{'fraction_non_alphanumeric': 0.03430859254696643, 'fraction_numerical': 0.009608869725900832, 'mean_word_length': 4.358415841584159, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Boltzmann equation in special and general relativity

Relativistic field equations for a gas in special and general relativity are determined from the Boltzmann equation. The constitutive equations are obtained from the Chapman-Enskog methodology applied to a relativistic model equation proposed by Anderson and Witting. Two applications in general relativity are considered: one refers to a gas in a homogeneous and isotropic Universe where irreversible processes are present during its evolution; in the other it is analyzed a gas under the influence of a spherically symmetrical non-rotating and uncharged source of the gravitational field.


INTRODUCTION
We may say that the development of the relativistic kinetic theory goes back to 1911 when Jüttner [1] succeeded to derive an equilibrium distribution function for a relativistic gas, which in the non-relativistic limiting case reduces to the Maxwellian distribution function. The relativistic Bose-Einstein and Fermi-Dirac distribution functions were also derived by him in 1928 [2]. At the beginning of the 1940s the covariant formulation of the Boltzmann equation was proposed by Lichnerowicz and Marrot [3], while the determination of the transport coefficients from the Boltzmann equation by using the Chapman-Enskog method was obtained by Israel [4] and Kelly [5] in the 1960s. The general relativistic formulation of the Boltzmann equation goes back also to the sixties of the last century when Chernikov [6] analyzed a relativistic gas in a gravitational field.
The aim of this work is to analyze the Boltzmann equation within the framework of special and general relativity and to derive the corresponding macroscopic field equations.
First the Boltzmann equation in special relativity is analyzed and the balance equations for the particle four-flow and energy-momentum tensor are obtained. As in the case of non-relativistic fluid dynamics these equations are useful to describe a relativistic fluid with five scalar fields of particle number density, four-velocity and temperature. The constitutive equations for the non-equilibrium pressure, pressure deviator and heat flux, as well as the corresponding transport coefficients of bulk and shear viscosities and thermal conductivity, are determined by using the Chapman-Enskog methodology applied to the Anderson and Witting model of the Boltzmann equation [7]. From the knowledge of the constitutive equations, the system of balance equations turns out to become a system of field equations for the basic fields and problems related to relativistic fluids can be analyzed.
Next, relativistic fluids in general relativity are analyzed within the framework of Boltzmann equation. In the first application a gas in a homogeneous and isotropic Universe is considered where irreversible processes are present during its evolution. The irreversible processes are characterized by a non-equilibrium pressure which is determined from the Anderson and Witting model by using the Chapman-Enskog methodology. Another application refers to a gas -which is not the source of the gravitational field -under the influence of a Schwarzschild metric, which is the solution of Einstein's field equations for a spherically symmetrical non-rotating and uncharged source of the gravitational field. For this case, the expression for the momentum density balance equation is obtained in the non-relativistic limiting case and for weak gravitational fields.
(x α ) = (x 0 = ct, x) and by the momentum four-vector (p α ) = (p 0 , p), where c is the speed of light. The momentum four-vector has a constant length mc which implies that p 0 can be expressed in terms of p by p 0 = |p| 2 + m 2 c 2 . The state of the relativistic gas -in the phase space spanned by the space-time and momentum coordinates -is characterized by the one-particle distribution function f (x α , p α ) = f (x, p,t), where the quantity f (x, p,t)d 3 3 gives the number of particles at time t in the volume element d 3 x about x and with momenta in a range d 3 p about p.
In relativity we have to take into account the behavior of the quantities under Lorentz transformations. The volume element d 3 xd 3 p is an invariant under Lorentz transformations, i.e., d 3 xd 3 p = d 3 x ′ d 3 p ′ . Furthermore, the number of particles in a volume element is a scalar invariant, since all observers will count the same number of particles. Hence, we may conclude that the one-particle distribution function f (x, p,t) is also a scalar invariant. It can be proved that the volume element d 3 p is not an invariant but only the ratio d 3 p ′ /p ′ 0 = d 3 p/p 0 . The evolution of the one-particle distribution function in the phase space is ruled by the Boltzmann equation, which in the absence of external forces reads In the above equation we have introduced the abbreviations f ′ where p and p * denote the momenta of two particles before a binary collision and p ′ and p ′ * are the corresponding momenta after collision. The pre and post collisional momentum four-vectors are connected by the energy-momentum conservation law p α + p α is the invariant flux, which in the non-relativistic limiting case it is proportional to the modulus of the relative velocity. The differential cross-section and the element of solid angle that characterize a binary collision are denoted by σ and dΩ, respectively.
The multiplication of the Boltzmann equation (1) by an arbitrary function ψ(p β ) and integration of the resulting equation with respect to the invariant d 3 p/p 0 , yields the following general equation of transfer which is only a function of the space-time coordinates. As in the non-relativistic case the right-hand side of the above equation is obtained by evoking the symmetry properties of the collision operator. When the underlined term in (2) vanishes, ψ becomes a summational invariant. The necessary and sufficient condition to be a summation invariant is that ψ must be given by ψ = A + B α p α , where A is an arbitrary scalar and B α an arbitrary four-vector that do not depend on the momentum four-vector p α . The macroscopic description of a relativistic gas may be characterized by the two first moments of the distribution function, namely which are known as the particle four-flow and the energy-momentum tensor, respectively. Their balance equations are obtained from the transfer equation (2) by choosing ψ = c and ψ = cp β , yielding respectively. They represent the conservation laws of the particle four-flow and of the energy-momentum tensor.
In the analysis of relativistic fluids it is usual to decompose the particle four-flow and the energy-momentum tensor in terms of quantities that appear in non-relativistic fluid dynamics, like particle number density n, energy per particle e, hydrostatic pressure p, non-equilibrium pressure ϖ, pressure deviator P αβ (the traceless part of the pressure tensor) and heat flux q α . There exist two decompositions for the particle four-flow and energy-momentum tensor as functions of the above mentioned quantities and they are known as the decompositions of Eckart and of Landau-Lifshitz (see e.g. [8]). Here we shall adopt the Landau-Lifshitz decomposition, namely The above decomposition is written in terms of the four-velocity U α (such that U α U α = c 2 ) and of the projector ∆ αβ = η αβ −U α U β /c 2 (such that ∆ αβ U β = 0). Furthermore, we have introduced the enthalpy per particle h = e + p/n.
As in the non-relativistic case, the equilibrium is characterized by the condition f ′ * f ′ = f * f which renders a vanishing collision term of the Boltzmann equation (1). This last relationship implies that ln f ′ * + ln f * = ln f * + ln f and ln f represents a summational invariant whose expression is given by ln f = A + B α p α . The terms A and B α are determined from the equilibrium values of the particle four-flow and energy-momentum tensor and it follows that the equilibrium distribution function is represented by the Maxwell-Jüttner distribution function denoting the modified Bessel function of second kind. Note that the distribution function depends on ζ = mc 2 /kT which represents the ratio of the rest energy mc 2 of a particle and the thermal energy of the gas kT , where k is the Boltzmann constant and T the temperature of the gas. The limiting case ζ ≫ 1 represents a non-relativistic regime of the gas, while ζ ≪ 1 its ultra-relativistic regime. For example, for an electron of rest mass m ≈ 9.1 × 10 −31 kg we have that ζ = 5.9 × 10 9 /T , indicating that only for very high temperatures the electron might be considered in the relativistic regime.
If we are interested in the description of the relativistic gas as a five-field scalar theory characterized by the fields of particle number density n, four-velocity U α and temperature T , we have to determine the constitutive equations for the energy per particle e, hydrostatic pressure p, non-equilibrium pressure ϖ, pressure deviator P αβ and heat flux q α as functions of (n,U α , T ). The determination of the constitutive equations for the energy per particle and hydrostatic pressure follows from the insertion of the Maxwell-Jüttner distribution function (6) into the definition of the energymomentum tensor (3) 2 and the integration of the resulting equation, yielding Hence, the expression for the hydrostatic pressure is the same as the one for a non-relativistic gas. In the non-relativistic and ultra-relativistic limiting cases the energy per particle becomes e = mc 2 + 3 2 kT 1 + 5 4ζ + . . . , e = 3kT + . . . , respectively. Note that the first term in the non-relativistic limiting case (8) 1 is the rest energy of a particle mc 2 , the second one 3kT /2 represents the classical value of the energy density, while the third one is a relativistic correction. By considering only the first two terms in this expansion it follows the well-known relationship between the pressure and the energy density of a non-relativistic gas, namely, p = 2n(e − mc 2 )/3. Furthermore, in the ultra-relativistic limiting case the following relationship p = ne/3 holds.

Model equation and Chapman-Enskog method
The constitutive equations for the non-equilibrium pressure ϖ, pressure deviator P αβ and heat flux q α can be obtained from the Boltzmann equation by using the Chapman-Enskog methodology. Here we shall use the simple model equation proposed by Anderson and Witting [7], which is a generalization of the BGK model for a relativistic gas. The model equation for the Boltzmann equation in this case is written as where τ is a characteristic time of order of the mean free time. Above we have introduced the differential operators D and ∇ α through the decomposition ∂ α = (U α D/c 2 + ∇ α ). The differential operators D and ∇ α reduce to a partial time derivative and a spatial gradient in a Lorentz rest frame, respectively. For processes close to equilibrium we may write f = f (0) (1 + φ ), where the deviation φ from the Maxwell-Jüttner distribution function is considered as a small quantity. Now by using the Chapman-Enskog methodology we insert the equilibrium distribution function on the left-hand side of (9) and the representation f = f (0) (1 + φ ) into its right-hand side and get Next we perform the derivatives which occur in the left-hand side of above equation and eliminate the derivatives Dn, DT and DU α by using the balance equations for the particle number density, energy density and momentum density of an Eulerian relativistic gas (where ϖ = 0, P αβ = 0 and q α = 0), namely, Hence, the deviation φ from the Maxwell-Jüttner distribution function can be determined, yielding In the above equation we have introduced the heat capacity per particle at constant volume c v and the traceless part of the velocity gradient ∇ α U β defined by Note that the deviation φ is a function of the thermodynamic forces which are underlined in (12): the divergence of velocity ∇ α U α , the traceless part of the velocity gradient ∇ α U β and a combination of the temperature and pressure gradients ∇ α T − T nh ∇ α p . The contribution of the pressure gradient refers only to a relativistic correction. The constitutive equations for the non-equilibrium pressure, pressure deviator and heat flux can be obtained now from the insertion of f = f (0) (1 + φ ) together with (12) into the definitions of the particle four-flow and energymomentum tensor (3) and integration of the resulting equations. Hence it follows The above equations represent the laws of Navier-Stokes and Fourier and we may identify the scalar coefficients η, µ and λ as the coefficients of bulk viscosity, shear viscosity and thermal conductivity, respectively. Their expressions read where Ki 1 denotes the following integral for the modified Bessel function Ki 1 (ζ ) = ∞ 0 e −ζ cosht dt/ cosht. In the non-relativistic limiting case (ζ ≫ 1) the transport coefficients become while in the ultra-relativistic limiting case (ζ ≪ 1) they read In the non-relativistic limiting case the ratio λ /µ ≈ 5k/2m has the same value as that of the BGK model. Furthermore, we note that the bulk viscosity coefficient is important only in the relativistic case, since it vanishes in the nonrelativistic and in the ultra-relativistic limiting cases. Once the constitutive equations (7) and (14) are known, the balance equations (4) become a system of field equations for the five scalar fields of particle number density n, four-velocity U α and temperature T and problems concerning relativistic fluids in special relativity could be analyzed.

BOLTZMANN EQUATION IN GRAVITATIONAL FIELDS
For the analysis of a relativistic gas in the presence of a gravitational field we have to consider a Riemannian space characterized by the metric tensor g µν . Here the invariant elements in the phase space are is the determinant of the metric tensor. Moreover, the constant length of the momentum four-vector g µν p µ p ν = m 2 c 2 implies that The Boltzmann equation in gravitational fields is written as where Γ σ µν = g σ τ ∂ µ g µτ + ∂ ν g µτ − ∂ τ g µν /2 are Christoffel symbols. Following the same methodology of the special relativistic case, we may define the two first moments of the distribution function and obtain their respective balance equations from the Boltzmann equation (20), which read In the above equations the semicolon refers to a covariant derivative.
For the solution of a problem concerning a relativistic gas in the presence of a gravitational field, it is necessary to determine the ten components of the metric tensor g µν , which is achieved by using Einstein's field equations represent the Ricci tensor and the curvature scalar, respectively. Moreover, G denotes the gravitational constant. In the right-hand side of Einstein's field equations the energy-momentum tensor represents the sources of the gravitational field, while its left-hand side corresponds to the geometry of the curved space-time. The conservation of the energy-momentum tensor T µν ;µ = 0 follows also from Einstein's field equation thanks to the Bianchi identity R µν − 1 2 Rg µν ;µ = 0.

Relativistic gas in a Robertson-Walker metric
The search for cosmological solutions of our Universe are founded on Einstein's field equations and on the hypothesis that the Universe is spatially homogeneous and isotropic at large scales, i.e., of order of 100 Mpc ≈ 3.1 × 10 24 m ≈ 3.26 × 10 8 light-years. The isotropy and homogeneity assumptions imply that the line element ds is given by the Friedmann-Lamaître-Robertson-Walker (FLRW) metric, which for a spatially flat Universe reads ds 2 = c 2 dt 2 − a(t) 2 (dx) 2 + (dy) 2 + (dz) 2 . This metric is a function of only one unknown parameter which is the cosmic scale factor a(t). In addition from the hypothesis of homogeneity and isotropy of the space-time, the pressure deviator P αβ and of the heat flux q α must be absent from the energy-momentum tensor, and in this case it reduces to T µ ν = diag(en, −p − ϖ, −p − ϖ, −p − ϖ). For the FLRW metric the non-vanishing components of the metric tensor and of the Christoffel symbols are given by g 00 = 1, g i j = −a 2 δ i j , g = −a 6 , Γ 0 i j =ȧa δ i j , Γ i 0 j = δ i jȧ /a, where the dot denotes the derivative with respect to x 0 = ct.