. $$

This holds in the rest frame of the circuit $S'$. Does it also hold in the frame $S$ where the circuit moves relativistically with velocity $v$?

Unfortunately, not in general. From the Lorentz transformation, we know that a piece of current-carrying perfect conductor with zero charge density in its rest frame ($\rho' = 0$) will have, in general, non-zero charge density $\rho$ proportional to current density $j_x'$ in direction of the velocity. In a DC scenario, where $j_x'$ is constant in time, $\rho$ will be constant in time as well, so $\nabla \cdot \mathbf j = 0$ and KCL holds in $S$. But when currents change in time (AC circuits), $j_x'$ is changing in time, and $\rho$ is no longer constant in time, and thus $\nabla \cdot \mathbf j$ no longer vanishes and thus KCL does not hold.

Nevertheless, all physics variables in both frames seem to be related via linear transformations, so the moving circuit should still be correctly described by some set of linear differential equations. It is just not possible in general to obtain those equations from KVL and KCL applied in the frame where the circuit moves.

Answer 3

The essential issue here is that we must draw a distinction between the textbook cartoon version of Kirchhoff's laws and the real world version. In reality, we patch up the cartoon as needed to account for time delay, displacement current, and induction. Yes, the cartoon won't work at relativistic speed, but it doesn't even work on the bench. If you use Kirchhoff's laws, you are implicitly promising to account for effects beyond the cartoon as required in practice.

Answer 4

This is not a full answer to my question and it is too long for a comment but it is a start.

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In this excellent note, Prof. Errede derives two results as to how the fields of a capacitor and of a coil transform from one inertial system to another. First I just summarize a few relevant results described there. For the capacitor:

$$\vec{E}=\gamma_0 E_0\hat y \tag{1e}$$ $$\vec{B}=-\frac{\gamma_0}{c}\beta_0 E_0\hat z \tag{1b}$$

Here $E_0=\frac{V}{d_0}$ is the homogeneous field intensity between plates separated by $d_0$ and voltage drop $V$.

For the coil: $$\vec{E}=0 \tag{2e}$$ $$\vec{B}=\mu_0 \mathcal {N} I \hat x \tag{2b}$$ where $\mathcal {N}$ is the number of turns per unit length.

For quasi-stationary approximation to hold, one that ignores all radiation effects, we must assume that ideally lumped circuits are point-like. In practice this means that their characteristic length must not be larger than a few percent of the free-space wavelength.

Errede's notes do not discuss the field transformation of the coil for lateral motion, but that is addressed for a single loop in Panofsky-Phillips with the result that along the perimeter of rectangular loop, 2 long and 2 short sides, carrying a current $I$ there develops a static dipole, so that one side is negatively charged the other is equally positively charged. The charge separation is within the confines the rectangular loop and the charge, and thus the dipole moment is proportional to the current. I have not been able to derive the same for a circular loop let alone for a coil but I am confident that a similar result will come about, just as I am confident that a similar development will happen if the capacitor is not a pair of planar plates but are made of a more complicated structure resulting in a magnetic field orthogonal to both the electric field inside, thus the voltage between the terminals, and to the motion.

The import of all this is that both the capacitor and the coil stay lumped that is all the relativistic effects stay inside the body of the element and the effect is a linear function of the voltage and current, respectively. An engineer modeling this would say that a capacitor is to be augmented in series with a voltage controlled current source that is an inductor; while the coil is to be augmented in parallel with a current controlled voltage source that is a capacitor. All this modification is completely within the classical lumped element version of Kirchhoff's KCL and KVL.

All this must include the proviso that the free space wavelength be much larger than any of the circuit elements. Here the motion towards the observer must be taken into account because the Doppler effect shrinks the wavelength already unrelated to relativity. That is we have a limit as to how fast we can move towards the observer and still maintain quasi-stationarity.

Geometry comes into play by noticing that the relativistic effect on each element depends on said point-like element's disposition relative to the motion. In the stationary system this is completely ignored by the KCL/KVL but it shows up in the controlled sources attached to the elements.

Where KCL/KVL may run into trouble is the connecting wires that are completely ignored by them, again as a result of assuming that the whole circuit is essentially point-like, no geometry, only topology. Here I can only offer what in my original question I already alluded to in that its resolution will be accounting for the wires as multi-coupled transmission lines. An example for which one must do something like that already is in the high frequency PC boards where the spurious coupling between connecting microstrip transmission lines are to be accounted for. (The fact that these PC boards usually have a common ground plane is incidental to this argument, it makes the system work better but coupled wires do not have to have a common ground plane.)

Of course, once we introduce transmission lines connecting the lumped elements we are not in the realm of the KCL/KVL as usually are formulated. To avoid misunderstanding I use the term transmission line, for lack of a better term, in a more general sense than just having two parallel Lecher wires. To me a transmission line is anything that guides EM waves. And when understood in this sense a transmission line can be represented as an infinite long set of L-s and C-s starting between the terminals of the driver source (no need for a common ground) and extending between the various wires. The said distributed not necessarily homogeneous L-s and C-s can be lumped to a few finite elements and thereby simplify the circuit but they all follow KCL/KVL and I believe that their relativistic effects can be handled same as the normal elements.

In summary, I still believe in Kirchhoff!