Boltzmann's $H$-theorem, entropy and the strength of gravity in theories with a nonminimal coupling between matter and geometry

In this paper we demonstrate that, contrary to recent claims, Boltzmann's $H$-theorem does not necessarily hold in the context of theories of gravity with nonminimally coupled matter fields if the effective gravitational constant increases with time. We also suggest a possible link between the high entropy of the Universe and the weakness of gravity in the context of these theories.


I. INTRODUCTION
General relativity has been extremely successful in describing the gravitational interaction from cosmological scales down to the sub-millimetre scale [1]. However, this success comes at the price of having to postulate an exotic dark energy component (dubbed dark energy), violating the weak energy condition, in order to explain the observed acceleration of the Universe [2][3][4]. A similar component, albeit with a much larger energy density, is also required for primordial inflation to occur, providing a solution to some of the most fundamental problems of the standard cosmological model (see, for example [5] and references therein).
The search for extensions of general relativity which can more naturally explain the early and late dynamics of the Universe is an extremely active area of research. Broad classes of modified theories of gravity incorporate the possibility of a nonminimal coupling (NMC) between geometry and matter [6][7][8][9][10][11][12][13]. In these theories the energymomentum is not usually covariantly conserved, which results on the non-geodesic motion of point particles (the same applies to solitonic particles with fixed energy and structure, independently of the details of their composition [14,15]). The resulting forces are dependent on the individual velocity of the particles, and provide an extra contribution to their linear momentum evolution. This contribution is not only responsible for the tight cosmic microwave and primordial nucleosynthesis constraints on the strength of the NMC coupling between the gravitational and matter fields [16], but can also have profound implications to the evolution of the entropy and energy * Electronic address: pedro.avelino@astro.up.pt † Electronic address: rplazevedo@fc.up.pt densities of the Universe which we shall explore in the present paper.
In the late nineteenth century Boltzmann developed, almost single-handedly, the foundations of modern statistical mechanics. One of his major contributions, the Boltzmann's H-theorem, implies that, under generic conditions, the entropy of a closed system is a non-decreasing function of time [17]. In a recent work [18], the authors claimed that Boltzmann's H-theorem is preserved in the context of NMC theories of gravity. In this paper we shall demonstrate that this is not necessarily the case by adequately modeling the impact of the NMC to gravity on particle motion. We shall also explore the cosmological consequences of the contribution to the entropy and energy densities of the Universe resulting from the NMC between the gravitational and matter fields, suggesting a possible link between the high entropy of the Universe and the weakness of gravity.
The outline of this paper is as follows. In Sec. II we briefly describe the simplest possible theory allowing for a NMC coupling between the gravitational and the matter fields. The velocity dependent 4-force on point-particles associated with the NMC to gravity is computed in Sec. III. In Sec. IV we consider Boltzmann's equation in the context of NMC gravity and study its implications for the evolution of Boltzmann's H in a homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker Background. We also provide sufficient conditions for the violation Boltzmann's H theorem. In Sec. V we discuss the evolution of the entropy of the universe in NMC gravity, exploiting the relationship between Boltzmann's H theorem and the second law of thermodynamics. The connection between the evolution of the entropy of the Universe and the strength of gravity is explored in Sec. VI. Finally, we conclude in section VII.
Throughout this paper we use units such that 16πG = c = k B = 1, where G is Newton's gravitational constant, c is the value of the speed of light in vacuum and k B is the Boltzmann's constant. We also adopt the metric signature (−, +, +, +). The Einstein summation convention will be used whenever a greek or a Latin index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position. Greek and latin indices take the values 0, · · · , 3 and 1, ..., 3, respectively.

II. NONMINIMALLY COUPLED GRAVITY
Consider a theory with a NMC between gravity and matter described by the action where g is the determinant of the metric, L m is the Lagrangian of the matter fields, and F > 0 is a generic function of the Ricci scalar R -note that general relativity is recovered if F (R) = 1. Assuming a Levi-Civita connection, the equations of motion for the gravitational field are given by where g µν are the components of the metric, G µν = R µν − 1 2 g µν R and R µν are, respectively, the components of the Einstein and Ricci tensors, ∆ µν ≡ ∇ µ ∇ ν − g µν , ≡ ∇ µ ∇ µ , ∇ µ is the covariant derivative with respect to the coordinates x µ and a prime represents a derivative with respect to R. The energy-momentum tensor of the matter fields may be computed as Taking the covariant derivative of Eq. (2) and using the Bianchi identities one obtains Equation (6) implies that the energy momentum-tensor is, in general, not covariantly conserved. In the following section we shall discuss the implications of this result for the motion of point particles.

III. 4-FORCE ON POINT PARTICLES
In this section we shall compute the 4-force on point particles associated to the NMC to gravity. Consider the action of a single point particle with energy momentum tensor where δ 4 (x − ξ(τ )) denotes the four-dimensional Dirac delta function, ξ µ (τ ) represents the particle wordline, τ is the proper time, u α are the components of the particle 4-velocity (u µ u µ = −1) and m is proper particle mass. If one considers its trace T α α = T αβ g αβ and integrates over the whole of space-time, we obtain which can be immediately identified as the action for a single massive particle, and therefore implies that is the particle Lagrangian. Note that the single point particle Lagrangian given in Eq. (9) may be obtained from the action of solitonic particles, with fixed rest mass, structure, and negligible self-induced gravitational field, independently of their constitution [15] (see also [19,20]). The covariant derivative of the energy-momentum tensor may be written as Hence, Eq. (5) becomes where h µν = g µν + u µ u ν is the projection operator. The equation of motion of the point particle is then given by where is the velocity dependent 4-force on the particles associated to the NMC to gravity and a µ is the corresponding 4-acceleration (see [21] for an analogous calculation in the context of growing neutrino models where the neutrino mass is non-minimally coupled to a dark energy scalar field).

A. A note on the 4-acceleration of a fluid element
If the particles are part of a fluid, then the 4acceleration of the individual particles does not, in general, coincide with the 4-acceleration of the fluid element to which they belong. This may be shown explicitly by considering a perfect fluid with energy-momentum tensor where ρ and p are respectively the proper energy density and pressure of the perfect fluid and U µ are the components of its 4-velocity. The 4-acceleration equation of a perfect fluid element may be written as where [10] a µ where L m is now the Lagragian of the perfect fluid (and not of its individual particles). Hence, not only the 4velocity of the fluid U is in general very different from the velocity 4-velocity u of the individual particles at any point on the fluid, but the same is also true in the case of the 4-acceleration (this point has been overlooked in [18]). Also note that the on-shell Lagrangians L m = p and L m = −ρ found in Brown's work on the action functionals for relativistic perfect fluids [22] (see also Schutz [23]) do not in general account for the microscopic properties of the fluid. Hence, their use may be inadequate to describe gravitational and fluid dynamics in the context of NMC gravity. In fact, the appropriate Lagrangian in the case of an ideal gas is L m = T [14,15]).

IV. BOLTZMANN'S H-THEOREM
The collisionless Boltzmann's equation given by expresses the conservation of the number of particles in a six-dimensional phase space composed of the six positions and momentum coordinates ( r, p) of the particles in the absence of particle collisions -f (t, r, p)d 3 rd 3 p is the number of particles in the phase space volume d 3 r d 3 p.
In a flat homogeneous and isotropic universe, described by the Lemaitre-Friedmann-Robertson-Walker metric, the line element is given by where a(t) is the scale factor and q are comoving cartesian coordinates. In this case, the Ricci scalar is a function of cosmic time alone [R = R(t)] and the i0 components of the projection operator may be written as h i0 = γ 2 v i , where γ = u 0 = dt/dτ and v i = u i /γ are the components of the 3-velocity. Therefore, Eq. (13) implies that the 3force on the particles is given by In a homogeneous universe f is independent of r [f = f (t, p)], and the collisionless Boltzmann equation may be simply written as where we have used the result given in Eq. (19) in order to obtain the first equality in Eq. (20).

Let us consider Boltzmann's H defined by
Taking the derivative of H with respect to time one obtains where d 3 r = a 3 d 3 q = a 3 V q , where V q is a constant comoving volume, and N is the number of particles inside V q (N = d 3 r d 3 p f ). Using Eq. (20), the integral which appears in the last term of Eq. (22) may be written as Integrating once by parts one obtains Equation (22) finally becomes where we taken N to be constant in the last equalityin the absence of particle collisions this is always a good approximation for sufficiently large V q .
In general relativity F is equal to unity and, therefore, Boltzmann's H is a constant in the absence of particle collisions. However, Eq. (25) implies that this is longer true in the context of NMC theories of gravity. In this case, the evolution of Boltzmann's H is directly coupled to the evolution of the universe. Boltzmann's H may either grow or decay, depending on whether if F is a growing or a decaying function of time, respectively (Eq. (25) implies that H ∝ F −3 ). This provides an explicitly demonstration that the Boltzmann's H theorem -which states that dH/dt ≤ 0 -may not hold in the context of NMC theories of gravity.

V. ENTROPY
Consider a fluid of N point particles with Gibbs' and Boltzmann's entropies given respectively by where P N ( r 1 , p 1 , . . . , r N , p N , t) and P ( r, p, t) are, respectively, the N-particle probability density function in 6Ndimensional phase space and the single particle probability in 6-dimensional phase space. P and P N are related by These two definitions of the entropy have been shown to coincide only if or, equivalently, if particle correlations can be neglected, as happens for an ideal gas [17]. In the remainder of the paper we shall assume that this is the case, so that S = S B = S G (otherwise S G < S B [17]). We shall also consider a fixed comoving volume V q .
Close to equilibrium f ( r, p, t) = N P ( r, p, t) holds to an excellent approximation and, therefore Again, assuming that the particle number N is fixed, Eq. (25) implies that or, equivalently, that S ∝ F −3 . Hence, the entropy S in a homogeneous and isotropic universe may decrease with cosmic time, as long as F grows with time. This shows that the second law of thermodynamics does not generally hold in the context of modified theories of gravity with a NMC between the gravitational and the matter fields (see also [24,25]). Adding a two-particle elastic scattering term to Boltzmann's equation results, under the rather general assumption of molecular chaos, in a non-negative contribution to the entropy increase with cosmic time -this contribution vanishes for systems in thermodynamic equilibrium. If the particles are non-relativistic, and assuming thermodynamic equilibrium, f ( p, t) follows a Maxwell-Boltzmann distribution. In a homogeneous and isotropic universe with a NMC to gravity the non-relativistic equilibrium distribution is maintained even if particle collisions are switched off at some later time, since the velocity of the individual particles would simply evolve as v ∝ (aF ) −1 in the absence of collisions (see Eq. (19)). On the other hand, if the particles are relativistic and have a negligible chemical potential, as happens in the case of the photons, then the equilibrium particle number density n is a function of the temperature T alone n(T ) ∝ T 3 ∝ (aF ) −1 , thus implying that N ∝ na 3 ∝ F −3 is not conserved -the case of photons in thermodynamic equilibrium has been studied in [24,25] where it has been shown that S ∝ F −3/4 . Hence, the equilibrium distribution of the photons is not maintained after the Universe becomes transparent at a redshift z ∼ 10 3 , given that the number of photons of the cosmic background radiation is essentially conserved after that. Hence, a direct identification of Boltzmann's H with the entropy should not be made in this case. The requirement that the resulting spectral distortions be compatible with observations has been used to put stringent limits on the evolution of F after recombination [15,16].

VI. THE STRENGTH OF GRAVITY
Let us consider a scenario in which the function F is a constant both at early and late times. In both limits the equations of motion of the gravitational field are just the Einstein field equations albeit with very different values of the effect gravitational constant defined by then gravity was much stronger at early times than it is today. More interestingly, the high entropy of the Universe and the weakness of gravity would be interrelated in this scenario. In both cases the evolution of G eff could result in values of the Planck time and length (both proportional to G −1/2 eff[early] ) orders of magnitude away from our present estimates based on a constant G eff (the same applies for all the other relevant Planck units). This could have profound implications in a quantum gravity description of the physics of the very early universe.

VII. CONCLUSIONS
In this paper we have demonstrated that, contrary to recent claims, a violation of Boltzmann's H theorem may occur in modified gravity models in which matter and gravity are non-minimally coupled, provided the effective gravitational constant increases with cosmic time. In this derivation we highlighted three crucial conditions for the correct computation of the evolution of Boltzmann's H in NMC theories of gravity which have been neglected in previous work: i) the use of the appropriate Lagrangian in the modeling of point particles ii) the correct determination of the velocity-dependent 4-acceleration of individual point particles associated to the non-minimally coupling to gravity, which except in the case of dust, is generally different from the 4-acceleration of the fluid element to which they belong iii) the appropriate modeling, using Boltzmann equation, of the effect of the velocity-dependent 4-acceleration of the particles associated to the NMC coupling to gravity on the particle distribution in phase space. We have also briefly explored the connection between the entropy of the universe and the strength of gravity in theories with a nonminimal mattergeometry coupling, showing, in particular, that the high entropy of Universe and the weakness of gravity at the present time may be interrelated.