Time dilation and thermodynamics

[Thebugger]

Hi guys, I was watching an episode of stargate and I got this idea, I've been trying to explain to myself. It has to do with time dilation and the first law of thermodynamics. So here it is. Imagine a small time dilation field, a small sphere or something, with an accelerated time. A simple heater is placed inside the field and an extension cord powers it from outside the field. Let's say the time inside the field progresses 60 times faster. The heater typically draws 1kW for instance. But that means the heater will dissipate 1kWh inside the field, but only draw 100Wh outside the field. Doesn't that violate the first law of thermodynamics, where an overunity device can't exist? Or am I thinking it through wrongly?
P.S. The other way around, also doesn't make sense. If time is slowed down inside the field, the heater will dissipate 100Wh, while drawing 1kWh, where the excess 900kWh doesn't dissipate as anything else (heat,motion etc.)

 

[jartsa]

The idea is unphysical, so therefore many laws of physics can be violated in that universe. We only know two things that have an effect on time: gravitational potential and fast motion, one of those must be used if we want to avoid being unphysical.

So let's see how the first law of thermodynamics is violated:

The heater normally draws 1 kw and heats a kettle full of water to boiling point in one minute, using 60 kJ energy.

When we put the heater in the magical sphere where time runs fast, the heater draws 60 kw, and heats a kettle full of water to boiling point in one second, using 60 kJ energy.

An observer next to the heater says the power of the heater is 1 kw, because his clock advances one minute while the water on the kettle is being heated to boiling point.

 

[BvU]

Hi bugger,

Your imagination got the better of you. Time dilation means time goes slower according to the outside observer.
But I like the idea of an extension cord between something that moves with respect to something else at 0.999861 times the speed of light

 

[Ibix]

The time dilation field is essentially magic, so you can have any rules you like. However, you can do things like put the heater deep in the gravity well of a black hole and transfer energy to it by laser beam.

I don't think there's an unambiguous way to define the power consumption of the heater as measured by the laser emitter. But the emitter will be able to predict the reading of a power meter attached to the heater and it will be completely consistent with the laser emission once all the gravitational blueshift is accounted for.

You could also use a huge extension cord, but the laser is easier to analyse.

 

[Thebugger]

As far as I know strong gravitational fields can also slow down time, so the extension cord and dilation bubble thing can be in reverse (the dilation field being stronger closer to the center of gravity than the power source is). Come to think about it, this happens constantly even on earth, even though the effect is veeery small. Purely theoretical speaking how does physics compensate in this example. Surely both laws are very strict, but they seem to contradict each other here.

 

[Dale]

Imagine a small time dilation field, a small sphere or something, with an accelerated time. A simple heater is placed inside the field and an extension cord powers it from outside the field.

There isn't a "time dilation field", but there is a gravitational field, and gravitation includes gravitational time dilation. To analyze your scenario in the simplest manner possible I would modify it slightly.

I would say that you have two blackbodies in radiative contact with each other, separated by some vertical distance in a uniform gravitational field.

 

[Thebugger]

Okay something a little more realistic then. Let's say I live in a house. The second floor is further from the gravity well, than the first floor therefore time passes the slightest of bits faster than the time in the first floor. The same example still applies, even though the energy difference will probably be pWh or nWh. It's the same deal, just the time difference is much much less.

 

[Dale]

The reason that I would use blackbodies in radiative thermal contact is that I know how to treat radiation relativistically. I don't know how to treat a power cord relativistically.

If you find a covariant generalization of circuit theory then I will be glad to help analyze it, but I myself don't know of such a generalization.

 

[Ibix]

Purely theoretical speaking how does physics compensate in this example.

You cannot talk about "at the same time, over there" in relativity without a lot of clarification. If you clarify and account carefully for the details, you'll find that the input and output powers are completely consistent. If you don't then your question is meaningless. So your worries about the power supply "over here" generating a different amount of power from that being used "at the same time, over there" aren't valid.

I don't think I can explain why without resorting to concepts way beyond a B level thread.

 

[Thebugger]

That's exactly my concern, I've never seen examples of power cords in a relativistic environment. They draw constant power from the source, yet the load behaves differently due to time dilation. Basically the power cord severs the imaginary line between different frame references or something. And come to think about it, this thing happens all the time, on a nano scale, as I mentioned in my ,,house example''

 

[Thebugger]

Yes the input and output powers are equal, but the load will consume more watts per hour, than the source gives off.

 

[Dale]

Sorry, I can help with a blackbody example, but I don't know how to do an electrical example. It must work out, but I cannot provide details other than to point to Maxwell's equations in covariant form.

 

[jartsa]

That's exactly my concern, I've never seen examples of power cords in a relativistic environment. They draw constant power from the source, yet the load behaves differently due to time dilation. Basically the power cord severs the imaginary line between different frame references or something. And come to think about it, this thing happens all the time, on a nano scale, as I mentioned in my ,,house example''

Is Kirchoff's law valid in DC-circuits in gravity fields?

Let's say we have an electric circuit in a gravity field, and we want to know the amperes and the voltages, at high altitude and low altitude. Let's say it's a DC circuit.

A low observer says: "Counting the electrons that pass this point of the circuit right next to the me can be used as a clock. Let's say that when million electrons has passed that point, then one second has passed."High observer says: That clock based on electron counting is time dilated, it takes ten seconds for million electrons to pass that low point on the circuit. Now if Kirchoff's law is valid in this circuit, then the high observer must count 0.1 million electrons passing any point on the circuit in one second. And the amperage on the circuit is 100000 negative elementary charges per second, according to high observer, ten times larger according to the low observer.

 

[jartsa]

Okay something a little more realistic then. Let's say I live in a house. The second floor is further from the gravity well, than the first floor therefore time passes the slightest of bits faster than the time in the first floor. The same example still applies, even though the energy difference will probably be pWh or nWh. It's the same deal, just the time difference is much much less.

Sadly the first law of thermodynamics is not violated even in the situation you described in the first post, because putting a heater in a sphere of fast flowing time is equivalent to turning up the power adjustment knob of the heater. See post #2.

But luckily putting some lukewarm stuff into a sphere of fast flowing time causes a violation of the second law of thermodynamics, by making the lukewarm stuff emit same kind of radiation as hot stuff.

But it happens to be so that in our universe moving some stuff to an area of fast flowing time always requires energy, which saves the second law of thermodynamics.

 

[mfb]

If you power your kettle upstairs (where time runs faster) with a laser, the light will arrive at a lower frequency => lower power. Energy is conserved.
If you power your kettle upstairs with an extension cord, the kettle will see a lower frequency of electrons passing => lower current => lower power. Energy is conserved.
If you do anything else, you'll still get the same result: energy is conserved.

 

[pervect]

Sorry, I can help with a blackbody example, but I don't know how to do an electrical example. It must work out, but I cannot provide details other than to point to Maxwell's equations in covariant form.

I can suggest a covariant approach, but I don't have a specific answer to the question. We can model the wire as some four-current density $J^a$. We can borrow the notation from the_wiki_article. For the benefit of the original poster, the 4-current density J simply consits of the charge density $\rho$ and a 3-current density j combined into a 4-vector formalism.

Wiki discusses the covariant continuity equations, which is what we need. Basically we used to say that the divergence of the 3-current j was zero, now instead we use the four-current continuity equation $\nabla_a J^a = 0$. (Wiiki writes this as $J^a{}_{;a}$, by the way). To be specific, I'd suggest initially using some variant of RIndler coordinates for an accelerating frame. This would correspond to performing the analysis of our 4-current model of a "wire" on Einstein's elevator. Then we could replace the Rindler metric with some variant of the Schwarzschild metric to explore a "true" gravitational field. I'd also physically interpret the results in an orthonormal basis.

For a covariant description of power, I imagine we'd use the Poynting vector - (I'm less certain about this part).

Unfortunately, this is a lot of work, and an A-level answer to what appears to be an I-level question. I'm not feeling ambitious enough to tackle it, and I'm not sure how useful the results would be to the OP if I did do the work anyway.

 

[Dale]

For a covariant description of power, I imagine we'd use the Poynting vector - (I'm less certain about this part).

I like your approach, and it seems right to me. I looked into it a bit and I think that the covariant object containing power is the EM stress energy tensor. It contains the energy density as the time time component, the Poynting vector as the time space component and the Maxwell stress tensor as the space space component.

 

[Thebugger]

Okay, this is getting a little complicated to me, I'm just an engineer, physics is more of a hobby to me, but basically I see this happening on a small scale all the time. I'm not talking about laser power, because as you already mentioned the doppler shift decreases the power and an equilibrium is maintained. I'm talking about electricity, specifically DC. Let's say the load consumes 1000 joules for 1s which is 1kW, but the source gives out these joules for 60s which is around 17W, where does the difference come from, and from what point of reference is it appropriate to view the situation from? Will the source supply more current than the load typically consumes, so that the equation evens out, or...? I just can't simplify it enough for a thought experiment.

 

[Dale]

Okay, this is getting a little complicated to me, I'm just an engineer, physics is more of a hobby to me,

I don't think that there is a simple answer to your question. I think that any answer will require a full covariant treatment of Maxwell's equations. You have been given an outline of an answer by @pervect and the key mathematical object has been identified by myself. I think you will have to take it from there and pursue the rest, if not on your own then at least as an active driver.

 

[jartsa]

Okay, this is getting a little complicated to me, I'm just an engineer, physics is more of a hobby to me, but basically I see this happening on a small scale all the time. I'm not talking about laser power, because as you already mentioned the doppler shift decreases the power and an equilibrium is maintained. I'm talking about electricity, specifically DC. Let's say the load consumes 1000 joules for 1s which is 1kW, but the source gives out these joules for 60s which is around 17W, where does the difference come from, and from what point of reference is it appropriate to view the situation from? Will the source supply more current than the load typically consumes, so that the equation evens out, or...? I just can't simplify it enough for a thought experiment.

A 12 V 100 Ah battery sits at the bottom of a gravity well. What kind of electricity does a high altitude electrical device get from the battery?

Answer is: At most 100 Ah of charge, voltage is 1.2 V if the redshift factor is 0.1.

We know this from the laws of conservation of charge and conservation of energy.

 

[Ibix]

A more manageable example: power the heater by batteries, which we charge at a station far from the black hole and shuttle down in free-fall orbits.

Now we have a shared time standard - the regular spacing of the batteries. And time dilation solved the problem. The charging station measures one hour between launches but the heater measures one minute between arrivals. Hence its (locally measured) power output is 60 times the charging station's output. Clearly the total energy output cannot exceed the input since it's limited by the number of launched batteries.

 

[pervect]

I did a 4-vector analysis of a relativistically boosted current loop in https://www.physicsforums.com/threads/boosting-a-current-loop.631446/. Assuming it's correct, it might be of some interest as to what sort of things can happen to circuit theory (in particular Kirchoff's current law) in a relativistic context.

It's not an answer to the original question though, and I don't think it was ever checked by anyone else.

 

[jartsa]

I did a 4-vector analysis of a relativistically boosted current loop in https://www.physicsforums.com/threads/boosting-a-current-loop.631446/. Assuming it's correct, it might be of some interest as to what sort of things can happen to circuit theory (in particular Kirchoff's current law) in a relativistic context.

It's not an answer to the original question though, and I don't think it was ever checked by anyone else.

Do we count the positive wire moving to the +x direction as a current?

I'm quite sure that in the boosted frame in one second the same number of electrons pass any mark attached to any part of the circuit. (Because that's intuitive)

Oh yes I have a more scientific argument too: the frequency of electron passing events time dilates, every part of the circuit time dilates the same amount.

 

[pervect]

Do we count the positive wire moving to the +x direction as a current?

Yes, any moving charge contributes to the current. So the positive charges moving in the +x direction are a positive current, and the negative charges moving in the x direction are a negative current. The sum of the currents of the positive and negative moving charges is the total current.

I'm quite sure that in the boosted frame in one second the same number of electrons pass any mark attached to any part of the circuit. (Because that's intuitive)

How is that intuitive? If I'm understanding what you say correctly, it'd be false even in Newtonian physics. In a Newtonian analysis, we would have some velocity v for the loop as a whole, and some velocity w for the electrons realtive to the loop, so in the upper and lower loops we'd have electron velocites of v+w and v-w (I'd have to look to see which was which). In this Newtonian analysis, the charge density $\rho$ would stay constant, so the number of electrons passing a point would be the density * area * velocity, (electrons / m^3) * (m^2) * (m/s) = electrons/second. So the electrons/second isn't constant for the upper and lower loop even in the Newtonian analysis, though the current is, because of the corresponding changes in positive charges/second counterbalancing due to linearity. But in the relativistic analysis, the density $\rho$ does change, and the velocities no longer add, so this argument fails.

I'm guessing that you didn't count the positive charges as contributing to the current - but they will.

The most non-intuitive part of the picture is that the density $\rho$ in the lab frame changes. This is discussed a bit in Purcell's treatment of the relativistic origins of the magnetic field, for instance - so while it's surprising and not particularly intuitive, it's well known.

 

[Dale]

I did a 4-vector analysis of a relativistically boosted current loop in https://www.physicsforums.com/threads/boosting-a-current-loop.631446/. Assuming it's correct, it might be of some interest as to what sort of things can happen to circuit theory (in particular Kirchoff's current law) in a relativistic context.

One of the basic assumptions of circuit theory is that none of the components have a net charge. So this analysis would seem to indicate that circuit theory is inherently non relativistic.

 

[pervect]

One of the basic assumptions of circuit theory is that none of the components have a net charge. So this analysis would seem to indicate that circuit theory is inherently non relativistic.

I basically agree. In particular, I think Kirchoff's current law has some issues - basically, it does not factor in the relativity of simultaneity. In case our OP is still with us, I'm going to go over some of the fundamentals as I see them. The key point as I see it is understanding how charge densities and currents densities transform between frames. Charge densities are just charge per unit volume, i.e. charge/m^3, current densities are charges/second across some area A, so they have units of charge / (area * second) or (charge / m^2 s).

Consider two frames, one of which is moving in the x direction with some velocity $\beta = v/c$ relative to the other. We will call the charge density $\rho$ or $\rho'$ depending on whether we are in the primed or unprimed frames, and we will write the current density components as $j_x, j_y, j_z$. In the non-relativistic Newtonian analysis would have for the relation between primed and unprimed frames:

$$\rho' = \rho \quad j_x' =j_x + \beta c \rho \quad j_y' = j_y \quad j_z' = j_z$$

You may see other versions with a minus sign in front of $\beta$, this is just a matter of sign convention, it indicates that we need to replace $\beta$ with $-\beta$ depending on how exactly we've defined the velocity between the primed and unprimed frames.

Basically what this says is that the charge density $\rho$ stays unchanged, and moving charge creates a current density of $\rho$ * velocity.

The relativistic analysis tells us, however, that

$$\rho' = \gamma (\rho + \beta j_x/c) \quad j_x' = \gamma(j_x + \beta c \rho) \quad j_y' = j_y \quad j_z' = j_z$$

where $\gamma$ is the relativistic factor $1/\sqrt{1-\beta^2}$.

Note that $\rho c$ has the same dimensions as j, as it is (charge / $m^3$) * (m/s) = charge / ($m^2 s)$

The short-hand version of explaining this is to say that charge and current transforms "as a 4-vector". Here I've written out the 4-vector transformations explicitly.

For the benefit of Jartsa, I would draw attention to the equation $\rho' = \gamma (\rho + \beta j_x/c)$. We can divide this into two terms: $\gamma \rho$ and $\gamma \beta j_x / c$ which add together to give the total charge density in the primed frame.

The origin of the first term can be explained by "length contraction". The origin of the second term is more mysterious, but it's absolutely necessary to include it to get a correct analysis. I would describe the ultimate origin of this term as being due to the "relativity of simultaneity".

So - to summarize the issue as I see it. The challenging part of the relativistic analysis is how the relativity of simultaneity affects the expression of the physics. A second challenge in communicating this issue - getting people to pay attention to the relativity of simultaneity. My working hypothesis is that the term "relativity of simultaneity" is very abstract, an tends not to be understood. The specific example of how charge density transforms between a moving and non-moving frame is less abstract, but as far as communication goes, it seems to have the opposite problem. It's very detail oriented, so the big picture gets lost. However, without a shared understanding of what the abstract term "the relativity of simultaneity" means, going over the details is the only way I can think of to try to explain the fundamental underlying issue.

 

[jartsa]

Yes, any moving charge contributes to the current. So the positive charges moving in the +x direction are a positive current, and the negative charges moving in the x direction are a negative current. The sum of the currents of the positive and negative moving charges is the total current.

I was talking about electrons passing a point on the circuit. The point and the circuit co-move.

If we are interested about a circuit, we want to know the currents through the components. So the current that does not go through any components we can ignore. For example if our circuit is charged and moves to the right, the charge and the motion form a current, which can be ignored.

 

[Dale]

So the current that does not go through any components we can ignore.

You can't ignore it in Maxwells equations. I am not sure what theory you would use to justify ignoring it.

 

[jartsa]

You can't ignore it in Maxwells equations. I am not sure what theory you would use to justify ignoring it.

Okay then, let's not ignore any current, but let's divide the confusing current into two components:

1: the current that goes through the wire, carried by slowly flowing electron gas.
2: the current that goes through empty space, carried by fast moving charged wire.

Then we consider the two currents separately. No current gets ignored, and hopefully we don't get confused.

 

[vanhees71]

Well, it boils down to the simple fact that charge and current densities build a Minkowski four-vector
$$(j^{\mu})=\begin{pmatrix} c \rho \\ \vec{j} \end{pmatrix}$$
and of course it transforms under Lorentz transformations (even under the full Lorentz group O(1,3)) as a four-vector.

Of course Kirchhoff circuit theory is not Lorentz covariant since it's an application of the quasi-stationary approximation, i.e., Maxwell's displacement current is ignored, and this violates Lorentz covariance. The right thing to do is to calculate everything in the restframe of the circuit and then Lorentz boost the charge-current-density four-vector (note that in the restframe there's no net charge density, while there is one in the boosted frame!).

 

[Dale]

Okay then, let's not ignore any current, but let's divide the confusing current into two components:

1: the current that goes through the wire, carried by slowly flowing electron gas.
2: the current that goes through empty space, carried by fast moving charged wire.

Then we consider the two currents separately. No current gets ignored, and hopefully we don't get confused.

Sure, you could do that. In the end, that would amount to transforming to the rest frame and doing standard circuit analysis in the rest frame.

 

[Dale]

Of course Kirchhoff circuit theory is not Lorentz covariant since it's an application of the quasi-stationary approximation, i.e., Maxwell's displacement current is ignored, and this violates Lorentz covariance.

Yes, that is clear and concise. Well said.