B Time dilation and thermodynamics

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1. Dec 20, 2016

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.)

2. Dec 20, 2016

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.

3. Dec 20, 2016

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

4. Dec 20, 2016

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.

5. Dec 20, 2016

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.

6. Dec 20, 2016

Staff: Mentor

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.

7. Dec 20, 2016

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.

8. Dec 20, 2016

Staff: Mentor

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.

9. Dec 20, 2016

Ibix

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.

10. Dec 20, 2016

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''

11. Dec 20, 2016

Thebugger

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

12. Dec 20, 2016

Staff: Mentor

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.

13. Dec 20, 2016

jartsa

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.

14. Dec 20, 2016

jartsa

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.

15. Dec 20, 2016

Staff: Mentor

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.

16. Dec 20, 2016

pervect

Staff Emeritus
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.

Last edited: Dec 21, 2016
17. Dec 20, 2016

Staff: Mentor

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.

18. Dec 21, 2016

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.

19. Dec 21, 2016

Staff: Mentor

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.

20. Dec 21, 2016

jartsa

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.

21. Dec 21, 2016

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.

22. Dec 21, 2016

pervect

Staff Emeritus
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.

23. Dec 21, 2016

jartsa

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.

Last edited: Dec 21, 2016
24. Dec 22, 2016

pervect

Staff Emeritus
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.

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.

25. Dec 22, 2016

Staff: Mentor

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.