Review Article | Open Access
J. Güémez, "Relativistic Thermodynamics: A Modern 4-Vector Approach", Physics Research International, vol. 2011, Article ID 387351, 18 pages, 2011. https://doi.org/10.1155/2011/387351
Relativistic Thermodynamics: A Modern 4-Vector Approach
Using the Minkowski relativistic 4-vector formalism, based on Einstein's equation, and the relativistic thermodynamics asynchronous formulation (Grøn (1973)), the isothermal compression of an ideal gas is analyzed, considering an electromagnetic origin for forces applied to it. This treatment is similar to the description previously developed by Van Kampen (van Kampen (1969)) and Hamity (Hamity (1969)). In this relativistic framework Mechanics and Thermodynamics merge in the first law of relativistic thermodynamics expressed, using 4-vector notation, such as ΔUμ = Wμ + Qμ, in Lorentz covariant formulation, which, with the covariant formalism for electromagnetic forces, constitutes a complete Lorentz covariant formulation for classical physics.
During the 1960s and 1970s many physicists devoted considerable effort to finding the most adequate relativistic formulation of thermodynamics . Yuen's 1970 paper  presents the state of the art on relativistic thermodynamics at this time. After the work by Van Kampen  and Hamity  introducing 4 vectors in thermodynamics and the clearly stated asynchronous formulation by Gamba , Cavalleri and Salgarelli , and Grøn  a consensual relativistic thermodynamics formalism should have been achieved. However, no agreement on the correct Relativistic thermodynamics was reached  (, pp. 303–305). Until recently, papers on this topic have been published , mainly on relativistic transformation of temperature [11, 12].
Let be a composite body, which moves, in a reference frame , under the action of external (conservative and nonconservative) forces , simultaneously applied during time interval , with resultant force , impulse , with nonzero work (only conservative forces perform work) and that experiences a certain thermodynamic process, with internal energy variation and heat . In classical physics, the complete description of this process, expressed in Galilean covariant form, is given by : (i) a vectorial equation (linear momentum-impulse equation) : and (ii) a scalar equation (first law of thermodynamics or energy equation) : From (1) the following equation can be obtained: or the center of mass equation , where is the displacement of the center of mass (cm) of and is its kinetic energy variation throughout the process.
For an observer in frame in standard configuration with respect to frame , with velocity (Appendix A), it has the corresponding equations where the corresponding magnitudes are measured in , the mass, force, interval of time, impulse , linear momentum variation , internal energy , and heat are Galilean invariants, and magnitude velocity , displacement , kinetic energy , and work , have their specific Galilean transformation .
It is interesting to note that when forces applied to have an electromagnetic origin, with some force obtained from “Lorentz force” equation ( electric charge, electric field, and magnetic field), the whole formalism is neither covariant under Galilean transformations (Lorentz force is not Galilean covariant ) nor covariant under Lorentz transformations (the previous thermodynamics formalism is not Lorentz covariant), in contradiction with Einstein's principle of inertia.
After these considerations about Galilean relativistic thermodynamics, not compatible with electromagnetic interactions, it seems necessary to obtain a formalism for the first law of thermodynamics expressed according to the principles of the special theory of relativity, that is, Lorentzian relativistic thermodynamics, compatible with electromagnetic interactions. As a result of this, it will be possible to obtain a Lorentz covariant formalism for exercises in classical physics that include concepts of mechanics, thermodynamics, and electromagnetism.
A modern view of a relativistic thermodynamics theory requires a clear definition of (i) the tensorial objects which characterize the equilibrium state of the system and of (ii) any tensorial object that characterizes the interaction of the system with its mechanical (work reservoir (, Chap. 3)) and thermal (heat reservoir (, pp. 89-90)) surroundings, with a prescription of the apparatus which measures it. The observables will depend, in general, on the physical system and on the observer (Appendix A), but the principle of relativity ensures that all inertial observers obtain equivalent descriptions of the same process. So, any relativistic formalism developed to describe a physical process must be according to this principle, that is, it must be Lorentz covariant. This is the course chosen in this paper, in which we solve an exercise on the isothermal (nonquasistatic) compression of an ideal gas in the reference frame in which the system is at rest and in a frame , in standard configuration to (Appendix A), using the Minkowski 4-vectors—related through Lorentz transformations —and a Lorentz covariant form for the first law of thermodynamics.
The paper is arranged as follows. In Section 2 the formalism, based on the principle of the inertia of energy (Einstein's equation) and on the asynchronous formulation, is developed. After that, in Section 2.3 the principle of similitude is enunciated, expressing the conditions under which the same equations can be used for an elementary particle and for a composite system. Section 3 presents the 4-vector energy function for different systems. The asynchronous formulation of 4-vector work is obtained in Section 4. In Section 5 thermal radiation 4-vector (heat) , based on photons, is introduced. In Section 6 the mathematical formulation of the relativistic thermodynamics first law is presented in Lorentz covariant form. In Section 7 the isothermal compression, by two pistons, of an ideal gas is solved by using the previously developed formalism in both frames , zero momentum frame, and , in standard configuration respect . Forces on pistons are described using an electromagnetic interaction, in its relativistic Lorentz covariant form. Finally, Section 8 proposes some conclusions regarding the possibility of solving exercises in classical physics in a complete Lorentz covariant form. Although we assume that the reader is familiar with the Minkowski 4-vector formalism, in Appendix A a brief review on 4-vectors and Lorentz transformation algebra is provided introducing the “metric tensor” and the “Lorentz transformation” used in the paper .
2. Relativistic Thermodynamics Formalism
Relativistic thermodynamics formalism is developed in two steps: (i) Einstein’s equation , expressed as the principle of the inertia of energy, which allows us to obtain energy function and the 4-vector energy function for a given system; (ii) the asynchronous formulation, that will allow us to obtain the work performed by forces acting on a system and the 4-vector work . As a consequence, the principle of similitude can be formulated, according to which, and under very general circumstances, a composite system behaves as a whole in its interactions with its surroundings and equations for an elementary particle can be used with a composite, deformable system.
2.1. Inertia of Energy
It could be considered, in a broad sense, that the main goal of relativistic thermodynamics is to reach a unified description on point dynamics and extended-body dynamics .
In order to ensure that an extended body behaves like a “single particle” interacting with its surroundings—work reservoirs or thermal bath—and so that it is physically meaningful to use Lorentz transformations, it is necessary that all forms of energy that make up the body contribute in the same way to its inertia . These forms of energy must include those related with the mass of its constituent elementary particles, binding—nuclear, chemical, and so forth—energies (Figure 1), internal kinetic energy (see Section 7.1.1), electrostatic energy , and so forth, and energy of thermal radiation in equilibrium with matter inside the system  (see Section 5).
Einstein's Equation for an extended body can be interpreted by relating its inertia—a body's reluctance to undergo a change in velocity —with energy function —energy content of the physical system or internal energy .
Principle of the Inertia of Energy
for an extended body in complete equilibrium, any kind of energy inside the system, relativistically expressed in reference frame in which the system as a whole is at rest, contributes to the energy function of the system . Considering that all forms of energy are convertible between them  the inertia of a system  in equilibrium is (, p. 163)
It is possible to define the inertia of a body (we prefer the term inertia, instead of mass , to avoid confusions when the system includes photons (Section 5)) as : the inertia of a composite body equals the sum of its elementary particles mass (protons, neutrons, and electrons) : with energy , minus the minimum energy , divided by , necessary to separate its elementary particles so that they are far apart (Section 7.1.1): with .
2.2. Asynchronous Formulation
For an extended, deformable body a relativistic theory cannot be directly formulated in an arbitrary inertial frame. It must be based on known prerelativistic descriptions. On the one hand, it seems necessary to maintain the classical concept that the resultant force on the body must be zero (zero total impulse) when the motion remains uniform and to assure that when no torque is applied to the system in a certain reference frame, no torque is applied to it in another frame . On the other hand, in classical mechanics forces on an extended system are applied simultaneously. This simultaneity occurs in all inertial frames. In thermodynamics, heat is a kind of interchanged energy with (assumed implicitly) zero linear momentum.
According to Cavalleri and Salgarelli, when forces on an extended, composite, body are applied, in order to develop a coherent formalism for relativistic thermodynamics, a privileged observer must exist, in reference frame , that performs experiments on the body that remains at rest (Figure 2) .
According to Gamba :
“in the Asynchronous Formulation, observers in frames and refer to the same experiment (the experiment performed in the privileged frame ) and obtain its own physical magnitudes, expressed as 4-vectors. In this formulation both descriptions of the experiment are connected by true Lorentz transformations .”
The observer in takes an ideal surface, at rest, which delimits the system considered and measures energy, work or heat, interchanged through the surface during time interval . An observer in obtains the same magnitudes by true Lorentz transformations, from the events considered by an observer in . The observer in does not perform a similar experiment to observer in (synchronous formulation ), it just translates the experiment performed in to its own physical magnitudes. Owing to the relativity of simultaneity, forces applied simultaneously in will not be simultaneous in (asynchronous processes).
The existence of frame guarantees the correspondence between the relativistic and the classical descriptions; an equivalence necessary in the low velocity limit.
In the asynchronous formulation, given the quantity , where is defined for the event and is defined for the event , with , but with , then quantity in is the same as in when all subindex A quantities are obtained from the corresponding quantities in through Lorentz transformations. Relativity gives rules to relate measurements made by observers in frame to measurements made by observers in frame only if this definition is adopted .
In the asynchronous formulation the 4-vector energy function is a time-like 4-vector in , with zero linear momentum components  (see Section 3): In frame , transforms under Lorentz transformation as .
As a generalization of this asynchronous formulation, in frame every flux of energy through the frontier of the system as thermal radiation (heat) is exchanged with zero total impulse. Thus, heat is exchanged with zero linear momentum in frame with a 4-vector related to thermal radiation exchange given by  (see Section 5)
In frame the same Lorentz transformation is common to , , with and to the 4-vector , that transforms as the energy (timelike) part of a (time-like) 4-vector, .
2.3. Principle of Similitude
The asynchronous formulation and the principle of the inertia of energy guarantee that the system can be described as a “single particle”  characterized by its energy function or its inertia . These considerations permit us to enunciate  the following.
Principle of Similitude
The mathematical expression for a physical law is the same when referred to an elementary particle, with tabulated mass , or when referred to a composite body, well characterized by its energy function , and inertia .
In the asynchronous formulation there is no difference between Lorentz transformations for an elementary particle and Lorentz transformations for an extended body, provided that the system is in equilibrium, that is, energy function of the body is well defined. In frames like , in which the system has velocity , differences between point dynamics and extended-body dynamics are due to the relativity of simultaneity , that is, forces applied simultaneously in but at different points of the body will not be simultaneous in .
The principle of similitude has the following meaning. Physics equations, such as the Lorentz force equation , Newton's second law of classical mechanics , or relativistic equations, such as or , are correct when they are applied to an elementary particle, with mass and charge , because every magnitude is well defined, for example, total energy , linear momentum , and so forth, as well as the electric field , the magnetic field , and so forth, and forces applied are local forces, all of them applied at the same point. Similarly, a 4-vector, like or , transforms between frames and in standard configuration, by using the Lorentz transformation, , with , and so forth, and where , and so forth, is in the same 4-vector in , because all of them are locally defined.
When one wants to apply these equations to a process described on a composite, deformable body (e.g., a Ni atomic nucleus, a gas enclosed in a cylinder-piston system, a macroscopic chunk of Fe, etc.) and one wants to use the Lorentz transformation between reference inertial frames to transform 4-vectors, it is necessary to have previously ensured that the body behaves as a whole and that the principle of inertia of energy is satisfied. Because on an extended body different forces are applied at different points, it is necessary to ensure previously that there exists a reference frame in which the center of mass does not move during the process. This goal is achieved when external forces are applied according to the asynchronous formulation and when the interval of time during which forces are applied on the mobile parts of the system is greater than the relaxation time of the system.
Consider a gas enclosed in a cylinder-piston system. If the force on piston is applied in such a way that the velocity of the piston is greater than the velocity of the sound in the gas , with a characteristic gas relaxation time given by , where is a characteristic linear dimension of the system, then the system does not behave as a whole during time intervals because there are parts of it that do not feel the perturbation and so do not contribute to the inertia of the system. In this case the description of the process cannot be made according to the relativistic formalism to be developed here, the principle of similitude is not applicable and another formalism must be used to describe the process .
When a process on a composite, extended body is carried out in such a way that the principle of inertia of energy is satisfied, the same set of equations valid for elementary particles can be used on the body.
Consider a macroscopic body with well-defined energy function . In general, this energy function is temperature dependent (see Section 7.1.1) ( dependence on volume will not be considered volume ) and also its inertia , according to the principle of inertia of energy (, p. 289). When moving with velocity (one-dimensional) in frame its linear momentum and total energy are given by As previously noted, these results can be obtained from . These equations constitute the generalization for an extended body of equations , , and for an elementary particle of mass and velocity . The total energy of the body can be expressed as Equation (13) is the generalization, in a thermodynamics context, of the equation for an elementary particle.
For an elementary particle mass and velocity , the kinetic energy is . The kinetic energy for an extended body, defined as , is The kinetic energy of the body in frame , in which its linear momentum is null, is zero.
3. Four-Vector Energy Function
Energy function of a composite body is obtained from the energy function of its components (Section 2.1). (1)Universal constants (, (Planck), (Boltzmann), , , etc.) are relativistic invariants having the same value for all inertial observers in relative motion.(2)For an elementary particle—proton, neutron, and electron—the inertia equals its tabulated mass—. , , respectively.(3)For a nucleus, N, with protons and neutrons, its inertia equals the sum of the inertia of the elementary particles—with all elementary particles at infinite separation as initial arrangement—minus its binding energy (strong interaction)  divided by (Figure 1): (4)For an atom, the inertia equals the sum of the inertia of its nucleus and electrons minus released energy (electromagnetic interaction)  divided by (Figure 1): For instance, energy function for a He atom (2 protons, 2 neutrons, and 2 electrons)  is given  by (5)For a molecule, formed by atoms, the inertia is the sum of the inertia of its individual atoms minus the energy released when chemical bonds are formed  divided by : Energy function of a composite, self-contained (stable) system is less than the sum of the energy function of its constituents  , .(6)For a system of free noninteracting components  like a gas of He atoms, the inertia equals the sum of the total energy of components —kinetic energy and energy function of the th component, respectively—divided by (see Section 7.1.1).(7)For thermal radiation (photons in a cavity with energy density proportional to fourth power of absolute temperature) filling a cavity  its total energy contributes to the total inertia of the system . The thermal radiation emitted by a body can be described as radiation in a cavity  (see Section 5).
As previously noted, in the zero-momentum frame of a composite system with energy function the 4-vector that denotes the state of the system is given by . For an observer in frame , 4-vector energy function is , and one obtains according to the principle of inertia of energy (Section 7.2.1).
for a completely isolated system that performs any kind of internal process, for example, annihilation or creation of particles, disintegration, inelastic collisions, and so forth, the inertia does not change along the process , according to the Principle of Inertia of Energy, with
4. Four-Vector Work
In order to obtain a complete characterization of forces applied to a thermodynamical system (i.e., based on a fundamental interaction), we will describe forces as the interaction between an electric charge located on the th piston and a (static) electric field . This procedure guarantees a detailed description of forces and of its relativistic transformation between reference frames (Appendix B).
Consider in frame a set of forces , with an electromagnetic origin, simultaneously applied, on different pistons, on an extended body (Figure 3) during the same interval of time , according to the previously discussed asynchronous formulation. Impulse and work for the th force are given by
The th field is represented by the -tensor : with the -tensor electromagnetic force : The th piston has a 4-vector displacement and a 4-vector velocity : where is the proper time of th piston displacement.
For the th piston, two 4-vectors can be obtained: (i) the 4-vector Minkowski force and (ii) the 4-vector work . (1)The 4-vector Minkowski force is given by (, Chap. 33) with .(2)The 4-vector work is given by a 4-vector with units of energy. The 4-vector can be obtained by deriving in respect to proper time of the th piston as
This obtention of the 4-vector shows that is a 4-vector itself (Appendix A). For a finite interval of time , with constant force , and 4-vector interval , the 4-vector work is
For the set of forces simultaneously applied to the body at different pistons in frame during the finite interval of time (Figure 4), the total 4-vector “force-displacement product” (work) is the sum of the 4-vector . The 4-vector total work is given by , with condition :
In frame , , with impulse and “force-displacement product” being
Adiabatic First Law (, Section 4.2)
A system that changes its energy function owing to forces applied to it, in an adiabatic process, is
5. Four-Vector Heat
Work is described in thermodynamics as oriented (nonrandom) internal energy transferred between a body and a work reservoir (Figure 4). However, heat is described as random (or nondirected) internal energy transferred between two bodies at different temperatures . Nondirected means “without linear momentum.”
According to Rindler, in the special theory of relativity any transfer of energy, being equivalent to a transfer of inertia, necessarily involves momentum [51,p. 91]. This assessment is valid for all forms of radiation and must be valid for heat , whatever definition of heat is being used.
The most direct argument on relativistic heat transformations is provided by Arzeliés . Based on the principle of equivalence work-heat, this author assumes that relativistic heat transforms as relativistic work.
The 4-vector heat, , is obtained in two steps. First, we obtain the 4-vector for thermal radiation (its frequency distribution fulfills Planck’s frequency distribution) enclosed in a cavity with walls at temperature , and then the thermal radiation exchanged by a body as heat is described as thermal radiation in a cavity.
In a generalization of the asynchronous formulation, we assume that in frame (zero momentum frame) heat is emitted or absorbed with zero linear momentum ( p. 173). With the 4-vector given in frame as , in frame , standard configuration, with , and , a linear momentum associated to , must be ( p. 1746) The relativistic linear momentum of heat in frame requires a physical interpretation—because of the contrast with no momentum for heat in classical thermodynamics . In order to provide the relativistic interpretation of heat and the description of a thermal bath, we will describe thermal radiation as an ensemble of emitted photons enclosed in a cavity .
A cavity with walls at temperature , measured with a gas thermometer at constant volume, and filled with photons that fit Planck’s frequency distribution—that is, thermal photons—constitutes a thermal bath. In frame in which cavity walls are at rest, the total linear momentum of the photon ensemble is zero. In the monochromatic approximation  to Planck’s distribution, every photon has the same frequency , with (Wien's Law), where is a constant (Figure 5).
For a given th photon, with frequency , , and moving in direction , there exists an energy 4-vector (), The norm of this 4-vector is . An individual photon has no inertia.
In frame , total linear momentum for the photons inside the cavity at temperature , , and its total energy are given by In the zero-momentum frame , total energy is the energy function of the system. The 4-vector thermal radiation , is In frame , The energy function is the norm of the 4-vector . This photons ensemble has nonzero inertia  .
Consider for a moment this cavity filled with thermal radiation containing one mole of atoms of a gas also. It is interesting to note that (i) photons of thermal radiation enclosed in a cavity, with Planck’s frequency distribution, the atoms of the gas, with its (ii) electrons distributed in electronic orbitals following Boltzmann's energy distribution, and (iii) atoms moving with Maxwell's (or Juttner distribution ) kinetic energy distribution, every distribution with the same parameter temperature , contribute to the energy function and to the inertia  of the system. As previously discussed, energy function for an ensemble of atoms and energy function for an ensemble of thermal photons transform between inertial frames in the same way. Thus, thermal equilibrium at temperature between matter and radiation is a relativistic invariant and every inertial observer will agree on that equilibrium (Figure 6).
After the obtention of a 4-vector for the contribution to its energy function by thermal radiation inside a cavity, it is necessary to characterize as a 4-vector heat the exchanged energy by a body as thermal photons.
First of all, systems thermally interacting with each other cannot be in equilibrium if they are in relative motion . In the Asynchronous Formulation generalization to heat, there exists a privileged frame in which the system is at rest with respect to the thermal bath and in thermal radiation (photons) is absorbed or emitted with zero total linear momentum.
The energy absorbed, or emitted, by a body as thermal radiation (heat) throughout a process can be modeled as photons inside a cavity. A thermal system can absorb or emit photons through its frontier except in adiabatic processes. A photon emitted by a body, with frequency and direction , contributes with to the linear momentum variation of the body and with to the total energy variation of the body that emits it. A photon absorbed by a body, with frequency and direction , contributes with to the linear momentum variation and with to total energy increment of the system.
Absorbed or emitted photons can be considered different phases in thermal equilibrium . Thus, there is not “force-displacement product” (work) associated with emission or absorption of thermal radiation (photons).
With the system and thermal bath mutually at rest, the ensemble of emitted photons (when the system is at higher temperature than thermal reservoir) is described as an ensemble of thermal photons in a cavity with zero total linear momentum, and so the ensemble of absorbed photons (when the system is at lower temperature than thermal reservoir).
In frame , the 4-vector thermal radiation (heat) associated when the body emits (−) or absorbs (+) photons with frequency is given by with , where is the flux of photons (net number of photons exchanged in unit time) and is the net number of photons exchanged by the body during time interval .
For an observer in frame , , from with linear momentum and total energy : Physical interpretation of linear momentum for heat in frame will be obtained from relativistic Doppler and aberration effects applied to photons (see Section 7.2.3). The norm of is with energy function and inertia relativistic invariants .
If heat is defined as the total energy associated with the emitted (or absorbed) photons as measured in the observer's frame, then , with . If heat is defined as the emitted (or absorbed) energy carried by photons in frame in which the interchange of photons with a thermal surrounding mutually at rest and zero linear momentum happens—as it is defined (implicitly) in classical thermodynamics—then is the norm of any 4-vector and it is a relativistic invariant. In any case, it is the 4-vector that possesses physical meaning, not its components.
For a system that changes its energy function without forces applied to it, by heating, or cooling, in diathermal contact with a thermal bath, system and bath at mutual rest, is
6. Relativistic Thermodynamics First Law
According to the generalized asynchronous formulation of relativistic thermodynamics, the description of a certain process on a composite, deformable system, and after the obtention of 4-vectors energy function , initial and final , work , and heat , is as follows.
Relativistic Thermodynamics First Law
Mathematical: the relationship between variations in energy function of a system after a certain process, during which it interacts with a mechanical reservoir, with forces simultaneously applied to it during that process, and a thermal reservoir, system and reservoir mutually at rest, with thermal radiation interchanged by the system during the process, with every magnitude expressed as a 4-vector, is [4, 61]
No matter whether a system is self-contained or free (confined in a container), any energy, momentum, , or , is always a 4-vector provided that one performs a covariant summation at constant time (simultaneously) in the frame in which the system is mutually at rest  (at least instantaneously) with its mechanical and thermal surroundings.
In the frame reference , the first law is expressed as Every 4-vector in can be obtained by a Lorentz transformation for the corresponding 4-vector in This circumstance guarantees the first law Lorentz covariance.
7. Ideal Gas Isothermal Compression
A horizontal cylinder (Figure 7), with thin metallic walls, section , length , containing 1 mol, atoms, of He gas, enclosed by two pistons, left (L) and right (R). We assume that helium behaves as an ideal gas, described by thermal equation of state . The gas is in equilibrium under pressure and at temperature , volume , . The limits of the system are the walls of the cylinder, considered diathermal. Pistons are considered adiabatic.
7.1. Compression in Frame
As privileged frame we take the frame in which cylinder walls, plate parallel capacitor and thermal reservoir walls are at rest. During the compression process, forces on gas are applied simultaneously, during time interval . Thermal radiation is interchanged with zero impulse in and the gas center of mass remains at the same point, with initial and final zero velocity.
7.1.1. Energy Function in
For simplicity, we assume that the atoms of He inside the cylinder possess only translational energy, that is, all atoms are in its ground electronic state. In general, one can assume that gas velocity distribution is Juttner distribution  (, pp. 289–293). For simplicity, one assumes that atoms are randomly distributed inside the container and that every atom has the same translational energy, that is, every He atom moves with the same speed (monokinetic approximation ), same modulus, but with different vectorial components , . In this approximation, . Constant is obtained by imposing where is the kinetic energy of a He atom and its energy function (Section 3).
Linear momentum (for simplicity we assume as movement of the atoms) and total energy for the th atom are Initial total linear momentum and initial total energy (energy function ) are given by The 4-vector initial energy function is then Energy function depends on temperature through temperature dependence on velocity . In (), total system energy (, Sec. 8.3) is, by definition, its energy function , sum of the kinetic energy of helium atoms , and .
For an ideal gas in an isothermal process, energy function remains constant, as well as the temperature, and also He atom speed. The 4-vector final energy function is then: with .
7.1.2. Work in
Forces on pistons are described as produced by the interaction of an electric charge with an electromagnetic field (Figure 7). Static electric positive and negative charges are located on right and left pistons, respectively. The whole device, gas plus pistons, is located inside the homogeneous electric field produced by a plane-parallel capacitor (, Chap. 13) with charge surface density .
In frame the capacitor is at rest. A horizontal electric field is created inside the capacitor. For this (uniform) electric field the potential 4-vector , where and are the vector potential and the scalar potential, respectively, is given by the contravariant 4-vector :
The electromagnetic -tensor —double contravariant—is given by (, Section 42) For the horizontal plane-parallel capacitor the -tensor is The mixed -tensor is obtained as
Initially, pistons are locked by a blocked mechanism (b in Figure 7). A laser (l in Figure 7) and a beam splitter (s in Figure 7) that is located just between the pistons, are used to release simultaneously both pistons. When the laser is turned on the split beams will arrive at the blocked mechanism of both left and right pistons and, at time , are unlocked allowing electric field charges interaction simultaneously on both L and R pistons .
When pistons are released, the Minkowski force on th piston is where is the electric charge on piston, is the -tensor electromagnetic field, and and are the th piston 4-vector velocity and proper time, respectively .
For the left piston (subindex L), displacement and velocity , 4-vector and 4-vector velocity , are respectively For the right piston (R), displacement and velocity , 4-vector and 4-vector velocity , are respectively
The Minkowski forces due to electromagnetic interaction are: With forces acting on pistons, and , 4-vectors work are, respectively, The total 4-vector work is then
7.1.3. Heat in
During (slow) gas compression, photons with frequency are emitted, with zero total linear momentum. Photons are emitted through the horizontal walls of the cylinder, photons are emitted in direction and photons in direction . Total linear momentum for this ensemble of photons is Total energy of these emitted photons , with , is its energy function : According to the principle of inertia of energy, these photons have inertia  . The 4-vector for heat (thermal radiation emitted from the point of view of the system) is then given by
7.1.4. First Law in
From first law , one obtains or The configurational work done on the gas provides the energy emitted as heat.
7.2. Compression in Frame
An observer in frame obtains the corresponding 4-vector , , , and by measuring different magnitudes—displacements , time intervals , velocities , forces , photon frequency , and so forth—in its own frame (Figure 8). With first law in frame expressed as , the Lorentz invariance of this equation assures that the same result as in , that is, , is obtained.
7.2.1. Energy Function in
Let there be one mol of He atoms moving inside the cylinder. In frame , with zero total linear momentum, velocities of atoms are measured simultaneously. During the same interval of time , displacement for the th atom is measured and its 3-vector velocity is given by . In order to ensure that in frame total linear momentum is zero, for every atom moving with velocity there must exist another atom moving with opposite velocity , such that and .
For every pair of opposite atoms, total momentum and total energy are easily obtained using the previous transformations: and total energy is given by For the total pairs of opposite atoms, one has This is the same result obtained from Lorentz transformation  on the 4-vector energy function in , given in (49), .
A similar description for 4-vector final energy function , with , is
7.2.2. Work in
By considering the locked-unlocked piston set, laser beam, splitter, blocked mechanism described previously in frame , it is evident that forces that are simultaneously applied in are not simultaneous in frame (Figure 8) .
To obtain the 4-vector in , relativistic transformations of time intervals, spatial displacements, and forces must be used.
In SA, 4-vector displacements are Spatial displacements and associated with forces FLA and FRA, respectively, are different in as well as time intervals: .
It is assumed that 4-vector forces acting on extended bodies are transformed in the same way as 4-vector forces acting on point particles [69, 70]. Force measured with respect to is given, in terms of force measured with respect to and the velocity of frame with respect to frame , by  For horizontal forces () and , Total impulse and work in frame are given by with This is the same result for obtained by using Lorentz transformation on 4-vector work in given by (60), .
The same result is obtained if one considers relativistic transformation of electric and magnetic fields (, Section 10.5).
The -tensor electromagnetic field in , , can be obtained as (Appendix A) (, p. 281) The electromagnetic field does not change in frame with respect to the field in (no magnetic field appears in ). The 4-vector work and can by obtained by applying its definition in :
In total impulse, is not zero and total work is The 4-vector work in is then a result previously obtained.
7.2.3. Heat in
From relativistic Doppler effect, frequency in frame for a photon emitted in frame with frequency and direction is given by The relativistic aberration effect  indicates that photon direction as measured in is
In frame , for a photon th emitted with angle there is another photon th emitted with angle (both with frequency , in order to assure zero total linear momentum for emitted photons). In frame photons are emitted with frequency , higher than frequency measured in (transverse Doppler effect). Photons in are emitted with angles larger than (in absolute value) (Figure 8).
In frame , total linear momentum and total energy are easy to obtain for this pair of opposite emitted photons (in ) and for the