In this paper we show that the on-shell Lagrangian of a perfect fluid depends on microscopic properties of the fluid, giving specific examples of perfect fluids with different on-shell Lagrangians but with the same energy-momentum tensor. We demonstrate that if the fluid is constituted by localized concentrations of energy with fixed rest mass and structure (solitons) then the average on-shell Lagrangian of a perfect fluid is given by ${\mathcal L}_m=T$, where $T$ is the trace of the energy-momentum tensor. We show that our results have profound implications for theories of gravity where the matter Lagrangian appears explicitly in the equations of motion of the gravitational and matter fields, potentially leading to observable deviations from a nearly perfect cosmic microwave background black body spectrum: $n$-type spectral distortions, affecting the normalization of the spectral energy density. Finally, we put stringent constraints on $f(R,{\mathcal L}_m)$ theories of gravity using the COBE-FIRAS measurement of the spectral radiance of the cosmic microwave background.
In this work we study how nonminimally coupled theories of gravity modify the usual Friedmann equation, and develop two methods to treat these. The ambiguity in the form of the Lagrangian density of a perfect fluid is emphasized, and the impact of different dominant matter species is assessed. The Cosmological Constant problem is also discussed.
We study the Vainshtein screening mechanism in Horndeski theories in the presence of a scalar field $ϕ$ nonminimally and kinetically coupled to ordinary matter field. A general interacting Lagrangian describing this coupling is characterized by energy transfer $f_1$ and momentum exchange $f_2$. For a spherically symmetric configurations on top of the cosmological background, we investigate the static perturbations in linear and nonlinear regimes with respect to the scalar field perturbation. In the former regime, the parametrized post-Newtonian parameter generally deviates from unity as long a…
In the standard model of cosmology, the background evolution of the Universe can in general be adequately described by general relativity and a uniform and isotropic metric minimally coupled with a collection of perfect fluids. These fluids are usually described by their energy-momentum tensor, which can be derived from the fluid's Lagrangian density. Under general relativity, the Lagrangian density is only relevant to the extent that it results in the correct energy-momentum tensor for a specific perfect fluid. This is not the case in theories that feature a nonminimal coupling (NMC) between …
In conventional Maxwell--Lorentz electrodynamics, the propagation of light is influenced by the metric, not, however, by the possible presence of a torsion T. Still the light can feel torsion if the latter is coupled nonminimally to the electromagnetic field F by means of a supplementary Lagrangian of the type l^2 T^2 F^2 (l = coupling constant). Recently Preuss suggested a specific nonminimal term of this nature. We evaluate the spacetime relation of Preuss in the background of a general O(3)-symmetric torsion field and prove by specifying the optical metric of spacetime that this can yield b…
In three dimensions, we consider the Einstein-Maxwell Lagrangian dressed by a nonminimally coupled scalar field in New Massive Gravity. For this theory, we provide two families of electrically charged Lifshitz black holes where their metric functions depend only on an integration constant. We calculate their masses using the quasilocal approach, as well as their entropy and electric charge. These charged configurations are interpreted as extremal in the sense that the mass vanishes identically while the entropy and electric charge are non zero thermodynamic quantities. Using these examples, we…
We provide a general framework for studying the dark energy cosmology in which a scalar field $ϕ$ is nonminimally and kinetically coupled to Cold Dark Matter (CDM). The scalar-graviton sector is described by the action of Horndeski theories with the speed of gravitational waves equivalent to that of light, whereas CDM is treated as a perfect fluid given by a Schutz-Sorkin action. We consider two interacting Lagrangians of the forms $f_1(ϕ,X)ρ_c (n_c)$ and $f_2 (n_c, ϕ,X) J_c^μ \partial_μϕ$, where $X=-\partial^μ ϕ\partial_μ ϕ/2$, $ρ_c$ and $n_c$ are the energy density and number density of CDM …
In arbitrary dimension, we consider a theory described by the most general quadratic curvature corrections of Einstein gravity together with a self-interacting nonminimally coupled scalar field. This theory is shown to admit five different families of Lifshitz black holes dressed with a nontrivial scalar field. The entropy of these configurations is microscopically computed by means of a higher-dimensional anisotropic Cardy-like formula where the role of the ground state is played by the soliton obtained through a double analytic continuation. This involves to calculate the correct expressions…
In this paper, a new generalised gravity-matter coupled theory of gravity is presented. This theory is constructed by assuming an action with an arbitrary function $f(T,B,L_m)$ which depends on the scalar torsion $T$, the boundary term $B=\nabla_μT^μ$ and the matter Lagrangian $L_m$. Since the function depends on $B$ which appears in $R=-T+B$, it is possible to also reproduce curvature-matter coupled models such as $f(R,L_m)$ gravity. Additionally, the full theory also contains some interesting new teleparallel gravity-matter coupled theories of gravities such as $f(T,L_m)$ or $C_1 T+ f(B,L_m)…
We extend the analysis of Chiba, Smith and Erickcek \cite{CSE} of Solar System constraints on $f(R)$ gravity to a class of nonminimally coupled (NMC) theories of gravity. These generalize $f(R)$ theories by replacing the action functional of General Relativity (GR) with a more general form involving two functions $f^1(R)$ and $f^2(R)$ of the Ricci scalar curvature $R$. While the function $f^1(R)$ is a nonlinear term in the action, analogous to $f(R)$ gravity, the function $f^2(R)$ yields a NMC between the matter Lagrangian density $\LL_m$ and the scalar curvature. The developed method allows f…
We revisit the issue of the correct Lagrangian description of a perfect fluid (pressure versus minus energy density) in relation with modified gravity theories in which galactic luminous matter couples nonminimally to the Ricci scalar. These Lagrangians are only equivalent when the fluid couples minimally to gravity and not otherwise; in the presence of nonminimal coupling they give rise to two distinct theories with different predictions.
We perform a phase space analysis of a generalized modified gravity theory with nonminimally coupling between geometry and matter. We apply the dynamical system approach to this generalized model and find that in the cosmological context, different choices of Lagrangian density will apparently result in different phases of the Universe. By carefully choosing the variables, we prove that there is an attractor solution to describe the late time accelerating universe when the modified gravity is chosen in a simple power-law form of the curvature scalar. We further examine the temperature evolutio…
In the context of general relativity, both energy and linear momentum constraints lead to the same equation for the evolution of the speed of free localized particles with fixed proper mass and structure in a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker universe. In this paper we extend this result by considering the dynamics of particles and fluids in the context of theories of gravity nonminimally coupled to matter. We show that the equation for the evolution of the linear momentum of the particles may be obtained irrespective of any prior assumptions regarding the form of t…
We present a novel theory of gravity, namely, an extension of symmetric teleparallel gravity. This is done by introducing a new class of theories where the nonmetricity $Q$ is coupled nonminimally to the matter Lagrangian. This nonminimal coupling entails the nonconservation of the energy-momentum tensor, and consequently the appearance of an extra force. We also present several cosmological applications.
We consider cosmological dynamics of nonminimally coupled scalar field in the scalar-torsion gravity in the presence of a hydrodynamical matter. Potential of the scalar field have been chosen as power-law with negative index, this type of potentials is usually used in quintessence scenarios. We identify several asymptotic regimes, including de Sitter, kinetic dominance, kinetic tracker and tracker solution and study conditions for their existence and stability. We show that for each combination of coupling constant and potential power index one of regimes studied in the present paper is stable…
We apply a recently developed 2+1+1 decomposition of spacetime, based on a nonorthogonal double foliation for the study of spherically symmetric, static black hole solutions of Horndeski scalar-tensor theory. Our discussion proceeds in an effective field theory (EFT) of modified gravity approach, with the action depending on metric and embedding scalars adapted to the nonorthogonal 2+1+1 decomposition. We prove that the most generic class of Horndeski Lagrangians compatible with observations can be expressed in this EFT form. By studying the first order perturbation of the EFT action we derive…
A Lagrangian formulation of perfect fluid due to a noncanonical three-form field is investigated. The thermodynamic quantities such as energy density, pressure and the four velocity are obtained and then analyzed by comparing with the k-essence scalar field. The nonrelativistic matter due to the generalized three-form field with the equation of state parameter being zero is realized while it might not be possible for the k-essence scalar field. We also found that nonadiabatic pressure perturbations can be possibly generated. The fluid dynamics of the perfect fluid due to the three-form field c…
The theory of perfect fluids is reconsidered from the point of view of a covariant Lagrangian theory. It has been shown that the Euler-Lagrange equations for a perfect fluid could be found in spaces with affine connections and metrics from an unconstrained variational principle by the use of the method of Lagrangians with covariant derivatives (MLCD) and additional conditions for reparametrizations of the proper time of the mass elements (particles) of the perfect fluid. The last conditions are not related to the variational principle and are not considered as constraints used in the process o…
In this paper we give a simple proof that when the particle number is conserved, the Lagrangian of a barotropic perfect fluid is $\mathcal{L}_m=-ρ[c^2 +\int P(ρ)/ρ^2 dρ]$, where $ρ$ is the \textit{rest mass} density and $P(ρ)$ is the pressure. To prove this result nor additional fields neither Lagrange multipliers are needed. Besides, the result is applicable to a wide range of theories of gravitation. The only assumptions used in the derivation are: 1) the matter part of the Lagrangian does not depend on the derivatives of the metric, and 2) the particle number of the fluid is conserved ($\na…
We present a novel theory of gravity by considering an extension of symmetric teleparallel gravity. This is done by introducing, in the framework of the metric-affine formalism, a new class of theories where the nonmetricity $Q$ is nonminimally coupled to the matter Lagrangian. More specifically, we consider a Lagrangian of the form $L \sim f_1(Q) + f_2(Q) L_M$, where $f_1$ and $f_2$ are generic functions of $Q$, and $L_M$ is the matter Lagrangian. This nonminimal coupling entails the nonconservation of the energy-momentum tensor, and consequently the appearance of an extra force. The motivati…
In this work, we consider different forms of relativistic perfect fluid Lagrangian densities, that yield the same gravitational field equations in General Relativity. A particularly intriguing example is the case with couplings of the form $[1+f_2(R)]{\cal L}_m$, where $R$ is the scalar curvature, which induces an extra force that depends on the form of the Lagrangian density. It has been found that, considering the Lagrangian density ${\cal L}_m = p$, where $p$ is the pressure, the extra-force vanishes. We argue that this is not the unique choice for the matter Lagrangian density, and that mo…
We develop a field theory description of non-dissipative string fluids and construct an explicit mapping between field theory degrees of freedom and hydrodynamic variables. The theory generalizes both a perfect particle fluid and pressureless string fluid to what we call a perfect string fluid. Ideal magnetohydrodynamics is shown to be an example of the perfect string fluid whose equations of motion can be obtained from a particular choice of the Lagrangian. The Lagrangian framework suggests a straightforward extension of the perfect string fluid to more general anisotropic fluids describing h…
Cosmological models with an SU(2) Yang-Mills field are studied. For a specific model with a minimally coupled Yang-Mills Lagrangian, which includes an arbitrary function of the second-order term and a fourth-order term, a corresponding reconstruction program is proposed. It is shown that the model with minimal coupling has no de Sitter solutions, for any nontrivial function of the second-order term. To get de Sitter solutions, a gravitational model with nonminimally coupled Yang-Mills fields is then investigated. It is shown that the model with non-minimal coupling has in fact a de Sitter solu…
We consider a cosmological scenario where the dark sector is described by two perfect fluids that interact through a velocity-dependent coupling. This coupling gives rise to an interaction in the dark sector driven by the relative velocity of the components, thus making the background evolution oblivious to the interaction and only the perturbed Euler equations are affected at first order. We obtain the equations governing this system with the Schutz-Sorkin Lagrangian formulation for perfect fluids and derive the corresponding stability conditions to avoid ghosts and Laplacian instabilities. A…
We study Friedmann--Robertson--Walker models with a perfect fluid matter source and a scalar field nonminimally coupled to matter. We prove that a general class of bounded from above potentials which fall to minus infinity as the field goes to minus infinity, forces the Hubble function to diverge to $-\infty$ in a finite time. This finite-time singularity theorem is true for arbitrary coupling coefficient, provided that it is a bounded function of the scalar field.
The first law of black hole mechanics (in the form derived by Wald), is expressed in terms of integrals over surfaces, at the horizon and spatial infinity, of a stationary, axisymmetric black hole, in a diffeomorphism invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter and Hawking for an Einstein-perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surfaces. When applied to the Einstein-perfect fluid system, however, Wald's methods yield restricted results. The…
The clebsch potential approach to fluid lagrangians is developed in order to establish contact with other approaches to fluids. Three variants of the perfect fluid approach are looked at. The first is an explicit linear lagrangian constructed directly from the clebsch potentials, this has fixed equation of state and explicit expression for the pressure but is less general than a perfect fluid. The second is lagrangians more general than that of a perfect fluid which are constructed from higher powers of the comoving vector. The third is lagrangians depending on two vector fields which can repr…
It is widely acknowledged that, for formal purposes, a minimally coupled scalar field is equivalent to an effective perfect fluid with equation of state determined by the scalar potential. This correspondence is not complete because the Lagrangian densities P and -rho, which are equivalent for a perfect fluid, are not equivalent for a minimally coupled scalar field. The exchange between these two Lagrangian densities amounts to exchanging a canonical scalar field with a phantom scalar field.
In the Rastall gravity a non-minimal coupling between geometry and matter fields is considered. Then the usual energy-momentum tensor conservation law is not valid. Here a Lagrangian formalism is proposed to the Rastall theory of gravity. The Gödel-type universe is studied in this gravitational model. Then it is studied whether this theory permits causality violation. The field equations do not exclude solutions with a breakdown of causality for a perfect fluid as matter content. In this case, an expression for the critical radius (beyond which the causality is violated) is determined. In addi…
We investigate the coupling of matter to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form $L_m\tilde{R}^2$, where $L_m$ is the ordinary matter Lagrangian, and $\tilde{R}$ is the Weyl scalar. The coupling explicitly satisfies the conformal invariance of the theory. By expressing $\tilde{R}^2$ with the help of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized, leading in the Riemann space to a conformally invariant $f\left(R,L_m\right)$ type theory, with the matter Lagrangian nonminimally coupled to the Ricci scalar.…
We prove that in Robertson-Walker space-times (and in generalized Robertson-Walker spacetimes of dimension greater than 3 with divergence-free Weyl tensor) all higher-order gravitational corrections of the Hilbert-Einstein Lagrangian density $F(R,\square R, ... , \square^k R)$ have the form of perfect fluids in the field equations. This statement definitively allows to deal with dark energy fluids as curvature effects.
We discuss conservation laws for gravity theories invariant under general coordinate and local Lorentz transformations. We demonstrate the possibility to formulate these conservation laws in many covariant and noncovariant(ly looking) ways. An interesting mathematical fact underlies such a diversity: there is a certain ambiguity in a definition of the (Lorentz-) covariant generalization of the usual Lie derivative. Using this freedom, we develop a general approach to construction of invariant conserved currents generated by an arbitrary vector field on the spacetime. This is done in any dimens…
In this proceeding, we review modified theories of gravity with a curvature-matter coupling between an arbitrary function of the scalar curvature and the Lagrangian density of matter. This explicit nonminimal coupling induces a non-vanishing covariant derivative of the energy-momentum tensor, that implies non-geodesic motion and consequently leads to the appearance of an extra force. Here, we explore the physical and cosmological implications of the nonconservation of the energy-momentum tensor by using the formalism of irreversible thermodynamics of open systems in the presence of matter crea…
We consider a model where particles are described as localized concentrations of energy, with fixed rest mass and structure, which are not significantly affected by their self-induced gravitational field. We show that the volume average of the on-shell matter Lagrangian ${\mathcal L_m}$ describing such particles, in the proper frame, is equal to the volume average of the trace $T$ of the energy-momentum tensor in the same frame, independently of the particle's structure and constitution. Since both ${\mathcal L_m}$ and $T$ are scalars, and thus independent of the reference frame, this result i…
We study the late time evolution of negatively curved Friedmann--Le\-ma\^ıtre--Robert\-son--Walker (FLRW) models with a perfect fluid matter source and a scalar field nonminimally coupled to matter. Since, under mild assumptions on the potential $V$, it is already known that equilibria corresponding to non-negative local minima for $V$ are asymptotically stable, we classify all cases where one of the energy components eventually dominates. In particular for nondegenerate minima with zero critical value, we rigorously prove that if $γ$, the parameter of the equation of state is larger than $2/3…
The connection is established between two different action principles for perfect fluids in the context of general relativity. For one of these actions, $S$, the fluid four--velocity is expressed as a sum of products of scalar fields and their gradients (the velocity--potential representation). For the other action, ${\bar S}$, the fluid four--velocity is proportional to the totally antisymmetric product of gradients of the fluid Lagrangian coordinates. The relationship between $S$ and ${\bar S}$ is established by expressing $S$ in Hamiltonian form and identifying certain canonical coordinates…
Relativistic hydrodynamics of an isentropic fluid in a gravitational field is considered as the particular example from the family of Lagrangian hydrodynamic-type systems which possess an infinite set of integrals of motion due to the symmetry of Lagrangian with respect to relabeling of fluid particle labels. Flows with fixed topology of the vorticity are investigated in quasi-static regime, when deviations of the space-time metric and the density of fluid from the corresponding equilibrium configuration are negligibly small. On the base of the variational principle for frozen-in vortex lines …
In this short note, we obtain the necessary and sufficient condition for a class of non-canonical single scalar field models to be exactly equivalent to barotropic perfect fluids, under the assumption of an irrotational fluid flow. An immediate consequence of this result is that the non-adiabatic pressure perturbation in this class of scalar field systems vanishes exactly at all orders in perturbation theory and on all scales. The Lagrangian for this general class of scalar field models depends on both the kinetic term and the value of the field. However, after a field redefinition, it can be …
We reformulate the standard Lagrangian formalism to a reparameterisation invariant Lagrangian formalism by means of Finsler and Kawaguchi geometry. In our formalism, various types of symmetries that appears in theories of physics are expressed geometrically by symmetries of Finsler (Kawaguchi) metric, and the conservation law of energy-momentum is a part of Euler-Lagrange equations. The application to scalar field, Dirac field, electromagnetic field and general relativity coupled to perfect fluid (added: ver.3) are discussed. By this formalism, we try to propose an alternative definition of en…
We derive a condition on the Lagrangian density describing a generic, single, non-canonical scalar field, by demanding that the intrinsic, non-adiabatic pressure perturbation associated with the scalar field vanishes identically. Based on the analogy with perfect fluids, we refer to such fields as perfect scalar fields. It is common knowledge that models that depend only on the kinetic energy of the scalar field (often referred to as pure kinetic models) possess no non-adiabatic pressure perturbation. While we are able to construct models that seemingly depend on the scalar field and also do n…
The variational theory of the perfect fluid with an intrinsic hypermomentum is developed. The Lagrangian density of such fluid is stated and the equations of motion of the fluid and the evolution equation of the hypermomentum tensor are derived. The expressions of the matter currents of the fluid (the metric stress-energy 4-form, the canonical energy-momentum 3-form and the hypermomentum 3-form) are obtained.
We show that in modified $f(R)$ type gravity models with non-minimal coupling between matter and geometry, both the matter Lagrangian, and the energy-momentum tensor, are completely and uniquely determined by the form of the coupling. This result is obtained by using the variational formulation for the derivation of the equations of motion in the modified gravity models with geometry-matter coupling, and the Newtonian limit for a fluid obeying a barotropic equation of state. The corresponding energy-momentum tensor of the matter in modified gravity models with non-minimal coupling is more gene…
Recently a Lagrangian formulation for Rastall Gravity (RG) has been proposed in the framework of f(R, T) gravity. In the present work we obtain Lagrangian formulation for the standard Rastall theory assuming the matter content is a perfect fluid with linear Equation of State (EoS). We therefore find a relation between the coupling constant of the Lagrangian, the Rastall gravitational coupling constant and the EoS parameter. We also propose a Lagrangian for the Generalized Rastall Gravity (GRG). In this case the Rastall parameter which is a constant is replaced by a variable one. More exactly, …
The variational theory of the perfect hypermomentum fluid is developed. The new type of the generalized Frenkel condition is considered. The Lagrangian density of such fluid is stated, and the equations of motion of the fluid and the Weyssenhoff-type evolution equation of the hypermomentum tensor are derived. The expressions of the matter currents of the fluid (the canonical energy-momentum 3-form, the metric stress-energy 4-form and the hypermomentum 3-form) are obtained. The Euler-type hydrodynamic equation of motion of the perfect hypermomentum fluid is derived. It is proved that the motion…
We consider the generalized set of theories of gravitation whose Lagrangians contain the term $R^{2}$ : $L=\sqrt{-g}(R+βR^{2})$. Inserting the RW metric with an imposed non-singular and inflationary behaviour of the scale factor $a(t)$, and using a arbitrary perfect fluid, we study the properties of $ρ$ and $p$ in this context. By requiring the positivity of the energy density, as well as real and finite velocity of sound, we can obtain the range of values of $β$ that ensure the inflationary behaviour and absence of singularity.
In this paper, we investigated the theoretical and cosmological effects of the matter Lagrangian degeneracy in an extension of the Symmetric Teleparallel Equivalent of General Relativity, denoted as $f (Q, T )$ gravity. This degeneracy comes from the fact that both $\mathcal{L}_m = p$ and $\mathcal{L}_m = -ρ$ can give rise to the stress-energy tensor of a perfect fluid. The $f(Q,T)$ equations depend on the form of the matter Lagrangian and hence they also have a degeneracy that has influence on the density evolution of dust matter and radiation, on the form of the generalized Friedmann equatio…
For perfect fluids with equation of state $ρ= ρ(n,s)$, Brown gave an action principle depending only on their Lagrange coordinates $α^i(x)$ without Clebsch potentials. After a reformulation on arbitrary spacelike hypersurfaces in Minkowski spacetime, the Wigner-covariant rest-frame instant form of these perfect fluids is given. Their Hamiltonian invariant mass can be given in closed form for the dust and the photon gas. The action for the coupling to tetrad gravity is given. Dixon's multipoles for the perfect fluids are studied on the rest-frame Wigner hyperplane. It is also shown that the sam…
A variational theory of a perfect spin fluid with intrinsic non-Abelian color charge is constructed with allowance for spin-polarization chromomagnetic effects in Riemann-Cartan space with curvature and torsion. The spacelike nature of the spin is taken into account explicitly in this theory by including the Frenkel condition in the Lagrangian. The equations of motion, the laws that govern the evolutions of the spin and color-charge tensors, and the expression for the energy-momentum tensor for the fluid in question are obtained. In the limiting case, the theory goes over to the well-known the…
We study the late time evolution of flat and negatively curved Friedmann-Robertson-Walker (FRW) models with a perfect fluid matter source and a scalar field arising in the conformal frame of $f(R)$ theories nonminimally coupled to matter. Under mild assumptions on the potential V we prove that equilibria corresponding to non-negative local minima for V are asymptotically stable, as well as horizontal asymptotes approached from above by V. We classify all cases of the flat model where one of the matter components eventually dominates. In particular for a nondegenerate minimum of the potential w…
We investigate the appropriateness of the use of different Lagrangians to describe various components of the cosmic energy budget, discussing the degeneracies between them in the absence of nonminimal couplings to gravity or other fields, and clarifying some misconceptions in the literature. We further demonstrate that these degeneracies are generally broken for nonminimal coupled fluids, in which case the identification of the appropriate on-shell Lagrangian may become essential in order characterize the overall dynamics. We then show that models with the same on-shell Lagrangian may have dif…
The late time evolution of Friedmann-Robertson-Walker (FRW) models with a perfect fluid matter source is studied in the conformal frame of $f(R) $ gravity. We assume that the corresponding scalar field, nonminimally coupled to matter, has an arbitrary non-negative potential function $V(ϕ) $. We prove that equilibria corresponding to non-negative local minima for $V$ are asymptotically stable. We investigate all cases where one of the matter components eventually dominates. The results are valid for a large class of non-negative potentials without any particular assumptions about the behavior o…
We study the evolution of cosmological perturbations in f(G) gravity, where the Lagrangian is the sum of a Ricci scalar R and an arbitrary function f in terms of a Gauss-Bonnet term G. We derive the equations for perturbations assuming matter to be described by a perfect fluid with a constant equation of state w. We show that density perturbations in perfect fluids exhibit negative instabilities during both the radiation and the matter domination, irrespective of the form of f(G). This growth of perturbations gets stronger on smaller scales, which is difficult to be compatible with the observe…
