# Cooling of Identical Bodies in SR: A Peculiar Situation

• A
• Pelion
In summary: In the scenario where the plates are at rest with respect to each other and the observer in S, then the temperature gauges attached to them will always display identical readings as a function of time.
Pelion
TL;DR Summary
Two identical bodies are cooling at the same rate in frame S while an observer in frame S' has some explaining to do.......
Hi all,
Consider the situation depicted in the illustration. Two identical 'square plates' are situated at rest, in frame S, as shown: Plate A has its thickness 'a' parallel to the x-axis and its sides 'L' parallel to the y and z axes, while plate B has its thickness parallel to the y-axis and its sides parallel to the x and z axes. They are heated to the same temperature and then are accelerated until they attain a velocity 'v' along the x-axis with respect to an observer in frame S'. Now, in S', plate A will only have its thickness 'a' contracted while plate B will only have the side 'L' parallel to the x-axis contracted. This means that, for the observer in S', plates A and B have different total surface areas and, if we presume that a<<L and that v is in the relativistic range, this difference in total surface area will be dramatic. For the observer at rest with plates A and B, in frame S, the two plates will display the same temperature for all times as they cool (assuming, in this thought experiment, that we are in the vacuum of outer space and that outer space is homogeneous and isotropic). The identically dropping temperature readings, at all times, are an objective fact...how does the observer in S' explain this fact despite his perception of drastically different surface areas associated with plates A and B ??

Cheers...

Pelion said:
Consider the situation depicted in the illustration.
WHAT illustration?
how does the observer in S' explain this fact despite his perception of drastically different surface areas associated with plates A and B ??
Well, presumably, he's smart enough to know that he has to use length contraction to know what size the plates are in their own frame, which is all that matters to their temperature.

Relativistic thermodynamics is fun, for complicated values of "fun".

Remember that if, in their rest frames, the plates are at uniform temperature then in the frame where they are moving the relativity of simultaneity means that the temperature isn't uniform (edit: unless it's also constant, which it isn't in your scenario). So you don't even expect uniform temperatures across a single plate, let alone identical ones between the plates - contrary to your "objective fact" claim.

Furthermore, since the plates are moving we would expect the emission to be non-uniform (even for the perpendicular-to-velocity plate) due to Doppler shifts and relativistic beaming. And time dilation would mean that the expected rate of change of temperature measured in the moving frame would be different from that measured in the rest frame (at rest, you could use a thermometer as a clock).

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PeterDonis and Dale
Classically, there are several different definitions of temperature, all giving the same temperature. In SR they do not. This makes any discussion of temperature in SR problematic. (It's also a non-problem experimentally, as two objects at large relative velocity won't be in thermal equilibrium)

Ibix said:
Relativistic thermodynamics is fun, for complicated values of "fun".

Remember that if, in their rest frames, the plates are at uniform temperature then in the frame where they are moving the relativity of simultaneity means that the temperature isn't uniform (edit: unless it's also constant, which it isn't in your scenario). So you don't even expect uniform temperatures across a single plate, let alone identical ones between the plates - contrary to your "objective fact" claim.

Furthermore, since the plates are moving we would expect the emission to be non-uniform (even for the perpendicular-to-velocity plate) due to Doppler shifts and relativistic beaming. And time dilation would mean that the expected rate of change of temperature measured in the moving frame would be different from that measured in the rest frame (at rest, you could use a thermometer as a clock).
I tend to agree with what you have stated except for your claim that the temperature readings for the observer at rest with the plates in frame S isn’t an objective fact—it has to be: in S both plates are at rest with respect to each other and the observer in S and if they read the same temperature at some initial time and they are identical bodies in S, then the temperature gauges attached to them will always display identical readings as a function of time in this scenario; the gauge readings are the same for all cranes, but moving frames have to explain why they are identical since in the moving frames the two plates have different surface areas. Perhaps the effects you mentioned account for this, perhaps they don’t.

Pelion said:
I tend to agree with what you have stated except for your claim that the temperature readings for the observer at rest with the plates in frame S isn’t an objective fact
I think it depends on what you mean by it being an objective fact that the temperature readings are identical. If, for example, your two plates in their rest frame are at the same temperature and you measure this with a mercury thermometer in contact with the centre of each plate, and I observe this from a fast moving rocket then to me the temperature readings will not be identical due to the relativity of simultaneity. But if you co-locate a clock with each thermometer and make a claim like "the thermometers both read 300K when their local clocks show 12.00" then this is a direct observable and will be agreed by everyone, yes. We just won't agree that the clocks read 12.00 simultaneously.

Generally, the details of how you account for another frame's sensor behaviour depends on how the sensor works. If you are using the emitted EM to determine the temperature then you need to consider what the radiation looks like in a moving frame (Doppler, beaming, non-uniform sources) and think about whether a moving sensor measures the energy of incoming EM alone, or if it depends also on its own velocity in the frame. On the other hand, if you are just going with mercury thermometers in direct contact with the plates then it's probably relatively straightforward. I don't think you've specified your temperature gauge types or their locations.

Pelion and PeterDonis
Pelion said:
the gauge readings are the same for all cranes, but moving frames have to explain why they are identical
I assume you mean “frames” not “cranes”, but I think you’re not grasping the implications of the relativity of simultaneity in this scenario. You have at least two devices measuring the temperature of something at two different locations in space. You can make a statement about how they change identically as time is passing, but this is implicitly a frame dependent statement since it involves two spacially separated things happening at the same time—this phrase “at the same time” always requires a single frame of reference in which it is to apply. It then cannot apply in any frames moving relative to that one.

How much of relativity have you studied? I ask because you marked this as an “A” level thread, and relativity of simultaneity is an introductory concept.

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In this case number of atoms is important, not number square meters.

Million atoms radiate more than half million atoms, right?(We could say that there is a reflective layer under the outer layer of radiating atoms, now every photon that leaves an atom goes into the outwards direction, even if atoms are squeezed and radiate non-isotropically)

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Pencilvester said:
I assume you mean “frames” not “cranes”, but I think you’re not grasping the implications of the relativity of simultaneity in this scenario. You have at least two devices measuring the temperature of something at two different locations in space. You can make a statement about how they change identically as time is passing, but this is implicitly a frame dependent statement since it involves two spacially separated things happening at the same time—this phrase “at the same time” always requires a single frame of reference in which it is to apply. It then cannot apply in any frames moving relative to that one.

How much of relativity have you studied? I ask because you marked this as an “A” level thread, and relativity of simultaneity is an introductory concept.
OK, I thought it was implied that the gauges can be independent of any SR effects: the two gauges are assumed to be situated on the y-axis and have the same x location, and thus they are not subject to the SR effects of the relative motion between frames S and S’, which is along the x-axis only.

Pelion said:
OK, I thought it was implied that the gauges can be independent of any SR effects: the two gauges are assumed to be situated on the y-axis and have the same x location, and thus they are not subject to the SR effects of the relative motion between frames S and S’, which is along the x-axis only.
Relativistic thermodynamics is not introductory, the purpose of this thought experiment is to see if people can spot fallacies easily or whether the scenario is worthy of more detailed mathematical analysis.

Pelion said:
the two gauges are assumed to be situated on the y-axis and have the same x location
In that case, yes, the readings between the two devices will be the same in all frames of reference with zero relative velocity in the ##y## and ##z## directions. But even then, so what? All that tells you is that the edge of plate B that touches plate A has the same temperature as A, but that says nothing of the rest of plate B.

Ibix
Pelion said:
whether the scenario is worthy of more detailed mathematical analysis.
I don't think you've specified it in enough detail to judge. What kind of temperature sensor did you envision? If you said, I missed it. If it's just a mercury thermometer which is essentially in point contact with a plate then a point doesn't have a length to contract and the only issues are around the relativity of simultaneity. On the other hand, if you are using a calorimeter of some kind, then this isn't making a point measure and you do need to worry about all the usual relativistic issues.

Also, although you will be able to predict the sensor output, you may or may not agree that the moving sensor actually measures temperature, in much the same way that a moving simultaneity detector doesn't detect simultaneity.

Pencilvester
I'd suggest looking at Derakhshani, "Black Body radiation in moving frames", https://arxiv.org/abs/1908.08599. It covers the approaches I've seen, and more that I haven't, and discusses the history of the topic.

For the two approaches I've seen:

MTW takes the route of defining temperature in the rest frame of a relativistic fluid, so it's basically a non-relativistic definition of temperature.

Nakumura, https://arxiv.org/abs/physics/0505004 building on work of Van Kampen, defines inverse temperature as a 4-vector ##\beta## and, in cases where we can ignore volume change (which complicate things and he discussed later) writes:

$$dS = \beta_u dP^u$$

to replace

$$\Delta S = \frac{\Delta Q }{T}$$

The difference is that ##\Delta Q## is a scalar, representing a change in energy. Energy is a frame dependent quantity, and not covariant, but the exchange of energy-momentum, described by ##dP^u## is a 4-vector, and is hence covariant.

In this formulation, 1/T becomes another 4-vector, and the dot product of the two 4-vectors yields the scalar change in entropy.

Understanding Nakumura's approach requires understanding 4-vectors, and why they are covariant objects, aka geometric objects. MTW's approach doesn't really address the relativistic aspects at all, it simply insists that temperature only needs to be defined in the rest frame of some fluid.

Other approaches than these two are possible and mentioned by Derakhshani, I'm simply not familiar with them.

Derakhshani paper seems fairly new, I can't say what sort of impact it has or if it's been cited much, but it covers more than what I know about the topic and seems like a reasonable place to start.

Pelion
Pelion said:
Relativistic thermodynamics is not introductory, the purpose of this thought experiment is to see if people can spot fallacies easily or whether the scenario is worthy of more detailed mathematical analysis.
Ibix said:
I don't think you've specified it in enough detail to judge. What kind of temperature sensor did you envision? If you said, I missed it. If it's just a mercury thermometer which is essentially in point contact with a plate then a point doesn't have a length to contract and the only issues are around the relativity of simultaneity. On the other hand, if you are using a calorimeter of some kind, then this isn't making a point measure and you do need to worry about all the usual relativistic issues.

Also, although you will be able to predict the sensor output, you may or may not agree that the moving sensor actually measures temperature, in much the same way that a moving simultaneity detector doesn't detect simultaneity.
Hi Ibix... and thanks for your insightful comments so far; I will have to think more about them. OK, let's assume 'point contact' (eg mercury) temperature gauges--good that that you brought up the interesting fact that in such a scenario the nature of the measuring device is important.
I do insist however that the observer S, at rest with plates A and B, will certainly find that the two identical plates are cooling at identical rates (assuming homogeneity & isotropy of space)...so whatever complicated relativistic effects are generated within the perception of the situation by the observer in S', the question is that S' will have to explain the identical readings for each time instant (which is an observable that is independent of the length contraction + time dilation transformations of the various variables involved). Do these Lorentzian effects (appearing in frame S') on the geometry, the Doppler effect, the blackbody radiation, etc., account for the observation of identical readings?

pervect said:
I'd suggest looking at Derakhshani, "Black Body radiation in moving frames", https://arxiv.org/abs/1908.08599. It covers the approaches I've seen, and more that I haven't, and discusses the history of the topic.

For the two approaches I've seen:

MTW takes the route of defining temperature in the rest frame of a relativistic fluid, so it's basically a non-relativistic definition of temperature.

Nakumura, https://arxiv.org/abs/physics/0505004 building on work of Van Kampen, defines inverse temperature as a 4-vector ##\beta## and, in cases where we can ignore volume change (which complicate things and he discussed later) writes:

$$dS = \beta_u dP^u$$

to replace

$$\Delta S = \frac{\Delta Q }{T}$$

The difference is that ##\Delta Q## is a scalar, representing a change in energy. Energy is a frame dependent quantity, and not covariant, but the exchange of energy-momentum, described by ##dP^u## is a 4-vector, and is hence covariant.

In this formulation, 1/T becomes another 4-vector, and the dot product of the two 4-vectors yields the scalar change in entropy.

Understanding Nakumura's approach requires understanding 4-vectors, and why they are covariant objects, aka geometric objects. MTW's approach doesn't really address the relativistic aspects at all, it simply insists that temperature only needs to be defined in the rest frame of some fluid.

Other approaches than these two are possible and mentioned by Derakhshani, I'm simply not familiar with them.

Derakhshani paper seems fairly new, I can't say what sort of impact it has or if it's been cited much, but it covers more than what I know about the topic and seems like a reasonable place to start.
Thanks for these references, appreciate it.

It's a big mess in the history of thermodynamics/statistical mechanics in relativity. It's even worse than the mess with the socalled "relativistic mass". As the latter has been abandoned in more recent times, so has the mess cleaned also in thermodynamics: One defines all thermodynamic quantities simply in the (local) rest frame of the substance in question.

To make statistical mechanics manifestly covariant one must just realize that the equilibrium state introduces a physically distinguished frame of reference, which is the rest frame of the heat bath. The manifestly covariant statistical operator for black-body radiation thus is
$$\hat{\rho}=\frac{1}{Z} \exp(-u_{\mu} \hat{P}^{\mu}/T),$$
where ##u^{\mu}## is the four-velocity of the heat bath and ##\hat{P}^{\mu}## the operator for the total four-momentum of the em. field, making necessarily the temperature ##T## a Lorentz scalar (I've used natural units with ##k_{\text{B}}=1##).

Pelion
OK, in light of the good comments so far, here is how I will try to make clear the 'peculiarity' of the scenario:

(a) {I claim} The observer in the rest frame, S, of the two identical plates, A & B, will find identical time-dependent (due to cooling) temperature readings for all times.

(b) {For dramatic effect} We presume that the thickness, a, is much smaller than the length, L. This means that we are essentially dealing with 'square sheets' of side L.

(c) Because of the conditions specified in (b), as v-->c, we have that: The total effective surface area of plate A (whose thickness,a, is parallel to v) will approach (LxL)+(LxL)--essentially what it is at rest; the total effective surface area of plate B (whose side, L, is parallel to v) will approach an infinitesimal value (!)

Thus, here is the 'paradox':
For the observer in S there is nothing unusual going on--two identical 'sheets' are cooling down at the same rate...sure, no problem; for the observer in S' we are stuck with the situation wherein the essentially infinitesimal surface area (when v-->c) of plate B will be exhibiting some extremely dramatic physical processes such that it manages to radiate energy at the same rate as plate A, whose effective surface area has barely changed from what it is in the rest frame.
The two plates are identical in a 'very real' sense (identical material and geometry)...how can they behave so differently just because of relative motion? Could it be that SR is simply not applicable to such a scenario?

Pelion said:
OK, let's assume 'point contact' (eg mercury) temperature gauges ... Do these Lorentzian effects (appearing in frame S') on the geometry, the Doppler effect, the blackbody radiation, etc., account for the observation of identical readings?
The fact that the two thermometers have no separation in the direction of ##v## (meaning there are no Lorentzian effects that S’ needs to consider in this context) accounts for S’ observing identical readings.

If you have a thermometer on the edge of plate B that’s opposite plate A, then S’ will not see identical readings. That thermometer will show that that edge of B is warmer than A.

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Pelion
Pelion said:
for the observer in S' we are stuck with the situation wherein the essentially infinitesimal surface area (when v-->c) of plate B will be exhibiting some extremely dramatic physical processes such that it manages to radiate energy at the same rate as plate A, whose effective surface area has barely changed from what it is in the rest frame.
Maybe the extreme density of the plate has something to do with it. Oh, the other plate is dense too, never mind then.Hmm well, let's say an astronaut is clipping his toe nails in a small cubical spaceship, each wall is equally likely to get hit by a piece of nail. How do some pieces of nail manage to avoid hitting the front wall and the rear wall, which are very close to each other in some suitably chosen frames?

Well, on the average the transverse velocity of pieces of nails must be larger than the longitudinal velocity. The old-fashioned longitudinal relativistic mass is: ##\gamma^3 m_0## while the old-fashioned transverse relativistic mass is: ##\gamma m_0##. That kind of suggests that there exists a large velocity difference between longitudinal and transverse velocities.

The average transverse velocity is gamma times larger than the longitudinal. (There is a difference between transverse and longitudinal forces too.)

Now I should tell what this has to do with radiative cooling. Well, a photon with transverse velocity can avoid hitting some atoms, like pieces of nails can avoid hitting some walls. So, although the plate is extra dense, radiation can leave from inside the plate, the same way as normally.A dense gas cloud radiates more from its surface, less from its inner parts, compared to a less dense gas cloud. Note that if we concentrate on the surfaces, then more density means more radiation.Oh, there is a difference between photons and pieces of nail, I forgot that. Anyway, quite many randomly launched photons manage to reach the contracted side wall, just like the randomly launched pieces of nail. Maybe readers can explain how exactly that happens.

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Pelion
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Pencilvester said:
The fact that the two thermometers have no separation in the direction of ##v## (meaning there are no Lorentzian effects that S’ needs to consider in this context) accounts for S’ observing identical readings.

If you have a thermometer on the edge of plate B that’s opposite plate A, then S’ will not see identical readings. That thermometer will show that that edge of B is warmer than A.
Skeptical because even though this may be the case--where one places the gauges results in different 'visual' perceptions of the readings, these effects do not seem to relate to the physics of what we are talking about (the actual cooling of the two plates in a relativistic setting). The effects you mention seem to be incidental to the scenario, much like the Terrell rotation of a fast-moving body--the contraction is 'real' according to SR, but the rotation is just optics for high velocities. I assume that temperature can be measured and read in a non-problematic manner in this scenario but there is a chance that is not the case.

Pelion said:
where one places the gauges results in different 'visual' perceptions of the readings
No, this is incorrect. Well, I mean sure, that’s true, but that is not the phenomenon I was talking about. If S’ is compensating for the travel time of light in determining what each thermometer reads at a single given time in his frame (as any good physicist would do), then even after that, he will still find that the thermometer readings don’t match. Again, this is the relativity of simultaneity.

I feel should ask again, how much relativity have you studied? I don’t doubt you’ve studied plenty of physics, it’s just that you don’t seem to be grasping this one introductory concept. I’m really not trying to insult you in any way, I just want to make sure you’re receiving answers that are suited to your current understanding.

Doctoral level physics...it is just that I do not shy away from discussing something I find interesting even though I am not an expert in the particular topic (relativistic thermodynamics). If I am mistaken in my comprehension and an expert points that out then so be it, I have gained in understanding! So, don't worry that I am not equipped to process your input...by the way, I still think your comments are 'incidental' to the setup and do not pertain to the actual 'problem' being discussed; the others that have posted replies do not seem concerned about the issues you are raising--it is assumed that measuring and reading temperature is non-problematic between inertial frames. By the way: temperature is a Lorentz scalar, so it is invariant, so the effects you illustrate must in fact be 'superficial' to what is being discussed. This does not mean that I don not appreciate your input, of course, and you may in fact be right, by I still cannot see why that may be the case.

Pelion said:
I still think your comments are 'incidental' to the setup and do not pertain to the actual 'problem' being discussed; the others that have posted replies do not seem concerned about the issues you are raising
Others have not been responding to this specific statement, but I have been:
Pelion said:
I tend to agree with what you have stated except for your claim that the temperature readings for the observer at rest with the plates in frame S isn’t an objective fact—it has to be: in S both plates are at rest with respect to each other and the observer in S and if they read the same temperature at some initial time and they are identical bodies in S, then the temperature gauges attached to them will always display identical readings as a function of time in this scenario; the gauge readings are the same for all cranes, but moving frames have to explain why they are identical
@Ibix did respond saying it depends on how S is measuring the temperature, but it turned out you were envisioning what I was assuming. You’re correct in that I am not responding to what your primary question seems to be—I am simply addressing what you said here in this quote.

Pelion said:
By the way: temperature is a Lorentz scalar, so it is invariant...
Sort of, yes. For definiteness, we will imagine that we've covered the surface of both objects with infinitesimally small mercury thermometers. Now we can make statements of the form "This particular thermometer on the surface of the object had this value as it passed thought this particular point in spacetime" and that is a Lorentz scalar. The values of the other thermometers or even of this thermometer at any other time and place are different Lorentz scalars.

Note also that every thermometer reads any given vaue only once and at most one thermometer passes through any given point in spacetime, so we can label the points in spacetime (which are conventionally called "events") by a thermometer reading: "This thermometer reads ##T##" identifies a particular event.

And with those preliminaries out of the way:
...so the effects you illustrate must in fact be 'superficial' to what is being discussed.
That does not follow.

Back in the first post of this thread, we specified that the objects "are heated to the same temperature", which is to say that we heated them in such a way that all the thermometers read the same temperature (which we'll call ##T_0##) at the same time, or equivalently for all the thermometers the events "thermometer reads ##T_0##" are simultaneous.

This is where the relativity of simultaneity comes in. In general, if two events are simultaneous in one frame they wil not be simultaneous in any other frame (this is why the words "at the same time" in any relativity thought experiment should set off alarm bells). Thus, our initial specification that the objects are heated to the same temperature is frame-dependent - if they are the same temperature in one frame, they will not be at the same temperature in any other frame.

Thus, the premise in your original post is mistaken. There's no need to explain the "The identically dropping temperature readings, at all times" using frame S' because they aren't identically dropping in that frame; both the initial and the ongoing temperature distribution is non-uniform. Of course that raises the question of how an observer at rest in frame S' does explain the thermal behavior of the two objects, and there are two ways of attacking that problem:
(We'll assume that all cooling is due to the emission of thermal radiation)
1) The easy way: working in the frame in which the plates are at rest and the temperature distribution is uniform so all the thermometers read the same thing at the same time and no relativity is involved, calculate the drop in temperature over time. Then Lorentz transform the coordinates of each "thermometer reads T" event into the coordinates of the primed frame.
2) The hard way: Working in the frame in which the plates are moving and the initial temperature distribution is non-uniform calculate the drop in temperature at each point over time. Be careful to include the effects of relativistic aberration of the thermal radiation emitted from each point, relativistic beaming, and maybe some other effects that I'm forgetting because only a total masochist woud ever want to attack the problem this way.

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vanhees71, Pelion and Pencilvester
Thanks much Nugatory for coming up with (and devoting the time to write) this great analysis of the situation! Excellent perspective on the problem, and one that delineates the issues...let me think about what you claim.

## 1. How does special relativity affect the cooling of identical bodies?

Special relativity states that the laws of physics are the same for all inertial frames of reference. This means that the cooling of identical bodies will be the same regardless of the observer's frame of reference.

## 2. What is the peculiar situation in the cooling of identical bodies in special relativity?

The peculiar situation refers to the fact that, according to special relativity, time is relative and can be experienced differently by different observers. This means that the cooling process of identical bodies may appear to be different for different observers.

## 3. How does the speed of the bodies affect their cooling in special relativity?

In special relativity, the speed of an object affects its perception of time. This means that as the speed of the bodies increases, the rate of their cooling may appear to slow down for an observer in a different frame of reference.

## 4. Can the cooling process of identical bodies be reversed in special relativity?

No, the laws of thermodynamics still apply in special relativity, so the cooling process cannot be reversed. However, the perception of time and the rate of cooling may appear to be different for different observers.

## 5. How does the concept of time dilation play a role in the cooling of identical bodies in special relativity?

Time dilation is a consequence of special relativity, where time appears to slow down for objects moving at high speeds. This means that the rate of cooling for identical bodies may appear to be slower for an observer in a different frame of reference due to time dilation.

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