A Relativistic Paradox and its Resolution or Who drives the entropy current between two bodies in relative motion?
AA Relativistic Paradox and its Resolution or Who drives the entropy current between two bodies in relative motion?
Friedrich Herrmann and Michael Pohlig
Abteilung für Didaktik der Physik, Karlsruhe Institute of Technology, D-76128 Karlsruhe
E-mail: [email protected], [email protected]
Abstract
We discuss a paradox from the field of relativistic thermodynamics: Two heat reservoirs of the same proper temperature move against each other. One is at rest in reference frame S A , the other in reference frame S B . For an observer, no matter in which of the two reference frames he is at rest, the temperatures of the two reservoirs are di ff erent. One might therefore conclude that a thermal engine can be operated between the reservoirs. However, the observers in S A and S B do not agree upon the direction of the entropy flow: from S A to S B , or from S B to S A . The resolution of the paradox is obtained by taking into account that the “drive” of an entropy current is not simply a temperature di ff erence, but the di ff erence of a quantity that depends on temperature and on velocity. Keywords:
Relativistic thermodynamics, thermal equilibrium, driving force, paradox __________________________________________________________________________ 1. Introduction
In physics, paradoxes are popular topics. They motivate to deal with a matter more intensively. Among the best known are Olbers’ Paradox, Gibbs’ Paradox and the paradoxes of special relativity. There is no introduction to the theory of relativity in which the twin paradox is not discussed. A paradox will always be resolved in the end, but in order to resolve it a good understanding of the corresponding phenomenon is required. We would like to present a problem that can be formulated as a paradox and that has to do with relativistic thermodynamics.
In every formulation of a paradox there is a hidden error that must be identified. In our case, too, there is an error. But we will not reveal it yet.
Here is the story:
Two heat reservoirs A and B with the protagonists Alice and Bob move against each other. S A is the reference frame in which Alice and reservoir A are at rest, Bob and reservoir B are at rest in S B . The “proper temperature” of the two reservoirs, i.e. the temperature measured by a thermometer that is at rest in the respective reservoir, has the same value for both reservoirs. Like the values of many other physical quantities, temperature depends on the reference frame. Temperature is not a relativistic invariant. Alice knows relativity and she has learned that according to Einstein [1] the temperature of B is lower than that of A. This gives her the idea of operating a heat engine between A and B. ntropy flows into a heat engine at a high temperature, and it comes out at a lower temperature. In the process, energy is released via the shaft of the heat engine. Alice therefore expects entropy to flow from her reservoir A through the still hypothetical engine into Bob’s reservoir B, Figure 1a.
Abb. 1. (a) For Alice, the temperature T A is higher than T B . She concludes that a thermal engine can be operated between her heat reservoir A and that of Bob B. Entropy would flow from A to B. (b) For Bob, T B is is higher than T A . In his opinion, entropy should flow from B to A. But Bob also knows the theory of relativity, and he has an idea that is similar to Alice’s. However, he comes to the conclusion that the entropy should flow from B to A, Figure 1b.
Now entropy S , unlike temperature T , is a relativistic invariant. Whether the entropy flows from A to B or from B to A cannot depend on the reference frame. So something must be wrong. This is our paradox. To solve it, we’ll use the temperature-four-vector. We will introduce it in section 2. In section 3 we try to give some idea about the temperature four-vector. In section 4 we look at the question of what one could understand by thermal equilibrium between two bodies that move against each other, or what replaces the concept of thermal equilibrium. By doing so we can also resolve our paradox.
Usually, the energy balance is placed in the foreground when describing a heat engine. We prefer to argue with entropy, see also Job [2] and Fuchs [3]. We give reasons for our approach in Appendix A. For our reasoning we also use an enlarged Gibbs’ fundamental equation. Since it is rarely found in this form, we derive the additional terms in Appendix B. In appendix C we draw a comparison: In bodies moving against each other, temperature and velocity are coupled in a similar way as the electrical and chemical potential of electric charge carriers in a semiconductor diode are coupled to form the electrochemical potential.
We argue as far as possible without referring to the contra- and covariant representation of four-vectors. All we need to know in this context is that a minus sign appears when a four-vector product is calculated.
We write four-vectors with bold characters, three-vectors are bold and italic, the magnitudes of vectors and scalar variables regular and italic.
For example, p denotes the momentum four-vector whereas p is the normal momentum vector in three-space. In the usual coordinate representation we thus have p = ( E/c , p ) with p = ( p x , p y , p z ). . The temperature four-vector In the following we need the Gibbs fundamental equation, which is usually written in the form (1)
It describes various energy exchange processes of a system. According to equation (1) the energy E of a system can be increased or decreased in three ways, depending on which of the extensive quantities entropy S , volume V or amounts of substance n i behind the di ff erential sign is changed. ( p is the pressure and μ i are the chemical potentials). The equation is by no means complete, because the energy can be altered in several other ways [4,5]: by exchanging the extensive quantities momentum p , angular momentum L , electric charge Q , etc.. If we add these quantities as variables of our system, equation (1) becomes (2) Here v is the velocity, U the electrical potential di ff erence and ω the angular velocity. (For an explanation of the last three terms see appendix B.) Depending on which of the terms on the right side of equation (2) is di ff erent from zero, one refers to an energy exchange in one or another form [6]. If all terms are zero except the entropy term, we say that the energy is exchanged in the form of “heat”. If only – pdV is di ff erent from zero, then “work is done”. μ i dn i characterizes an exchange of “chemical energy”, etc. In some cases, such as in the case of TdS , a special symbol is used for such a term. So
TdS is written dQ for short, sometimes 𝛿 Q to remind that it is not a total di ff erential. This should not detract from the fact that the summands on the right side of equations (1) and (2) are not physical quantities in the usual sense. In particular, no value can be specified for the heat, compression energy or chemical energy contained in a system [7].
Another important aspect in our context is the role of the intensive quantities, which are placed before the di ff erential sign in equations (1) and (2). In dissipative transport processes they act as driving force in the following sense: a temperature di ff erence acts as a drive for an entropy flow in a heat conductor, a pressure di ff erence for a volume flow, e.g. when water flows through a pipe, a chemical potential di ff erence for a substance flow in a di ff usion process, a velocity di ff erence for a momentum transport in a viscous fluid, an electric potential di ff erence for an electric current in a conductor that has a resistance. We first consider a process in which only the first term of equation (3) is di ff erent from zero: We heat a body, or in other words we add entropy to it together with energy. Then equation (2) reduces to dE = TdS (3)
We bring the entropy increase dS to the left dS = (1 /T) · dE . (4) In this form the equation can be generalized relativistically. At first we replace the energy by the four-momentum E → p = ( E/c , p ) , thus, in our case d E = Td S − p d V + ∑ i μ i d n i d E = Td S − p d V + ∑ i μ i d n i + v d p + UdQ + ω d L dE → d p = ( dE/c , d p ) Since entropy is a scalar and a relativistic invariant, the reciprocal temperature must also be replaced by a four-vector, which multiplied by the four-momentum vector results in the entropy change dS . The four-vector with this property is [8,9,10]: β = ( c/T , v /T ) Here, v is the velocity at which the heat reservoir moves. β is called the inverse temperature four-vector, or four-temperature for short. So we replace the right side of equation (4) with the dot product of the two four-vectors ( c/T , v /T ) and ( dE/c , d p ). When such a product is calculated, the contribution of the space-like components is preceded by a minus sign*. We obtain: dS = β d p = dE/T – ( v /T ) d p or: dE = TdS + v d p (5) The comparison with equation (3) shows that an additional term v d p has emerged, which also appears on the right side of Gibbs’ fundamental equation (2). In fact, such a term is to be expected: equation (3) alone would not correctly describe our process, because with the entropy we also supply energy. Because of the energy-mass equivalence, the momentum of our system also changes during the process. The term v d p expresses exactly this. We therefore see that processes which are described by the temperature in the non-relativistic limiting case are now better described by the temperature four-vector. * ((footnote)) The dot product of x = ( x , x ) and y = ( y , y ) is x · y = x y – xy For the square we get correspondingly x = ( x ) – x
3. The new “drive”
Let us try to get an idea of the temperature four-vector. As with other four-vectors, it is a good idea to look first at its magnitude. We calculate the four-vector product of β with itself: We see that is a function of v and T : β = β ( v,T ) . Next, we write the expression β = ( cT , v T ) = c – v T β = β = c – v T (6) that will play a crucial role in resolving our paradox, and we give it its own symbol: We thus have (7)
Like every square of a four-vector, β and thus also β and Θ are relativistic invariants. This means: If the state of one and the same body is described in di ff erent reference frames, the value of Θ is always the same. But it also means: If di ff erent bodies of the same proper temperature move at di ff erent velocities in one given reference frame, Θ has the same value for all these bodies. This last statement is important for us. It means that for moving bodies, i.e. heat reservoirs, heat conductors or heat engines Θ ( T , v ) takes over the role that the temperature plays for bodies at rest. In particular, a di ff erence in the values of Θ ( T , v ) represents a “drive” for an entropy flow. We will come back to this. For the time being we are still interested in how the temperature of a moving body is related to its proper temperature.
According to equation (7), for the rest frame, i.e. for v = 0, Θ is equal to the proper temperature: Θ ( T ,0) = T (8) Since Θ is Lorentz invariant, we have: Θ ( T ,0) = Θ ( T , v ) and with (7) and (8) follows (9) Equation (9) tells us that the temperature T of a system is lower than or equal to its proper temperature. This is what we had assumed in our initial story.
4. Does a thermal equilibrium exist between bodies moving against each other? - The resolution of the paradox
We look at two heat reservoirs A and B, Fig. 2. Neither of them is moving. If we establish a thermally conductive connection between them, an entropy will flow from the body with the higher temperature to that with the lower temperature, because the “drive variable” temperature has di ff erent values. This entropy current has the tendency to reduce the drive, i.e. to reduce the temperature di ff erence. The process is irreversible, i.e. additional entropy is created, so that more entropy arrives at the body with the lower temperature than has flowed o ff at the body with the high temperature. c β ( T , v ) = T v c Θ ( T , v ) ≔ c β ( T , v ) Θ ( T , v ) = T v c T = T v c Abb. 2. A does not move against B. If A and B have di ff erent temperatures, an entropy current flows through the heat-conducting connection. If T A > T B , it flows from A to B; if T B > T A , it flows from B to A. One could also have installed a heat engine between the two reservoirs. In this case, the entropy flow between A and B would have taken place without further entropy production.
If the temperatures of both reservoirs are the same, there is no drive and consequently no entropy flows and no heat engine can be operated.
Now the reservoir A is supposed to move relative to B, whereby the proper temperatures of A and B are to be equal. We are now in the situation of our paradox. Again we ask in which direction the entropy flows. But the question arises how we can establish a heat-conducting connection. The problem is that it must be frictionless, because friction would cause the bodies to come not only into thermal but also in mechanical equilibrium, i.e. the velocities would equalize. This would also happen in case the proper temperatures are equal, and of course also in the non-relativistic case.
Fig. 3 shows a possibility: between the bodies there is a rotating, heat-conducting cylinder that rolls on the bodies without friction.
Abb. 3. The thermal connection is established by means of a thermally conductive cylinder.
We first assume that A moves to the right and B to the left, so that the cylinder rotates but stays in place. At the two contact points, thermal equilibrium will establish between the cylinder and the respective body. There is no problem with this, because the two components have the same velocity. It is clear that in this case no entropy current flows between A and B, for reasons of symmetry alone.
Let us now describe the same situation in the reference frame in which reservoir B is at rest, Fig. 4. The fact that there is no entropy flow cannot change. What must change is our description of the situation. According to equation (9) the temperatures are now di ff erent. The fact there is still no entropy flow means that the temperature di ff erence alone cannot be responsible for the drive. Abb. 4. The temperatures of A and B are di ff erent, but not the drive variables Θ . owever, with Θ ( T , v ) (equation (7)) we have known a variable which depends on temperature, but which is invariant upon a change of the reference frame. Only if Θ ( T , v ) is di ff erent for A and B, there is a drive for an entropy flow, and thus the possibility to operate a heat engine. It is not surprising that temperature alone is no longer the driving force, because a momentum transport is coupled to the entropy transport. The drive for the flow of the two firmly coupled quantities can no longer simply be the temperature di ff erence. In addition to the temperature, the drive variable must also comprise the velocity. If one of the bodies moves with velocity v A and the other with v B , the drive for the entropy momentum current is the di ff erence In case that Θ ( T A , v A ) = Θ ( T B , v B ) there is no dissipative entropy current anywhere on the path between A and B. Θ ( T , v ) therefore has the same value everywhere, i.e. also at every point of the cylinder that connects the two bodies thermally. So there is also no “local drive”. This state of the system substitutes the state of thermal equilibrium. (Within the cylinder we have a cyclic convective entropy transport because of the rotational movement. However, this is irrelevant in our context.) We thus also have the solution to our paradox. Alice had assumed that the drive variable responsible for her engine should be the temperature. After some thought, she realizes that this is not true: In order for her engine to run, she needs a di ff erence of the quantity Θ , but there is no such di ff erence. Θ has the same value for S A and S B . Bob, of course, comes to the same conclusion.
5. Conclusion
The seeming contradiction in our paradox is caused by the expectation that one can operate a heat engine between two heat reservoirs that are at di ff erent temperatures. According to this expectation, the temperature di ff erence should be the driving force of the engine. However, it turns out that not the temperature but another variable Θ acts as the drive variable when the heat reservoirs move against each other. Θ depends not only on the temperature but also on the velocity. Not a temperature di ff erence, but a di ff erence of the quantity Θ is required to drive a heat engine or to cause a heat flow through a heat conductor. The concept of thermal equilibrium, where two (or more) temperatures are equal, turns into an equilibrium, where Θ takes upon the same value at the subsystems. Appendices A. Why entropy and not heat
Fig. 5 a shows schematically the energy and entropy balance of a heat engine.
Fig. 5 b shows the balance of the same variables in case the heat reservoirs are connected by means of a heat conductor. In this case, the work that a heat engine could have done is not used. Δ Θ = T A v A c – T B v B c Abb. 5. (a) Reversible heat engine: The amount of entropy that arrives from heat reservoir A flows away to B. Only a part of the energy coming from A flows to B. The rest leaves the engine as work. (b) Heat conductor: The amount of energy that arrives from heat reservoir A flows away to B. Not only the entropy that is transferred from A to the heat conductor arrives at B, but also the entropy produced in the heat conductor.
Usually the focus is on only one of the two balances. We want to compare them:
Description with the entropy:
In the heat engine, entropy goes from high to low temperature and thereby performs work. Entropy thus behaves in the same way as Carnot's caloric. This was pointed out soon after the Clausius’ introduction of the entropy. A comprehensive overview of the history of this finding is given by Hirshfeld [11]. Usually Callendar [12] is quoted as the first one who noticed this agreement. In fact there is an even earlier source, namely 1908 in Wilhelm Ostwald’s book
Die Energie [13].
The temperature di ff erence plays the role of a driving force for the entropy current. This is also suggested by the structure of Gibbs’ fundamental equation [14]. The entropy balance for heat conduction looks somewhat more complicated: Entropy also enters the heat conductor at the higher temperature and exits at the lower temperature. Since the heat conduction process is irreversible, i.e. entropy is produced in the process, more entropy comes out at low temperature than has flowed in at high temperature. The outflowing entropy is the sum of the inflowing entropy and the produced entropy.
Description with the energy:
More energy goes into the heat engine at high temperatures than out at lower temperatures. The di ff erence is the work done by the heat engine. The same amount of energy goes into the heat conductor and out of it. So we can say that for the description of heat conduction the energy is the more convenient quantity. In fact, in the literature about the thermodynamics of irreversible processes it is usually preferred.
Why do we prefer the entropy balance in our context? It has the great advantage over heat Q of being a state variable. Since there is no heat density, no local balance can be formulated for Q . With entropy, however, this is possible [15,16]. In addition, Gibbs’ fundamental equation suggests that a temperature gradient should be interpreted as a drive for the entropy flow and not for the energy flow. . Supplementing Gibbs’ fundamental equation
We justify the last three terms on the right side of equation (2) for the energy change dE . We start with v d p . It describes the energy change of a body which moves with the velocity v , if the momentum d p , for example in a collision process, is supplied to it. We begin with the kinetic energy of the body: and get
The same result is obtained with the relativistic energy-impulse relation. Starting from we get
The derivation for the rotational contribution is analogous.
The rotational kinetic energy is:
Here L is the angular momentum and I the moment of inertia. We get where ω is the angular velocity. Finally the last term. We consider a capacitor whose charge is Q and change the charge by dQ . The energy within the field of the capacitor is
Here C is the capacitance. It follows: U is the electric potential di ff erence between the plates of the capacitor. In an axiomatic approach, one would interpret all terms of Gibbs’ fundamental equation in the same way as it is done traditionally with the terms
TdS and μ dn . I is common practice to define the absolute temperature as E = p md E = 2 p d p m = p m d p = v d p E = E + c p d E = c d ( p ))2 E + c p = 2 c p d p E = pm d p = vd pE = L Id E = 2 L d L I = L I d L = ω d L E = Q Cd E = 2
QdQ C = QC dQ = UdQT := ( ∂ E ∂ S ) V , n i ,... nd the chemical potential as With corresponding expressions one can also define pressure, velocity and electrical potential.
C. No surprise for solid state physicists
We had identified the quantity as the variable responsible for the drive of the entropy flow.
Here, solid-state physicists recognize a situation that is familiar to them. The transport of charge carriers in metals or semiconductors is not simply caused by an electrical potential gradient or di ff erence. Since electrical charge and the intensive quantity “amount of substance” (or particle number) are firmly coupled to each other, the gradient of the sum of the electrical and chemical potential is responsible for the flow of the electrons. This sum is called the electrochemical potential. If a piece of copper and a piece of aluminum are in contact, the electrical potential in copper is 1.68 volts higher than in aluminum. Because of the charge of the electrons, this di ff erence represents a driving force, from aluminum to copper. Nevertheless, no current flows, because the di ff erence of the chemical potentials of the electrons in copper and aluminum represents a drive for the electrons in the opposite direction, i.e. from copper to aluminum. There are numerous other situations in which two or more extensive quantities are coupled so that a “combined potential” is responsible for a transport of particles or of a substance [17,18,19]. Back to our bodies moving against each other. There is no entropy current between A and B without a momentum current (just as there is no electric current without a flow of the amount of substance n ). The drive variable for this combined current of entropy and momentum is a combination of temperature and velocity, namely the expression Θ . So it would not be unreasonable to give this quantity not only its own symbol but also its own name. Just as the combination of electrical and chemical potential is called electro-chemical potential, Θ might be called thermo-kinetic potential. The equilibrium between the two bodies would accordingly be called thermo-kinetic equilibrium. We see that for bodies that move against each other, the term thermal equilibrium must be replaced by thermo-kinetic equilibrium.
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