I Energy and reference frames

[Ivan Nikiforov]

I am interested in a question related to energy for different reference frames. Let's say we have two inertial reference frames A and B. The reference frame B moves relative to the reference frame A at a high speed at which relativistic effects are manifested. In reference frame B there is a diesel electric generator that operates and outputs a power of 2200VA, voltage 220V, current 10A. The terminals of this generator are connected using wires and sliding contacts to the reference frame A, which contains a load of 2200VA, 220V, 10A.

According to the special theory of relativity, for reference frames A and B, there is a relativity of simultaneity, that is, time slows down in the moving reference frame B. In this case, reference frames A and B are connected by an electric circuit, the processes in which occur at the speed of light, that is, at the maximum possible speed, the same for all reference frames.

Is it true that the following results are obtained? An observer in the frame of reference A (load) sees in the frame of reference B a generator that runs slower (consumes less fuel), but at the same time outputs power, voltage and current, as in normal operation. The observer in reference frame B sees in his own reference frame a normally operating generator that consumes a normal amount of fuel and outputs normal power, voltage and current. After returning to the frame of reference A, this observer discovers that the generator, which worked for him for 1 day, supplied energy to the frame of reference A for 2 days. Are these correct?

 

[PeroK]

In this case, reference frames A and B are connected by an electric circuit, ...

Reference frames are not physical objects that can be connected by an electric circuit! A reference is essentially a system of coordinates that assigns time and space coordinates to every event.

 

[Ibix]

First of all, you are misusing the term "reference frame". Reference frames are just coordinate systems - they can't be connected by circuits because they're just ideas. You can connect objects that are at rest in different frames, which is what you appear to be trying to describe.

Second, the relativity of simultaneity is not the same as "time slows down". It is a related effect, but it means that clocks that show the same time according to one frame of reference may not do so according to another.

To answer your actual question, I suspect that your setup is contradictory. You have a generator that produces some amount of power as measured in its own rest trame, and a load that consumes that amount of power in its own reference frame. With special relativity in play, the two devices see each other time dilated, so don't measure the same power production/consumption as in the devices' respective rest frames. So your scenario can't work as described.

It's also not clear to me whether you're imagining the devices moving in straight lines or one orbiting the other (as in the diagram). That makes a difference to the analysis.

 

[Ivan Nikiforov]

Reference frames are not physical objects that can be connected by an electric circuit! A reference is essentially a system of coordinates that assigns time and space coordinates to every event.

I agree with you. You can change the model a bit. Let's say there is a laboratory with which the reference frame A is connected and there is an object, for example, a capsule, which moves relative to the laboratory and with which the reference frame B is connected.

 

[PeroK]

I agree with you. You can change the model a bit. Let's say there is a laboratory with which the reference frame A is connected and there is an object, for example, a capsule, which moves relative to the laboratory and with which the reference frame B is connected.

You still can't "connect" reference frames like they were physical objects.

 

[Ivan Nikiforov]

To answer your actual question, I suspect that your setup is contradictory. You have a generator that produces some amount of power as measured in its own rest trame, and a load that consumes that amount of power in its own reference frame. With special relativity in play, the two devices see each other time dilated, so don't measure the same power production/consumption as in the devices' respective rest frames. So your scenario can't work as described.

The generator is located in a moving capsule. The capsule is connected to the laboratory by means of wires and sliding contacts. The load is in the laboratory. That is, a process is obtained in which, on the one hand, there is a time dilation between the capsule and the laboratory, and on the other hand, the capsule and the laboratory are connected by an electrical circuit, in which processes occur simultaneously for the capsule and the laboratory.

 

[Ivan Nikiforov]

You still can't "connect" reference frames like they were physical objects.

Let's say the laboratory is a very long building with two rails installed on the roof. A load, such as an incandescent lamp, is connected to these rails. A capsule is flying relative to the laboratory, inside of which there is a diesel generator. Wires are stretched from the capsule to the laboratory rails, which are connected through sliding contacts. It is quite possible to create such a model in reality.

 

[Ibix]

The generator is located in a moving capsule.

But moving in a straight line, or in circles? Or something else?

That is, a process is obtained in which, on the one hand, there is a time dilation between the capsule and the laboratory,

Yes, and vice versa, although details depend on your answer to the path the capsule follows.

the capsule and the laboratory are connected by an electrical circuit, in which processes occur simultaneously for the capsule and the laboratory.

This is rather vague. What processes?

 

[Ivan Nikiforov]

But moving in a straight line, or in circles? Or something else?

Yes, and vice versa, although details depend on your answer to the path the capsule follows.

This is rather vague. What processes?

This refers to an electric current, which is known to propagate in wires at the speed of light. The speed of light is the same for all reference frames.

 

[Ibix]

This refers to an electric current, which is known to propagate in wires at the speed of light.

It doesn't - electrical signals travel at a comparable speed to light, but not at light speed.

That's not really relevant to your problem, though, which is that you have a system that produces power apparently at a different rate from which it is being consumed. Am I correct in understanding that this is your problem?

 

[Ivan Nikiforov]

It doesn't - electrical signals travel at a comparable speed to light, but not at light speed.

That's not really relevant to your problem, though, which is that you have a system that produces power apparently at a different rate from which it is being consumed. Am I correct in understanding that this is your problem?

Yes, I would like to understand the energy ratios for this case, when the generator is in a capsule and the load is in the laboratory. For the capsule, there is a time dilation, that is, the processes in the capsule and the laboratory do not occur simultaneously, while the electric current in the capsule and the laboratory flows simultaneously.

 

[Nugatory]

This refers to an electric current, which is known to propagate in wires at the speed of light.

They do not, which unnecessarily complicates the analysis.

You may find the problem easier to analyze if you put it in a different form: We have a light source with a particular fuel consumption and power output, as measured using a frame in which it is at rest. It illuminates a photocell that converts light to electrical power with 100% efficiency. The photocell and the light source are moving relative to one another.

In any case, if the photocell and light source are initially colocated and at rest relative to one another, separate by moving relative to one another, and eventually reunite: The energy received between separation and reunion will be equal to the energy emitted between separation and reunion.

 

[Ibix]

Yes, I would like to understand the energy ratios for this case, when the generator is in a capsule and the load is in the laboratory.

The first thing to think about is the changing distance between the generator and the load. That on its own means that the absorbed power is different from the emitted power. Also, you will need to factor in the different energy content of the fuel in the two frames.

 

[Ivan Nikiforov]

The first thing to think about is the changing distance between the generator and the load. That on its own means that the absorbed power is different from the emitted power. Also, you will need to factor in the different energy content of the fuel in the two frames.

The distance between the rails and the capsule does not change. How can the energy of diesel fuel change during the transition to the reference frame B? Relativistic mass increase, linear size reduction, time dilation?

 

[Ibix]

The distance between the rails and the capsule does not change

The distance between the generator and the load does, though.

How can the energy of diesel fuel change during the transition to the reference frame B?

Energy is frame dependent, even in Newtonian mechanics. It matters more often in relativity.

Relativistic mass increase

Don't use relativistic mass - outside pop sci, it's been deprecated for decades because it just causes confusion.

 

[Ivan Nikiforov]

They do not, which unnecessarily complicates the analysis.

You may find the problem easier to analyze if you put it in a different form: We have a light source with a particular fuel consumption and power output, as measured using a frame in which it is at rest. It illuminates a photocell that converts light to electrical power with 100% efficiency. The photocell and the light source are moving relative to one another.

In any case, if the photocell and light source are initially colocated and at rest relative to one another, separate by moving relative to one another, and eventually reunite: The energy received between separation and reunion will be equal to the energy emitted between separation and reunion.

Yes, your model can be used to address this issue. As for the periods between separation and reunification, I think the question is that these periods are different for different frames of reference. Of course, I need some time to think about your comment.

 

[Ivan Nikiforov]

The distance between the generator and the load does, though.

The capsule moves parallel to the rails and the distance between them does not change. The load, an incandescent lamp, is connected to the rails and such a system, in fact, is an ordinary electric line.

 

[DaveE]

This all sounds like a red shifted photon to me, but with a bunch of complex hardware. No?

 

[Ibix]

The capsule moves parallel to the rails and the distance between them does not change.

Either the load is moving relative to the generator or it isn't. In the first case the distance between the load and generator must be changing. In the other case, they're at rest in the same frame.

 

[Ibix]

This all sounds like a red shifted photon to me, but with a bunch of complex hardware. No?

You need to worry about the energy content of the fuel in the two frames too, but basically yes.

 

[Ivan Nikiforov]

This all sounds like a red shifted photon to me, but with a bunch of complex hardware. No?

Devi, forgive me, I don't quite understand what a redshifted photon means.

 

[jbriggs444]

Devi, forgive me, I don't quite understand what a redshifted photon means.

You have energy in transit in the rails. Accounting for the energy in transit may be complicated. A red shifted photon is a less complex way of accounting for energy in transit.

 

[Ivan Nikiforov]

Either the load is moving relative to the generator or it isn't. In the first case the distance between the load and generator must be changing. In the other case, they're at rest in the same frame.

The load is an incandescent lamp connected to the rails. The distance between the rails and the capsule does not change. The rails are structurally part of the load.

 

[phinds]

View attachment 356680 The load is an incandescent lamp connected to the rails. The distance between the rails and the capsule does not change. The rails are structurally part of the load.

That's not exactly irrelevant, but it's far less important than the fact that the distance between the source and the load DOES change.

 

[DaveE]

Devi, forgive me, I don't quite understand what a redshifted photon means.

https://en.wikipedia.org/wiki/Redshift

Change your generator to two groups of hydrogen atoms moving wrt each other (in any single RF). A photon is made by a pair in one group, travels and is absorbed by atoms in the other group (in any single RF).

If you observe the system in a different RF, the photon energy will appear to be different, but so will the energy of the atoms.

It's not that one lab "has" a RF and the other "has" a different one. The system is two labs with generators and loads that have some relative velocity. The system can be observed in different reference frames, with different results for the internal pieces.

 

[Ivan Nikiforov]

That's not exactly irrelevant, but it's far less important than the fact that the distance between the source and the load DOES change.

Thank you for your comment. Could you explain how this affects the amount of energy? And what are the ratios for energy in such a system?

 

[Ivan Nikiforov]

https://en.wikipedia.org/wiki/Redshift

Change your generator to two groups of hydrogen atoms moving wrt each other (in any single RF). A photon is made by a pair in one group, travels and is absorbed by atoms in the other group (in any single RF).

If you observe the system in a different RF, the photon energy will appear to be different, but so will the energy of the atoms.

It's not that one lab "has" a RF and the other "has" a different one. The system is two labs with generators and loads that have some relative velocity. The system can be observed in different reference frames, with different results for the internal pieces.

Devi, thank you for your comment. I'll need some time to think about it. Unfortunately, I'm not very good at processes involving atoms and photons. I'm just an electrical engineer.

 

[PeroK]

I'm just an electrical engineer.

So is @DaveE

 

[Ivan Nikiforov]

Thank you for your comment. Could you explain how this affects the amount of energy? And what are the ratios for energy in such a system?

Sorry, if that's how the load is presented, then the distance to it probably doesn't change.

 

[SiennaTheGr8]

You have a generator that produces some amount of power as measured in its own rest trame, and a load that consumes that amount of power in its own reference frame. With special relativity in play, the two devices see each other time dilated, so don't measure the same power production/consumption as in the devices' respective rest frames.

If I'm not mistaken, it's a little more subtle than that, since both the time and the energy transform. In general, the "coordinate power" is given by $P = \gamma^3(\vec v \cdot \vec a)m + \gamma \dot{m}$, and by time dilation and the chain rule we have $\gamma \dot{m} = dm/d\tau$ (i.e., the proper power). So in the special situation where there's power but no acceleration, the power is in fact Lorentz-invariant (the magnitude of the spacelike four-force). So, technically it might depend on the generator?

(Then there's a further subtlety, which is that for a body to lose energy without accelerating, it's got to radiate "equally" from its spatially separated extremities, and then the relativity of simultaneity complicates things.)

 

[Ivan Nikiforov]

Excuse me if I misunderstand something, but in this book, on page 295, it literally says the following: "Since energy manifests itself only when it changes, we can put it without prejudice to the physical meaning.:

Thus, we come to the conclusion that energy, measured by the amount of "work that a body is capable of doing," and mass, determined by momentum at a given speed, are interrelated concepts. If there is one, then there is another. Neither mass nor energy are invariant; each quantity, according to (16.42), depends on the observer's frame of reference."

 

[Ivan Nikiforov]

View attachment 356684
Excuse me if I misunderstand something, but in this book, on page 295, it literally says the following: "Since energy manifests itself only when it changes, we can put it without prejudice to the physical meaning.:
View attachment 356683
Thus, we come to the conclusion that energy, measured by the amount of "work that a body is capable of doing," and mass, determined by momentum at a given speed, are interrelated concepts. If there is one, then there is another. Neither mass nor energy are invariant; each quantity, according to (16.42), depends on the observer's frame of reference."

It seems to me that the following conclusion can be drawn from this statement. If the energy depends on the reference frame, then the energy in the capsule (reference frame B) differs from what energy an observer in the laboratory identifies in it (reference frame A). If two reference frames are connected by a process occurring at the speed of light, which is equal for all reference frames, then the energy of the reference frame B can be obtained in the reference frame A...

 

[DaveE]

then the energy in the capsule (reference frame B) differs from what energy an observer in the laboratory identifies in it (reference frame A).

yes

the energy of the reference frame B can be obtained in the reference frame A...

The energy observed in reference frame B can be obtained by observations made in reference frame A if you know the relationships between the reference frames.

Also, the total energy observed in any reference frame is invariant (the same)*.

*In a non-expanding universe.

 

[Ivan Nikiforov]

yes


The energy observed in reference frame B can be obtained by observations made in reference frame A if you know the relationships between the reference frames.

Also, the total energy observed in any reference frame is invariant (the same)*.

*In a non-expanding universe.

That's the whole question. Is it true that being in the frame of reference A, we observe one amount of energy in the frame of reference B, and on the sliding contacts coming from the system B, we measure another amount of energy?

 

[DaveE]

That's the whole question. Is it true that being in the frame of reference A, we observe one amount of energy in the frame of reference B, and on the sliding contacts coming from the system B, we measure another amount of energy?

Maybe. It is only the total energy of a closed system that is invariant.

The energy observed in reference frame B can be obtained by observations made in reference frame A if you know the relationships between the reference frames.

Also, the total energy observed in any reference frame is invariant (the same)*.

*In a non-expanding universe.

Sorry, I don't know how to say this more simply.

 

[Ivan Nikiforov]

Maybe. It is only the total energy of a closed system that is invariant.


Sorry, I don't know how to say this more simply.

Devi, thank you. I know what you're talking about.

 

[jbriggs444]

Maybe. It is only the total energy of a closed system that is invariant.

The total energy of a closed system is conserved but not invariant.

That is to say that the energy is the same, no matter when you look.
But the energy can change depending on the frame of reference you adopt.

 

[berkeman]

Devi, thank you.

LOL, it's funny how the translation software translates the @DaveE username.

 

[Ivan Nikiforov]

LOL, it's funny how the translation software translates the @DaveE username.

I'm so sorry. Maybe I should have copied and pasted @DaveE. I'm sorry if my text is incorrect due to the translator.

 

[DaveE]

The total energy of a closed system is conserved but not invariant.

That is to say that the energy is the same, no matter when you look.
But the energy can change depending on the frame of reference you adopt.

Yes thanks. I'm being sloppy again.
Real Physicists know the difference better than "just" Engineers. I still expecting to get kicked out of PF any day now for have forgotten all of my physics classes, of which there weren't that many anyway .

 

[Ivan Nikiforov]

The total energy of a closed system is conserved but not invariant.

That is to say that the energy is the same, no matter when you look.
But the energy can change depending on the frame of reference you adopt.

Thank you. I would like to understand the relations for energy depending on the frame of reference. By the way, a separate question is whether the system in question is closed, due to the fact that time is heterogeneous?

 

[DaveE]

LOL, it's funny how the translation software translates the @DaveE username.

I'm so sorry. Maybe I should have copied and pasted @DaveE. I'm sorry if my text is incorrect due to the translator.

No, it's fine. I kind of like it better in a way. Dave, Mike, Chris, Mary, Scott, Lisa... they're all pretty boring names to native English speakers.

 

[Dale]

Oh wow, this is a long thread. I guess I will make it a bit longer.

1) energy is conserved in every frame.
2) different frames will disagree about the total amount of energy.
3) circuit theory is non-relativistic, so you have to use Maxwell’s equations instead.
4) this problem is not well enough specified to solve it using Maxwell’s equations, but without substantial simplification or numerical methods it would be difficult to solve.
5) nevertheless, Poynting’s theorem holds in all frames so regardless of the details we know that point 1) holds

 

[Ivan Nikiforov]

Oh wow, this is a long thread. I guess I will make it a bit longer.

1) energy is conserved in every frame.
2) different frames will disagree about the total amount of energy.
3) circuit theory is non-relativistic, so you have to use Maxwell’s equations instead.
4) this problem is not well enough specified to solve it using Maxwell’s equations, but without substantial simplification or numerical methods it would be difficult to solve.
5) nevertheless, Poynting’s theorem holds in all frames so regardless of the details we know that point 1) holds

Dale, thank you for your comment and for joining the discussion. Your opinion is very important. Please tell me if the following statement is true: Is it true that, being in the frame of reference A, we observe one amount of energy in the frame of reference B, and on the sliding contacts coming from the system B, we measure a different amount of energy? How exactly do we understand that the total sum of energies will be different in different reference frames? Is there perhaps an example?

 

[PeroK]

Dale, thank you for your comment and for joining the discussion. Your opinion is very important. Please tell me if the following statement is true: Is it true that, being in the frame of reference A, we observe one amount of energy in the frame of reference B, and on the sliding contacts coming from the system B, we measure a different amount of energy? How exactly do we understand that the total sum of energies will be different in different reference frames? Is there perhaps an example?

Imagine a car of mass $m$ accelerating from rest to speed $v$. In the original rest frame it gains kinetic energy of $\frac 1 2 mv^2$. In the final rest frame, it started with a speed of $v$ and decelerated to rest. In that frame it lost kinetic energy of $\frac 1 2 mv^2$.

And, in a frame moving at $v/2$, the car started at speed $v/2$ and ended at speed $v/2$, so it neither gained nor lost kinetic energy.

As you can see from this example, your difficulties understanding energy and reference frames are fundamental and are not directly related to special relativity.

 

[Ivan Nikiforov]

Imagine a car of mass $m$ accelerating from rest to speed $v$. In the original rest frame it gains kinetic energy of $\frac 1 2 mv^2$. In the final rest frame, it started with a speed of $v$ and decelerated to rest. In that frame it lost kinetic energy of $\frac 1 2 mv^2$.

And, in a frame moving at $v/2$, the car started at speed $v/2$ and ended at speed $v/2$, so it neither gained nor lost kinetic energy.

As you can see from this example, your difficulties understanding energy and reference frames are fundamental and are not directly related to special relativity.

Thanks for the comment. I'll think about it.

 

[pervect]

I have not worked this particular problem, but I do have a problem that I worked that illustrates some of the point (already made) that circuit theory is not relativistic.

The thread is called "boosting a current loop".

https://www.physicsforums.com/threa...rrent-density-and-charge-distribution.631446/

It illustrates how Kirchoff's current law does not work without modifications. And it serves as an inspiration for the suggestion I will give as to how to formulate your problem in a relativistic manner.

I have not worked your particular problem - it's an interesting one, but I'm much slower (and more error prone) to work out problems nowadays so it's unlikely I will even make the attempt. But I can suggest how I might approach it in a relativistic manner.

There are two quantities of interest here, current and voltage, of which you need the relativistic versions. As other posters have mentioned, circuit theory is not relativistic so you'll have to put that aside for a bit.

The relativistic version of current is the "four-current density", see https://en.wikipedia.org/wiki/Four-current. This describes how charge and current densities transform relativistically. You will also need to include the efffects of relativistic length contraction in your problem formulation.

The relativistic model for voltage would be the "Electromagnetic 4-potential", https://en.wikipedia.org/wiki/Electromagnetic_four-potential. This will transform between the frames via the Lorentz transform. Taking the appropriate gradient as described in the will give you the E and B field in a given frame. There are some other approaches you could use, but I would think that the 4-current and the 4-potential would be enough, conceptually. I could be wrong.

Because this isn't circuit theory, you will need to keep tract of the magnetic fields as well as the electric fields.

I think that what I'd suggest as the first shot would be to imagine a transmission line with a sliding "resistor". This solves the problem of dealing with the magnetic fields. Ohm's law is not relativistic, so avoid it. One might consider finding a relativistic version of Ohm's law, but I am not going to pursue that route.

The goal is to write the 4-current vector through the sliding "resistor", in the lab frame, which I would define to be of such a magnitude that it did not cause any reflected waves in the transmission line. I am assuming here some knowledge of transmission line theory and how they need to be terminated to avoid reflections and standing waves. Describing this in detail would be too much of a digression from my thought processes and the actual problem.

If we know the voltage in the lab frame, and the impedance of the transmission line, we know the current in the lab frame, from which we can compute the appropriate relativistic generalizations.

If all this is worked out, one should then be able to confirm that the problem setup satisfies Maxwell's equations.

Given this lab-frame solution, it should then be reasonably simple to convert to the moving frame, and to keep tract of all the power in each frame and confirm that it is conserved. I'd expect the power to be frame dependent, but I have not done any of the actual work, I've just thought a bit about how to set it up.

 

[Dale]

Is it true that, being in the frame of reference A, we observe one amount of energy in the frame of reference B, and on the sliding contacts coming from the system B, we measure a different amount of energy?

The situation you described is too difficult for me to analyze and answer this explicitly. However, it certainly would be possible for other more simplified scenarios.

How exactly do we understand that the total sum of energies will be different in different reference frames? Is there perhaps an example?

Yes. The best examples are very simple ones, like an infinite sheet of charge or current oriented perpendicular to the direction of travel. These geometries change the problem into a 1+1D problem.

 

[Ivan Nikiforov]

I have not worked this particular problem, but I do have a problem that I worked that illustrates some of the point (already made) that circuit theory is not relativistic.

The thread is called "boosting a current loop".

https://www.physicsforums.com/threa...rrent-density-and-charge-distribution.631446/

It illustrates how Kirchoff's current law does not work without modifications. And it serves as an inspiration for the suggestion I will give as to how to formulate your problem in a relativistic manner.

I have not worked your particular problem - it's an interesting one, but I'm much slower (and more error prone) to work out problems nowadays so it's unlikely I will even make the attempt. But I can suggest how I might approach it in a relativistic manner.

There are two quantities of interest here, current and voltage, of which you need the relativistic versions. As other posters have mentioned, circuit theory is not relativistic so you'll have to put that aside for a bit.

The relativistic version of current is the "four-current density", see https://en.wikipedia.org/wiki/Four-current. This describes how charge and current densities transform relativistically. You will also need to include the efffects of relativistic length contraction in your problem formulation.

The relativistic model for voltage would be the "Electromagnetic 4-potential", https://en.wikipedia.org/wiki/Electromagnetic_four-potential. This will transform between the frames via the Lorentz transform. Taking the appropriate gradient as described in the will give you the E and B field in a given frame. There are some other approaches you could use, but I would think that the 4-current and the 4-potential would be enough, conceptually. I could be wrong.

Because this isn't circuit theory, you will need to keep tract of the magnetic fields as well as the electric fields.

I think that what I'd suggest as the first shot would be to imagine a transmission line with a sliding "resistor". This solves the problem of dealing with the magnetic fields. Ohm's law is not relativistic, so avoid it. One might consider finding a relativistic version of Ohm's law, but I am not going to pursue that route.

The goal is to write the 4-current vector through the sliding "resistor", in the lab frame, which I would define to be of such a magnitude that it did not cause any reflected waves in the transmission line. I am assuming here some knowledge of transmission line theory and how they need to be terminated to avoid reflections and standing waves. Describing this in detail would be too much of a digression from my thought processes and the actual problem.

If we know the voltage in the lab frame, and the impedance of the transmission line, we know the current in the lab frame, from which we can compute the appropriate relativistic generalizations.

If all this is worked out, one should then be able to confirm that the problem setup satisfies Maxwell's equations.

Given this lab-frame solution, it should then be reasonably simple to convert to the moving frame, and to keep tract of all the power in each frame and confirm that it is conserved. I'd expect the power to be frame dependent, but I have not done any of the actual work, I've just thought a bit about how to set it up.

Thanks for the comment. Your thoughts are very interesting. I will carefully consider your arguments.

 

[pervect]

There's a discussion of the parallel plate transmission line I found online at https://ws.engr.illinois.edu/sitemanager/getfile.asp?id=153. But I'm too rusty to run the numbers, the source was helpful though.

What I am thinking, is that the voltage seen in the frame of the "moving resistor" is lower than the voltage in the lab frame. *ONE* of the effects of using the full relativistic solution is that the "moving resistor" along the transmission line is cutting through magnetic field lines generated by the currents in the transmission line which generates an induced voltage, similar to how a wire moving through a magnetic field generates a voltage in a general. Because the wire is moving very quickly, this effect is rather large.

But it's really more complicated than that, and a proper analysis (which I haven't done) is needed to be sure. It would, for example, be wrong to say that you add this induced voltage to the lab voltage. The relativistic transformation laws don't work like that.

An alternative to using the electromagnetic 4-potential I suggested earlier is to consider how the electromagnetic fields transform as per

https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#E_and_B_fields

(If you don't trust Wiki on this, Griffith's book on E&M should cover this also). But I think the 4-potential approach would be less error prone for me personally, no cross products to get the sign wrong on :). Also, it's much more like the circuit theory concept of voltage.

I'm too rusty to work this out properly, but those are my thoughts on how to formulate the problem in a way it can be solved.