To conserve, or not to conserve: A review of nonconservative theories of gravity
aa r X i v : . [ g r- q c ] F e b To conserve, or not to conserve: A review of nonconservativetheories of gravity
Hermano Velten ∗ and Thiago R. P. Caramˆes † Departamento de F´ısica, Universidade Federal deOuro Preto (UFOP), Ouro Preto-MG, Brazil and Departamento de F´ısica, Universidade Federal de Lavras (UFLA), Lavras-MG, Brazil
Apart from the familiar structure firmly-rooted in the general relativistic fieldequations where the energy–momentum tensor has a null divergence i.e., it conserves,there exists a considerable number of extended theories of gravity allowing departuresfrom the usual conservative framework. Many of these theories became popular inthe last few years, aiming to describe the phenomenology behind dark matter anddark energy. However, within these scenarios, it is common to see attempts topreserve the conservative property of the energy–momentum tensor. Most of thetime, it is done by means of some additional constraint that ensures the validityof the standard conservation law, as long as this option is available in the theory.However, if no such extra constraint is available, the theory will inevitably carry anon-trivial conservation law as part of its structure. In this work, we review some ofsuch proposals discussing the theoretical construction leading to the non-conservationof the energy–momentum tensor.
Key-words : general relativity; cosmology; extended theories of gravity
I. INTRODUCTION
The principle of matter-energy conservation is one of the main pillars of GeneralRelativity (GR). Its importance when formulating a generally covariant gravitational ∗ Electronic address: [email protected] † Electronic address: thiago.carames@ufla.br theory is a matter of intense discussion since the first few years of GR. Indeed, theseminal work by E. Noether has its origins in the debate between F. Klein, E. Noether,D. Hilbert and A. Einstein about the mathematical relevance of energy conservation (seeRef. [1] for historical details). Throughout the 20th century, the features of conservationlaws in GR have been frequently discussed in the literature [2–6].Within a general relativistic based description of gravity, the covariant conservationlaw obeyed by ordinary matter is straightforwardly obtained when one applies the con-tracted Bianchi identities to the Einstein equations, which provides the well-known nullcovariant derivative of the energy–momentum tensor for the respective gravitating sys-tem. However, there is much more than a mere mathematical result in this importantaspect of General Relativity (GR). It reveals two paramount features of the Einstein’sgravity: the invariance under diffeomorphism and the minimal matter–curvature cou-pling. The former aspect means, in other words, that GR is a coordinate invarianttheory, whereas the latter reflects the clear separation between the geometrical andmatter sectors seen in the effective action of the theory.Thus, one expects that any extended gravity theory evading some of these proper-ties shall lead to a different conservation condition to be obeyed by a given energy–momentum tensor. It is possible, however, to have a deviation from the usual conserva-tion law by imposing it by hand. A famous example is the Rastall gravity [7]. In addi-tion, like Brans–Dicke theory, popular in some scalar tensor theories, non-conservationcan also be achieved if one works within the Einstein frame [8–10], where the dilatoncomes up as part of the matter sector. Many other works can be found in the literaturededicated to analyzing the arising of non-conservation within the context of alternativetheories of gravity. In this review, we shall revisit some of them.The breaking of diffeomorphism may be verified in the gravity sector of the so-calledStandard Model Extension (SME), in general accompanied also by a local violationof Lorentz symmetry [11]. As discussed in such a reference, the breaking is causedby the presence of a background field, which can either be endowed with dynamicsor not. When this field has a dynamical character, the standard conservation of theenergy–momentum tensor is naturally obeyed. On the other hand, when this field hasno dynamics, the breaking it induces is denoted “explicit” and leads to a deviationfrom the usual conservation law. In models where gravity is thought as an emergentphenomenon, i.e., a low energy manifestation of a fundamental higher energy theorywhere a background dependence shows up, diffeomorphism breaking is also verified, asdiscussed in [12]. In that work, as expected, the vanishing of the covariant divergence ofa given energy–momentum tensor is not automatically satisfied. Actually, the authorshad to impose it by assuming an additional constraint in the model.Another nonconservative gravity that has attracted recent attention is the Lazo’stheory, in which the Lagrangian density carries a dependence on the action itself [13].This theory consists of a covariant version of the Herglotz variational problem, which byits turn was an attempt to incorporate dissipative effects into the classical mechanicsvia a variational principle [14]. In Lazo’s approach, the non-conservation of energy–momentum is caused by the presence of a background four-vector that introduces intothe theory a preferred direction, thus breaking the diffeomorphism invariance.As mentioned above, models where matter and gravity are non-minimally coupledconstitute another realm where a departure of the usual conservation law is verified.In [15], the author discusses how this conservation condition should look for a wideclass of such modified gravity theories, both in the metric and Palatini formulations.Considering a family of models whose action carries both non-minimal coupled termsand arbitrary functions of the scalar curvature, he finds expressions for the modifiedconservation law for both of the variational formalisms. In addition, he shows thatit is possible to generalize the Bianchi identity so that the usual conservation law isensured. This result arises thanks to a specific choice of boundary conditions madeduring the process of extremization of the action under a given infinitesimal activecoordinate transformation. In [16], the authors obtain the energy–momentum conser-vation for an even wider class of theories of gravity, where the geometric dependence ofthe non-minimal coupling function is not restricted to scalar curvature, as it may alsodepend on the square of the Ricci and Riemann tensors. Furthermore, they also con-sider another family of non-minimal coupled theories where Lagrangian density has anarbitrary dependence on multiple curvature invariants. For both cases, they obtainedthe extended conservation law, generalizing previous results.An interesting case where such a non-minimal interaction between matter and cur-vature is also admissible, having expected consequences for the covariant conservationlaw, is the family of the so-called f ( R, T ) theories [17]. There are, however, specificfunctional forms for f ( R, T ) in which the standard conservation can be preserved[18].On the other hand, it is worth mentioning that, in this conservative subclass, there isno mixing involving the both dependencies on R and T . In other words, the densityLagrangian f ( R, T ) admits the particular form f ( R, T ) = f ( R ) + f ( T ). In this vein,it is possible to use the purely T -dependent term as part of a redefinition of the mattersector in a minimally coupled gravity [19]. This aspect helps us to illustrate the closerelation between the matter/curvature coupling and the conservation law to be obeyedby an energy–momentum tensor.Apart from proposals of modified gravitational theories, it is worth mentioning thatthe steady state cosmological model, proposed by T. Gold, H. Bondi, and F. Hoyle(see [20, 21]), proposes that the universe expands eternally, with continual creation ofmatter assuring a constant density of mass. It became clear that predictions of thesteady state model were not compatible with new observational data that supportedthe Big Bang cosmology. The concept of matter creation is still present in modernphenomenological models to mimic a possible interaction between dark matter anddark energy [22, 23]. Even in the context of modified gravity theories, the cosmologicalparticle creation process has been investigated [24–27].In the next section, we review the notion of energy–momentum conservation tensorin General Relativity. The interpretation of conserved quantities in a gravitational fieldbackground is a very subtle issue and deserves a proper discussion. In the subsequentsections, we present specific nonconservative theories. At the end of this work, werevisit the notion of energy conditions in modified gravity theories (Section XI) andthen present our final considerations. II. THE CONSERVATIVE LANDSCAPEA. From Special to General Relativity
The conservation principles are one of the most interesting aspects of physics. Theyhelp us to make predictions about the evolution of a given physical system, ensuringthat, despite the change, it undergoes certain aspects present in it that shall remainthe same. Already in our high school physics and early undergraduate classes, we madecontact with such an important property and learned that, under certain circumstances,the energy, linear, and angular momentum of physical systems are preserved (see [28]for a recent discussion).It is also usual to associate the notion of conservation with ideal physical situationsin which dissipative mechanisms do not take place. Indeed, one of the main pillarsof physics is the Hamilton’s principle of stationary action used to derive equations ofmotion for many conservative systems of varying degrees of complexity. It is worthnoting that only recently an extension of the Hamilton’s principle to nonconservativeclassical systems has been developed [29].From a strictly, but crude, mathematical standpoint, the notion of conservation isintrinsically related to the way one performs derivatives of physical quantities. Ourusual concept of conservation has its foundations in simple laboratory experiments inwhich classical systems e.g., hydrodynamics experiments, are tested. Starting with aflat spacetime metric with signature η µν = ( − , +1 , +1 , +1), one defines the derivativeof a scalar quantity ϕ simply as ∂ϕ/∂x ≡ ϕ ,x or, in a multi-dimensional spacetimewith coordinates denoted by index µ , i.e., ∂ϕ/∂x µ ≡ ϕ ,µ . Here, the symbol comma “ , “refers to an ordinary derivative. However, the formulation of currents and the energyconservation has revealed much more intricacies than that [30].The tensorial formalism is a more generic structure to represent fluid quantities. Inany spacetime, the energy–momentum tensor T µν can be decomposed in its rest framecomponents such that T = ρ = energy density; T i represents the internal heat-conduction; T i corresponds to the momentum transferred in the internal energy fluxprocess and T ij is the momentum flux. This tensor is symmetric so that T µν = T νµ . Fora fluid element occupying a non expanding volume subjected to energy/particle flowingacross its surface, the conservation of energy is stated by T µ,µ = 0 while momentumconservation by T iµ ,µ = 0 where i refers to the momentum component under analysis.This implies in the general conservation law T µν,µ = 0 . (1)Along the fluid flow, one also defines the particle quadriflux with components N = c × particle number density and N i = particle flux. The macroscopic descrip-tion of relativistic fluids demands the introduction of the four-velocity u µ for which byconvention one has u µ = η µν u ν with u = − u i = 0. Therefore, N µ = nu µ . Theparticle flux conservation is then expressed by the law N µ ,µ = ( nu µ ) ,µ = 0 . (2)Apart from vacuum solutions e.g., black holes, in which the intrinsic gravitationalaspects are studied, it is obvious that the universe is not empty. It is therefore manda-tory to set up an energy–momentum tensor for relativistic fluids. The simplest possibleconfiguration, and widely used as the standard starting point in the study of relativisticfluids, is to consider the so-called perfect fluids. They are basically non-viscous fluidconfigurations obeying the structure T µν = ( ρ + p ) u µ u ν + pη µν = ρu µ u ν + ph µν , (3)where ρ is the energy density and p the fluid pressure. The second equality of (3) statesthat the pure pressure contribution is associated with the symmetric projection tensor h µν = u µ u ν + η µν .In the context of a covariant gravitational theory as, e.g., GR, the manifestationof the gravitational interaction is seen as an effect of the curvature of the spacetime.Trajectories, flows, and the variation rates of physical quantities should obey new rulesthat take into account curvature. The mathematical mechanism used to address thisissue is the replacement of the flat spacetime metric η µν by the curved one g µν . Themetric g µν is adapted to the physical problem one wants to study and is written in sucha way that it describes the geometry of the curved spacetime. Now, the equivalenceprinciple implies that the conservation law (1) is replaced by its version in a curvedspacetime T µν ; µ = T µν,µ + T αν Γ µαµ + T µα Γ ναµ = 0 , (4)where the symbol “;” means covariant derivative. The additional contributions on theright-hand side brings the so-called affine connection, which, for a Riemannian manifold,coincide with the Christoffel symbols, defined as follows:Γ αµν = g αβ g βµ,ν + g βν,µ − g µν,β ) . (5)There are four different equations within (4) since ν = 0 , , ,
3. The ν = 0 equationdenotes conservation of energy while, for ν = i = 1 , ,
3, one has the conservation ofthe i th component of the momentum.According to our discussion so far, Equation (4) has been introduced as an extensionof the flat spacetime conservation to curved geometries. The gravitational interactionis not implicitly stated at this stage. However, we know that matter curves spacetimevia gravity. Then, prior to the appropriate introduction of the gravitational interactionin our discussion, let us continue to describe generic curved manifolds via the definitionof the Riemann curvature tensor R αβµν = Γ αβν,µ − Γ αβµ,ν + Γ ασµ Γ σβν − Γ ασν Γ σβµ . (6)Using the fact that second derivatives of the metric tensor are non-vanishing quan-tities and that partial derivatives commute, the following identity takes place: R αβµν + R ανβµ + R αµνβ = 0 . (7)Similarly, one can find symmetry properties of the Riemann tensor e.g., R αβµν = − R βαµν ; R αβµν = − R αβνµ ; R αβµν = R µναβ . Finally, with such results, one can find thedesired results for our discussion, the so-called Bianchi identities R αβµν ; λ + R βλµν ; α + R λαµν ; β = 0 . (8)Now, we can show the consequence of this identity to the conservation of T µν . Bycontracting Ref. (8) twice, firstly with g αµ , then, with g βν and using the symmetryproperties of the metric tensor, the Bianchi identity becomes(2 R µλ − δ µλ R ) ; µ = 0 , (9)where the Ricci tensor R αβ = R µαµβ = R βα and the Ricci scalar R = g µν R µν havebeen defined. In principle, there is nothing special with Equation (9). Let us analyzean important consequence of (9). As it is well known, the GR field equations may bederived from a variational principle. The starting point of this procedure is the totalaction below S = 12 κ Z d x √− gR + Z d x √− g L m . (10)The first term on the right-hand side is the so-called Einstein–Hilbert action, definedvia the Lagrangian L EH = R , whilst the second one is the matter action defined asusual, in terms of the Lagrangian density associated with the matter fields, L m . Theenergy–momentum tensor of arbitrary matter configurations is defined in terms of L m in the following way: T µν = 2 √− g δ ( √− g L m ) δg µν . (11)Taking (11) into account, the variation of the action (10) gives R µν − g µν R = κT µν , (12)where one immediately recognizes the left-hand side of this equation, also known as theEinstein tensor G µν = R µν − g µν R, (13)with the quantity appearing in (9). Therefore, the covariant derivative in (9) shouldalso apply to the right-hand side of (12) implying conservation of T µν . In the equationabove, κ ≡ πG is the gravitational coupling constant by assuming that we are workingwith c = 1 units.Alternatively, in the presence of a cosmological constant, the Einstein–Hilbert La-grangian would be redefined as L EH → L EH − G µν + Λ g µν = κT µν . (14)The above equation also has vanishing covariant divergence, and it is the only fieldequation of the generic type F µν ( g αβ , g αβ,δ , g αβ,δγ ) = T µν , where F µν is a tensor func-tional, derivable from an action principle in which the gravitational Lagrangian densityis a scalar invariant of the metric [31]. B. Diffeomorphism Invariance
A remarkable aspect of GR arises when we consider the invariance of the theoryunder diffeomorphism. Consider an infinitesimal active transformation generated by agiven vector field V µ . This corresponds to the following mapping x µ → x µ + V µ . (15)It is well known that such a transformation allows us to introduce a derivativeoperator which provides the rate of change of a given tensor along the integral curvesof V µ . This operator is the so-called Lie derivative, denoted as L V (where the indexrefers to V µ ). In any GR textbook, the reader can find a detailed discussion on howsuch an operator acts on arbitrary-rank tensors [32–34]. An interesting relation showsup when one applies such operator on the metric tensor. In this case, we have L V g µν = 2 ∇ ( µ V ν ) . (16)In order to proceed with our purpose, let us rewrite the total action (10) as follows: S = 12 κ S EH [ g µν ] + S m [ g µν , ψ i ] . (17)The theory is diffeomorphism-invariant if such a transformation implies in a variationof the action so that δS = 0. Thus, let us assume this property a priori and examineits consequences. By varying (17) and making it equal to zero, we have12 κ Z d x δ ( √− g L EH ) δg µν δg µν + Z d x δ ( √− g L m ) δg µν δg µν + Z d x √− g δS m δψ i δψ i = 0 (18)0For a diffeomorphism generated by V µ , the variation of the metric is simply its Liederivative along V µ , which is given by (16). We can check that the Einstein–Hilbertaction is itself invariant under diffeomorphism. To verify this, notice that the first termof (18) corresponds to δS EH = Z d x δ ( √− g L EH ) δg µν δg µν = Z d x √− gG µν δg µν = Z d x √− gG µν L V g µν = 2 Z d x √− gG µν ∇ µ V ν . (19)If we now integrate (19) by parts and use the covariant form of the Gauss law, wefind δS EH = Z d x √− g ( ∇ µ G µν ) V ν , (20)where we get rid of the surface integral, as it is assumed that the vector V µ vanishes onthe boundary, although V µ is arbitrary within the volume enclosed by such a bound-ary. Thus, for a generic V µ , (20) tells us that the invariance of diffeomorphism of theEinstein–Hilbert action, δS EH = 0, is ensured by ∇ µ G µν = 0, which is the contractedBianchi identity, previously presented in (9).It is implicit that the fields ψ i satisfy the matter equations of motion, which leadsto the third term vanishing. Thus, the remaining term has necessarily to obey Z d x δ ( √− g L m ) δg µν δg µν = 0 . (21)Using the definition (11) and (16) in (21), one finds Z d x √− g T µν L V g µν = 0 . (22)Analogous to the previous case, we can repeat here the procedure described by (19)and (20). This shall lead (23) to Z d x √− g ( ∇ µ T µν ) V ν = 0 . (23)Given an arbitrary V µ , (23) implies in the covariant conservation of T µν , namely (4) ∇ µ T µν = 0 . (24)1Now, we understand how the standard energy–momentum conservation emergeswithin GR as a product of an essential property of the theory, i.e., the diffeomor-phism invariance. In the first few years after General Relativity was formulated, anintense debate existed on whether or not energy conservation was a true mathematicalidentity of the theory [1]. Finally, in [35, 36], E. Noether showed that the conservationof energy, linear, and angular momentum of physical systems, as well as other physicalquantities, are justified by first principles. The Noether’s claim is that the conservationof a given quantity follows from a specific symmetry obeyed by the action. In this vein,the time translation invariance leads to the energy conservation, whereas the positiontranslation makes the linear momentum to be conserved, as it happens to the angularmomentum when the action exhibits invariance under space rotations. This featurereveals a universal aspect of conservative systems in nature, and its validation is evokedin all physical domains ranging from the quantum world to the cosmos. There is alsoa recent attempt to provide a unified view on the conservation laws in gravitationalsystems [37].The above discussion can not be seen as an argument to refute any alternative toequation (14) as long as g µν is the only field variable sourced by T µν . In the nextsections, we are going to show some remarkable examples in the literature. C. The Meaning of the Term “Energy-Momentum Conservation” in thePresence of a Gravitational Field
Although we refer to Equation (4) in most of this work as a conservation law forthe energy–momentum tensor, led mainly by the common usage of the term, it isimportant to emphasize that, roughly speaking, this classification may be misleadingand not strictly correct if we think of how to extract in practice from T µν the physicalquantities that will follow conservation laws, namely, the energy and momentum. Thisdiscussion is made by taking different paths in the various GR textbooks. As it is shown2in [38] in the Minkowski spacetime, the quantity below P µ = 1 c Z T µν dS ν (25)is a conserved quantity identified with the 4 − momentum of the system. The integrationis taken on the hypersurface S which contains all the three-dimensional space. It is wellknown that the conservation of P µ can be expressed in terms of the null divergence of T µν : ∂ µ T µν = 0 , (26)which makes it clear why the statement of (26) as a conservation law is totally consistent.It is natural to think of the extension of (25) in a curved spacetime as follows: P µ = 1 c Z S √− gT µν dS ν . (27)We may wonder if does lead to conservation of energy and momentum in a curvedmanifold as (25) does in the Minkowski spacetime. If this is true, the integral (27)would be conserved if the following condition applied: ∂ ( √− gT µν ) ∂x µ = 0 . (28)However, a mismatch is verified when we rewrite the equation (4) in the form below T µν ; µ = 1 √− g ∂ ( √− gT µν ) ∂x µ − ∂g µλ ∂x ν T µλ = 0 . (29)The above expression is different from (28), unless we are in a particular coordinatesystem, x ∗ , where ∂g µλ ( x ∗ ) /∂x ν = 0 holds. This result points to the need of reformu-lating the relation (27) in order to properly describe conservation of the energy andmomentum in a curved spacetime. In this vein, Landau and Lifshitz call our attentionto the fact that, although Equation (4) indeed expresses a local covariant conservationof T µν , it is not true globally, since the gravitational field itself carries energy, whosecontribution is missing in (27). This point is also illustrated by S. Weinberg in [39], Although we consider c = 1 throughout this work, particularly in this section we show c explicitly,in order to match the Ref. [38] that is used in the present discussion. t µν , which is constructed with the aid of the geometric termspresent in the Einstein equations. Thereby, they show that, instead of Equation (27),we shall have the following relation for the 4-momentum in order for us to properlydescribe a conservation law for energy and momentum: P µ = 1 c Z S ( − g )( T µν + t µν ) dS ν , (30)where the quantity t µν is given by t σκ = c πG [(2Γ νλµ Γ θνθ − Γ νλθ Γ θµν − Γ νλν Γ θµθ )( g σλ g κµ − g µκ g λµ )+ g σλ g µν (Γ κλθ Γ θµν + Γ κµν Γ θλθ − Γ κνθ Γ θλµ − Γ κλµ Γ θνθ )+ g κλ g µν (Γ κλθ Γ θµν + Γ σµν Γ θλθ − Γ σνθ Γ θλµ − Γ σλµ Γ θνθ ) + g κλ g µν (cid:0) Γ σλµ Γ κµθ − Γ σλµ Γ κνθ (cid:1) ] . (31)The expression above for t µν is achieved by means of a lengthy calculation which isprovided step by step in the aforementioned reference Landau & Lifshitz [38], to whichwe refer the interested reader for full details. Given (30), the equivalent equation thatindeed leads to a conservation law shall be ∂ µ [( − g )( T µν + t µν )] = 0 . (32)Notice that the nontensorial nature of t µν is evident due to its explicit dependenceon products of Christoffel symbols. However, it does behave as a tensor under linearcoordinate transformations, of which the Lorentz transformation is a particularly inter-esting case. In addition, notice that it vanishes in the locally inertial frame x ∗ , whereΓ αµν ( x ∗ ) = 0, at which the special relativity relation (27) is recovered. This enhances thefact that even the procedure leading to (32) fails in providing a meaningful local con-servation law that we could associate with the energy conservation, due to the absencein GR of a local meaning for the gravitational field.4In the R. Wald textbook [33], the physical meaning of (4) is also discussed, althoughin a bit of a different way. The author shows that, in special relativity, the meaning of ∂ µ T µν = 0 is unambiguous as a conservation expression for the energy–momentum ten-sor. To show this, let us consider a family of inertial observers with parallel worldlines, v α , which means ∂ β v α = 0 . (33)If we assume, for instance, a perfect fluid (although the reasoning actually holds forany matter and fields) described by T µν , it is known that the quantity J α = − T αβ v β (34)shall correspond to the mass–energy current density 4 − vector as measured by theseobservers. Given (26), from (34), it follows that ∂ α J α = 0 . (35)The Gauss law tells us that the integration of (35) over a four-dimensional volume, V , is equal to the surface integral below Z S J α n α dS = 0 , (36)where n α is the unit normal vector to S . The null flux defined by (36) clearly indicatesa conservation of energy, as it leads to the vanishing of the time variation of such aquantity inside the volume V . In other words, we can say that (26) is a requirementfor the energy conservation as measured by a family of inertial observers.On the other hand, in a curved spacetime, as it is known, the relation (26) is replacedby (4). However, the presence of curvature spoils the interpretation of such a equationas a conservation law, since in this case there is no well-defined notion of parallel vectorsat different points, which harms the introduction of a global family of inertial observersable to measure the energy of a distant particle. Let us see this feature in more detail.It is natural to think of the curvature-dependent extension of the condition (33) as ∇ ( β v α ) = 0 . (37)5Thus, the energy–momentum four vector (34) now is the one measured by observerssatisfying (37). Nevertheless, in this case, the covariant derivative of T µν does not leadnaturally to the covariant divergence of the current J µ , ∇ α ( T µν v ν ) = 0 , (38)which, by Gauss law, as we see before, this could ensure the conservation of energy.Because in the curved spacetime, in general, one is not able to define a family ofobservers satisfying v α v α = − v µ is a Killing vector that generates a one-parameter group of isometries in thespacetime. In this case, as it is well known, v µ shall obey Equation (37), which, in thiscontext, is called the Killing equation. Thus, this inconsistency between (37) and (38)prevents ∇ µ T µν = 0 to be interpreted as a requirement for a global energy conservation.In fact, let us recall that gravitational field itself, by means of tidal effects, can do workon a material system, thus altering locally its energy content. However, if we consider asmall region of the space where such effects can be neglected, the energy of the systemshall be conserved in a reasonable approximation. Thus, within such a small spacetimeregion, it is possible to define a vector field such that ∇ β v α ≈
0. Thus, Equation (4)would indeed reflect an approximate conservation of energy as seen by these observers.Therefore, it is plausible to say that Ref. (4) represents a local conservation of theenergy content of a physical system over small regions of spacetime.
III. RASTALL GRAVITY
The Rastall proposal extends GR causing a violation of the usual conservation law,making the covariant divergence of T µν proportional to the covariant divergence of thecurvature scalar R [7]. While non-trivial to explain the nature of such a new source,this can be phenomenologically seen as the emergence of quantum effects in curvedspacetimes as e.g., in the case of gravitational anomalies [40, 41]. This phenomenologicalapproach and the absence of any variational formalism from which the field equationsof Rastall theory could emerge have attracted the attention of many authors that tried6to formulate a variational principle for the Rastall gravity. Some efforts in this sensecan be found in the references [42–44].In Rastall gravity, the conservation law is replaced by the equation T µν ; ν = 1 − λ πG g µν R ; ν (39)where the GR framework is recovered by setting λ = 1. The associated field equationsaccording to the Rastall’s proposal are R µν − λ g µν R = κT µν . (40)Within the context of Riemannian geometry, there is no variational principle asso-ciated with this theory, but similar structures may be found in the context of Weylgeometry [45].Applications of the Rastall gravity to cosmology have been performed in Refs. [46–48]. Black holes and other exact solutions have been studied in Refs. [49, 50]. However,since Rastall theory should manifest mostly in high curvature environments, compactobjects like neutron stars are perfect laboratories to constrain the parameter λ . InRef. [51], by using realistic equations of state for neutron stars, interior conservativebounds on the non-GR behaviour of the Rastall theory have been placed at the 1%level i.e., λ < . T µνeff . This isalso the case of Rastall as discussed in [55].7 IV. BRANS–DICKE THEORY IN THE EINSTEIN FRAME
It is well known that string theory predicts a scalar partner of the graviton in thelow energy limits, the so-called dilatonic field (or dilaton). The Brans–Dicke theory isthe simplest model in which such an extra degree of freedom shows up [8]. There are,however, two approaches by which such a theory can be studied [56]. In the Jordanframe, the dilaton is a crucial piece of the geometrical sector in which it takes part asbeing co-responsible by the gravitational field, along with the metric tensor. However,in the Einstein frame, this scalar field is shifted to the matter sector, where it nowshall couple to the ordinary matter. Both frames are related merely by a conformaltransformation. As a consequence of this coupling, the paths followed by test particlesbecome non-geodesic for a given spacetime and the standard covariant conservation ofthe energy–momentum deviates from the usual one.In the Jordan frame, the gravitational action is S JF = 116 π Z d x √− g (cid:20) φR − ωφ ∇ µ φ ∇ µ φ (cid:21) + Z d x √− g L m , (41)where ω is the Brans–Dicke parameter and L m is the Lagrangian of the matter fields.By extremizing the action above with respect to the metric, one has the following setof field equations: R µν − g µν R = 8 πφ T µν + ωφ (cid:18) ∇ µ φ ∇ ν φ − g µν ∇ α φ ∇ α φ (cid:19) + 1 φ ( ∇ µ ∇ ν φ − g µν (cid:3) φ ) . (42)The variation of the action with respect to the scalar field gives the dynamics obeyedby φ (cid:3) φ = 8 πT ω , (43)where T is the trace of the energy–momentum tensor. Let us recall that the GR limit isachieved when ω → ∞ and φ → φ = G − [39]. In this frame, T µν conserves accordingto the standard condition: ∇ ν T µν = 0 . (44)It should be mentioned that there is an interesting version of the Brans–Dicke gravityin the Jordan frame in which the non-conservation of T µν shows up. In this alternative8scenario, the Brans–Dicke model is combined with the Rastall theory, thus inheritingits non-conservative aspect [57]. This model was called Brans–Dicke-Rastall gravity.In [57], the authors analyze some consequences of this theory both to the backgroundcosmology and the parametrized post-Newtonian formalism.It is well known, however, that an alternative formulation for the Brans–Dicke theoryis possible. It is achieved by means of the conformal transformation below g µν −→ ˜ g µν = Gφg µν , (45)along with the following redefinition φ −→ ˜ φ = Z r ω + 316 πG dφφ . (46)Equations (45) and (46) lead the theory to the so-called Einstein frame. In thisformulation, the gravitational action is rewritten as follows: S EF = Z d x p − ˜ g " ˜ R πG −
12 ˜ g µν ˜ ∇ µ ˜ φ ˜ ∇ ν ˜ φ + S M (cid:16) e − α ˜ φ ˜ g µν , ψ (cid:17) , (47)where α ≡ q πG ω +3 and ψ denotes the matter fields. Notice that, in this case, theredefined dilaton, ˜ φ , couples minimally to the curvature, and the geometric sector issolely the Einstein–Hilbert action with ˜ φ acting as a matter field. This novel aspectarising in the Einstein frame reinterprets the role of the dilaton in the Brans–Dickegravity, since now it indeed appears in the matter action, thus becoming able to couplewith a given matter configuration. The main consequence of that is a departure of theenergy–momentum conservation from its traditional expression. Now, this law is led tothe following nonconservative form: ∇ ν T µν = αT ∂ µ φ. (48)Notice that, leaving aside the trivial case of the GR limit (for which α → φ → φ ), the standard conservation can also occur for a given matter configurationwhose energy–momentum tensor is trace free.9 V. GRAVITY THEORIES FROM THE STANDARD MODEL EXTENSION
In the gravity sector of the Standard Model Extension (SME), it is also possibleto envisage physical properties with consequences to the energy–momentum conserva-tion law. The SME is an effective theory encompassing the Standard Model of theparticle physics and the General Relativity that incorporates possible deviations fromthe Lorentz and diffeomorphism symmetries [58–62]. The breaking of diffeomorphisminvariance is, in general, a remarkable feature of gravity theories arising from exten-sions of the standard model. Usually, this violation is induced by the presence of fixedbackground fields, which can break the local Lorentz and diffeomorphism symmetrieseither explicitly or spontaneously [11]. In the former case, the background fields showup explicitly in the Lagrangian of the model, whereas, in the latter case, this does nothappen, and the background just appears as a vacuum solution of the theory. It is wellknown, however, that, while the explicit breaking imposes difficulties for the gravitysector of the SME, with serious conflicts between the dynamical and the geometricalmodel’s constraints, the spontaneous one points towards a promising direction, free ofsuch drawbacks. Nevertheless, as Kosteleck´y shows in [59], these conflicts can be fixedin the context of Chern–Simons and massive gravity with an underlying Riemannianspacetime.An effective gravity theory exhibiting Lorentz and diffeomorphism breaking can begenerically represented by the action below (in the low energy limit) S = S EH + S LV + S LI , (49)where S EH is the Einstein–Hilbert action, S EH = 12 κ Z d x √− gR. (50)The Lorentz-violation term S LV is S LV = Z d x √− g L LV ( g µν , f ψ , ¯ k χ ) , (51)where f ψ means the usual matter fields and ¯ k χ denoting the diffeomorphism violatingbackground field. Moreover, ψ and χ represent all the component indices of the tensors0 f ψ and ¯ k χ , respectively. In addition, finally, the Lorentz- and diffeomorphism-invariantterm L LI is given by S LI = Z d x √− g L LI ( g µν , f ψ ) . (52)As mentioned above, when the diffeomorphism symmetry breaking is explicit, thefixed background field ¯ k χ is nondynamical. This means that( δS LV ) diff = Z d x √− g δ L LV δ ¯ k χ δ ¯ k χ = 0 . (53)In addition, the variation of the full action (49) gives G µν = κ ( T µνLI + T µνLV ) . (54)As clearly showed in Section II, the contracted Bianchi identity can be understoodas a consequence of the diffeomorphism invariance of (50) in isolation. As this sameinvariance is not respected by S LV , by taking the covariant divergence of (54), one isleft with ∇ µ ( T µνLI + T µνLV ) = 0 , (55)which must hold on-shell. In this equation, the energy–momentum tensors T µνLI and T µνLV are defined in terms of L LI and L LV , respectively, according to (11). Thus, in general,the conventional matter described by T µνLI will not conserve as usual due to the presenceof the inhomogeneous diffeomorphism breaking term in Equation (55). However, it isusual to require the additional constraint ∇ µ T µνLV = 0 in order to ensure the standardconservation condition [11].The equations above are presented in a quite general form, in order to cover arbitrarycases involving explicit spacetime symmetry breaking. However, following the spiritof the Ref. [11], we can look at specific examples. It is common that these modelspresent conflicts between its dynamical description and some geometrical constraints.Such conflicts are usually evinced when the Bianchi identity is imposed onto the fieldequations. Let us analyze two examples that will help us to understand this aspect.1 A. Spacetime-Dependent Cosmological Constant
For this model, the gravity is endowed with a spacetime-dependent cosmologicalconstant. The total action of this theory is given by S = Z d x √− g (cid:26) κ [ R − x )] + L m (cid:27) . (56)By comparing (56) with (49), we can set the mapping among the corresponding vari-ables. Here, the fixed background is ¯ k χ = Λ( x ), and the Lorentz-invariant Lagrangianis L LI = L m , whereas the symmetry-violating piece is L LV = − Λ( x ) /κ .Notice that, when Λ( x ) = 0 and is non-constant, the condition (53) applies, as thetheory is endowed with a fixed, non-dynamical, background field given by Λ( x ) thatbreaks explicitly the diffeomorphism, since this field appears explicitly in the action. Infact, for this case, Equation (53) becomes( δS LV ) diff = Z d x √− g δ L LV δ Λ( x ) L V Λ( x ) = − Z d x √− g κ V µ ∂ µ Λ( x ) = 0 , (57)since the Lie derivative on Λ( x ) is L V Λ( x ) = V µ ∂ µ Λ( x ). The corresponding fieldequations are G µν = − Λ( x ) g µν + κT µνm , (58)where the symmetry-breaking energy–momentum tensor in the present example is T µνLV = − g µν Λ( x ) /κ . By using the contracted Bianchi identity on (58), we obtainfor this case the corresponding form for (55), which is ∇ µ T µνm = (1 /κ ) g µν ∂ µ Λ( x ) . (59)Therefore, the presence of a non-constant Λ( x ) in the description of the gravityleads to a deadlock: if the standard conservation for T µνm is required, Equation (59)tells us that necessarily ∂ µ Λ( x ) = 0 implying in Λ( x ) = const., thus restoring thediffeomorphism invariance and contradicting the a priori assumption that Λ( x ) is non-constant. Notice that, in this case, the aforementioned conflict between the dynamicsand the geometrical identities (in the present case, the Bianchi identity) is not evadedunless one assumes the non-trivial conservation law above (59). One may wonder if2such a conflict is unavoidable, by raising the following question: is it possible to ensurethe standard conservation law for T µνm without necessarily restoring the diffeomorphisminvariance? The next example provides an affirmative answer for this question. B. Chern–Simons Gravity
The so-called Chern–Simons term was first introduced in the context of three-dimensional gauge field theory and gravitational models [63, 64]. Some years later,this same model was extended in order to represent a theory of gravity in the four-dimensional spacetime [65, 66]. The action for the Chern–Simons gravity in four di-mensions can be written as follows: S CS = Z d x (cid:20) κ (cid:18) √− gR + 14 θ ∗ RR (cid:19) + √− g L m (cid:21) , (60)where ∗ RR ≡ ∗ R κλµν R λκµν is the gravitational Pontryagin density and ∗ R κλµν = ǫ µναβ R κλαβ . The explicit breaking of diffeomorphism invariance occurs due to theembedding coordinate, v µ , which is related to the non-dynamical scalar θ ( x ) through v µ ≡ ∂ µ θ . The variation of the action (60) gives G µν + C µν = κT µνm , (61)where C µν is the four-dimensional Cotton tensor which has the following form: C µν = − √− g (cid:2) v σ (cid:0) ǫ σµαβ ∇ α R νβ + ǫ σναβ ∇ α R µβ (cid:1) + ∇ σ v τ ( ∗ R τµσν + ∗ R τνσµ ) (cid:3) . (62)Due to the dependence of C µν upon the embedding coordinate v µ , the Cotton tensorencodes the information of diffeomorphism breaking in the field equations, so that wecan set the following correspondence: T µνLV = C µν κ . (63)By computing the covariant divergence of C µν , we will have ∇ µ C µν = 18 √− g ( ∂ ν θ ) ∗ RR. (64)3Looking at the action (60) and considering diffeomorphism transformations like (15),Equation (53) shall be ( δS LV ) diff = Z d x √− g
14 ( ∗ RR ) V µ ∂ µ θ. (65)Thus, by the equation above, we find a twofold condition for the explicit breaking ofdiffeomorphism. The first condition is ∂ µ θ = 0 and the second one is that the Pontryagindensity is non-zero. We can examine the occurrence (or not) of the dynamics-geometryconflict by taking the covariant divergence of (61). If ∇ µ T µνm = 0 is required, the Bianchiidentity then obliges the relation below: ∇ µ C µν = 0 (66)to hold on shell. Since the divergence of C µν was also computed in (64), the condition(66) imposes that ∗ RR ∂ µ θ = 0. Thus, the conflict is evaded not only in the trivialsituation when ∂ µ θ = 0, implying in the restoration of the diffeomorphism invariance. Itis evaded as well when the Pontryagin density vanishes, obeying the so-called Pontryaginconstraint ∗ RR = 0 . (67)This is a constraint on the geometry that indicates a way to avoid the mentioneddynamics-geometry conflict by restricting the class of allowed geometries. It is known,for example, that any spacetime of Petrov types III, N, and O automatically satisfy(67) (See [67]).On the other hand, if the usual conservation law ∇ µ T µνm = 0 was somehow relaxed,when we took the covariant divergence of (61), we would have ∇ µ T µνm = 18 κ √− g ∂ ν θ ∗ RR, which means a deviation from the standard conservation law sourced by both the pres-ence of ∂ ν θ and the Pontryagin density.The formulation showed here for the Chern–Simons gravity is not the only one foundin the literature. In [67], one can see an alternative one, where it is possible to renderdynamics to the scalar field θ . In this case, such a field shall obey a Klein–Gordon like4equation of motion that is sourced both by stress-energy tensor and the curvature ofspacetime.To end this section, let us mention an important attempt of setting experimentalbounds on the non-dynamics Chern-Simons gravity. In [68], by studying gravitomag-netic effects within this theory, the authors place important bounds on the parameter m cs . In that work, they define such a parameter as m − cs ∝ ˙ θ and assume the scalarfield θ as being time varying but spatially homogeneous. They compute orbits of testbodies and the precession of gyroscopes in the linearized Chern–Simons gravity arounda massive spinning bod. Then, they use observation from the LAGEOS [69] and Grav-ity Probe B [70] satellites to restrict m − cs to be less than 1000 km, which correspondsto m cs ≥ × − GeV.
VI. EMERGENT GRAVITY THEORIES BREAKING GENERALCOVARIANCE
The so far weakly unexplored high energy limit of GR leaves room for investigationof emergent gravity theories i.e., approaches in which the low energy behavior appearsas a manifestation of some yet unknown fundamental theory.Small violations of diffeomorphism invariance can be introduced into a physical the-ory in order to explore the phenomenology behind emergent phenomena. As an example,one such approach has been discussed in Ref. [12]. The general action proposed in thisreference is of the form L = 12 κ " R + X i a i L i + L m , (68)where the L i terms involve contributions that induce a violation of diffeomorphisminvariance L = − g µν Γ αµλ Γ λνα , L = − g µν Γ αµν Γ λλα , L = − g αγ g βρ g µν Γ µαβ Γ νγρ , (69) L = − g αγ g βλ g µν Γ λµν Γ βγα , L = − g αβ Γ λλα Γ µµβ , L = − g µν ∂ ν Γ λµλ , L = − g µν ∂ λ Γ λµν , leading to the following field equations R µν − g µν R + a M µν = κT µν . (70)5The departure from GR is encoded in the new contribution M µν defined as M µν = B µν + D µν (71) B µν = − g µν g αβ g γδ g ǫη Γ αγǫ Γ βδη + g αβ g γδ g νφ g µǫ Γ ǫαγ Γ φβδ + 2 g φǫ g αγ g δǫ g φβ Γ βµα Γ δνγ D µν = Γ λαλ A αµν + A αµν,α , A αµν = g αβ g γµ Γ γνβ + g αβ g γν Γ γµβ − Γ αµν . which is clearly not invariant under general coordinate transformation.In order to implement a consistent condition on this set of equations in Ref. [12], ithas been imposed the constraint below: ∇ µ M µν = 0 (72)In that reference, the authors analyzed this theory by expanding Equations (70) alongwith the (72) in light of the recipe given by the PPN formalism. With this treatment,they aimed at constraining the diffeomorphism-breaking terms present in the model.As expected, this investigation shows that the parameters usually identified with thenon-conservation of energy and momentum will not be zero for this model; they willin fact depend on the dimensionless parameter a appearing in (70). In addition, theyfound a strong bound on this parameter coming from the absence of preferred-frameeffects in pulsars that leads a to be less than 10 − in gravitational strength. VII. ACTION DEPENDENT LAGRANGIAN THEORIES
This class of theory is based on the so-called Herglotz problem. The latter wasoriginally built within a classical mechanics scenario, and consists of generalizing theaction principle by introducing in the Lagrangian an action-dependence. Though nontrivial, this kind of construction reads S = Z L ( x, ˙ x, S ) dt. (73)This allows a proper description of dissipative phenomena in classical systems fromfirst principles. Recently, Lazo et al [13] found that there exists a covariant general-ization of this problem. Hence, a prototype of gravitational theory can be designed6from L = √− g ( R − λ α s α ) + L m (74)where quantity s α is an action-density field. The coupling term λ µ depends on the space-time coordinates. The interpretation employed in this approach refers to the action-dependence introduced in (74) associated with s α only with respect to the standardEinstein–Hilbert action. The matter action is not coupled to the s α field. Therefore,departures from standard gravity provided by this theory are purely of geometric nature.As a result, this approach leads to a geometrical viscous gravity model in which thedynamics of the theory is described by the generalized field equations R µν − Rg µν + K µν − Kg µν = κT µν . (75)By applying the Bianchi identities to the above equation, one finds a relation involv-ing K αβ , its trace, and the matter sector. By considering a constant G coupling, thesystem of field equations is sourced by the modified conservation law κT µν ; µ = K µν ; µ − K ; ν . (76)The new aspect here is clearly encoded in the quantity K µν given by K µν = λ α Γ αµν − (cid:0) λ µ Γ ανα + λ ν Γ αµα (cid:1) . (77)The quantity λ α plays the role of a background four-vector necessary in this non-conservative structure.In recent years, many gravitational problems have been investigated within thistheory. In [71], the authors performed a study both of the FLRW background cosmologyand the linear perturbative regime for this nonconservative gravity. They found that thebackground dynamics are equivalent to that one provided by the bulk viscous cosmology[72]. On the other hand, the evolution of the linear perturbations indicated a possibilityof avoiding, within the nonconservative theory, some of the problems present in theviscous scenarios [73, 74]. In [75], the authors deepen such a cosmological study. Inthat work, they submitted the cosmology emerging from this theory to the scrutiny ofsome important cosmological datasets, both at the background and perturbative levels.7This study revealed that the nonconservative cosmology was not viable, at least in theway it was originally formulated. However, the authors showed an interesting way outfor this issue, by assuming that the matter conserves as usual, whereas the dark energythat obeys the non-standard conservation law becomes able to be pressureless. Thisnew framework was revealed as a viable model in light of the analyzed cosmologicaldata. In this study, we have computed the f σ − . H < λ < − . H , which also revealed compatible with H ( z ) data, indicating aviable model both in the background and the perturbative levels.This theory was also used to study cosmic string configurations, where were in-vestigated the Abelian–Higgs strings as well as the phenomenological model of theHiscock–Gott string, by means both of analytical and numerical techniques [77]. Thesum rules formalism for braneworld models within this nonconservative theory was ex-amined in [78]. In Ref. [79], the authors discussed the conditions for the existence ofstatic spherically symmetric solutions in this gravity (see also [80]). VIII. NONMINIMAL CURVATURE–MATTER COUPLING
The principle of minimal coupling is evoked as one of the pillars to realize a gravita-tional theory. Alternative gravitational theories designed to stay close to GR maintainit. The consequences of abandoning this principle directly affect the way matter fieldsinteract with geometry. If this principle is abandoned, the resulting field equationsare non-trivial since the direct interplay between flat and curved spacetime, given bythe familiar principle of General Covariance and the Equivalence Principle, is dam-aged. A direct consequence of adopting non-minimally couplings between the matterand geometric sectors is the appearance of nonconservative features. Extensions of f ( R )theories involving non-minimal couplings between curvature and the matter Lagrangianrepresent a class of theories in which T µν does not conserve. The family of f ( R, L m )theories is the typical prototype for this situation [81] (see also [16, 82]).8Particularly, the particle creation phenomena is a simple mechanism leading to theidea of non conservation. Ref. [83] has discussed how one can associate this to a non-minimal coupling between the matter Lagrangian and curvature terms. This idea isdesigned by the following action: S = Z √− g (cid:20) f ( R )2 κ + f ( R ) L m (cid:21) , (78)where f ( R ) and f ( R ) are arbitrary functions of the Ricci scalar.The energy–momentum tensor is defined in terms of the matter Lagrangian accordingto (11). Then, one show that a general property of this type of theories is the non-conservation such that ∇ µ T µν = λF λF [ g µν L m − T µν ] ∇ µ R, (79)where F i = f i,R and λ is a coupling constant that measures how strong the interactionis between f ( R ) and the matter Lagrangian. Of course, the usual conservation law isrecovered with λ = 0.This interpretation has been criticized, however, in Ref. [84]. The reasoning of thecriticism contained in the latter reference is that the non-minimal coupling actuallyinduces a change in the particle–momentum on a cosmological timescale, which can notbe associated with the particle creation process.Observational constraints on this class of theories can be found in, e.g., Ref. [85]. IX. F ( R, T ) THEORIES
The non-minimal coupling between matter and curvature terms has been indeedwidely studied. One such proposal is the so-called f ( R, T ) theory where T = T µµ is thetrace of the stress–energy tensor. Of course, the GR limit of such theory correspondsto f ( R, T ) = R in the action S = S G + S m = 12 κ Z d x √− gf ( R, T ) + Z d x √− g L m (80)This approach has been introduced in Ref. [17] aiming to describe running cosmo-logical constant cosmologies. Therefore, this modification of gravity trying to explain9the accelerated expansion of the universe and dark matter seems to be a fundamentalingredient in the viable f ( R, T ) scenarios [86].By defining T µν as T µν = g µν L m − ∂ L m ∂g µν . (81)By varying the action with respect to the metric (as in the standard metric formal-ism), f R ( R, T ) R µν − f ( R, T ) g µν + ( g µν (cid:3) − ∇ µ ∇ ν ) f R ( R, T ) = [ κ − f T ( R, T )] T µν − f T Θ µν , (82)where Θ µν ≡ g αβ δT αβ δg µν = − T µν + g µν L m − g αβ ∂ L m ∂g µν ∂g αβ . (83)In addition, the conservation law to be obeyed by T µν in this case shall be ∇ µ [( κ − f T ) T µν − f T Θ µν ] = 0 . (84)This condition is easily obtained by taking the covariant divergence of (82), bear-ing in mind that the left-hand side of these equations has null divergence, as can bestraightforwardly verified (see [15]).In a FLRW background, an energy–momentum tensor of a perfect fluid is sourcedwritten in terms of its energy density ρ and the pressure p . From the above definitions,we can write the Lagrangian as L m = − p , while the tensor (83) reduces to Θ µν = − T µν − pg µν . Let us focus on a class of f ( R, T ) theories given by f ( R, T ) = f ( R ) + f ( T ). The background expansion obeys equations −
3( ˙ H + H ) f ′ − f − f H ˙ f ′ = κρ + f ′ (1 + w ) (85)and ( ˙ H + 3 H ) f ′ + f f − ¨ f ′ − H ˙ f ′ = κp, (86)where the prime and dot denote derivatives with respect to the argument, i.e., f ′ = d f ( R ) /dR , and to the cosmic time, respectively. With a minus sign with respect to (11). f ( R, T ) model gives rise to a deviation from the usual conservationlaw such that ˙ ρ + 3 Hρ (1 + w ) = − κ + f ′ (cid:20) (1 + w ) ρ ˙ f ′ + w ˙ ρf ′ + 12 ˙ f (cid:21) , (87)where p = wρ . Notice that (85)–(87) form a system of three independent differentialequations. The fourth-order derivatives of the metric appearing in these equations, ¨ f ′ ,gives rise to new degrees of freedom in f ( R, T ) theory, so that, along with the variables a and ρ , it is also necessary to consider ¨ a as an independent variable in order to providea solution for such system of equations.In the context of interacting dark energy models, a local violation of ∇ µ T µν =0 may be allowed by means of a possible exchange of either energy or momentum(or both) between the two dark components. Nonetheless, even in these models, thisexchange occurs in such a way as to preserve the conservation of the total dark fluid.Differently, Equation (87) shows a non-conservation of the matter–energy content asa whole, revealing a significant drawback of this class of f ( R, T ) theories. In light ofthis, in [18], the authors imposed by hand the fulfillment of (87) by setting to zero theexpression inside the bracket, e.g (1 + w ) ρ ˙ f ′ + w ˙ ρf ′ + ˙ f = 0. By using a chain rule,one can get rid of the time derivatives and write this constraint condition as a secondorder differential equation for the function f ( T ), whose integration provides a solutionin the form f ( T ) = σT w +12( w +1) + σ , (88)where σ and σ are integration constants. We may avoid the trivial case f ( T ) = const.by assuming the necessary condition ω = − /
3. In addition, ω = +1 / T = 0. Considering pressureless matter, ω = 0, the solutionbecomes f ( T ) = σT + σ . (89)Then, this is the only choice assuring conservation, as it constitutes the only case inwhich the standard conservation law is preserved, which implies automatically rulingout anyone else if one demands that conservation is required. It is not surprising1that the usual conservation condition is gained in “separable” f ( R, T ) models, namelythose ones obeying f ( R, T ) = f ( R ) + f ( T ). Let us recall that the departure fromthe traditional conservation law in arbitrary f ( R, T ) theories arises precisely becauseof the non-minimal coupling between curvature and matter. When these two sectorsappear in such a separated added up terms as f ( R, T ) = f ( R ) + f ( T ), it makes itpossible to obtain a differential equation only for f ( T ), decoupled from the function f ( R ), therefore free from any R -dependence. As we have seen in (87), this differentialequation constitutes a constraint leading to the usual conservation law. Thus, it issomehow expected that, when a non-coupling is imposed, the standard conservationappears as a particular case. It is also possible to redesign the f ( R, T ) theory in such away that it evades the continuity equation by adding the extra geometric terms to sumup the effective energy–momentum tensor [87].However, if one intends to assume the usual conservation, a stringent restriction on f ( R, T ) gravity applies. By taking this path, Refs. [18] as well as [86] bring the messagethat versions of f ( R, T ) models based on the separation f ( R, T ) = f ( R ) + f ( T ) aredisfavored in light of recent data and therefore cannot be interpreted as viable theories.Furthermore, there is a more profound argument to not take into consideration this classof f ( R, T ) models. In a recent discussion, it was claimed that the term f ( T ) may besimply incorporated into the matter Lagrangian L m , which means that it is not possibleto physically separate their effects as they depend on the same variables and are addedup in the total action [88]. Thus, these models would consist of a mere redefinitionof the matter sector without bringing any new information or physical effect to theproblem under study. Other authors, however, have disputed this claim. We direct thereader to Refs. [88, 89], where it is possible to follow the entire debate on it.The discussion on the conservation properties in f ( R, T φ ), a scalar field variantformulation, has been recently discussed in [90].2 X. NONCONSERVATIVE TRACELESS GRAVITY
Unimodular gravity is a well-known alternative gravitational theory. In this ap-proach, the cosmological constant appears naturally in the form of an integration con-stant. By obtaining the field equations from the Einstein–Hilbert Lagrangian imposingthe condition g µν δg µν = 0, one finds G µν + 14 g µν R = κ (cid:18) T µν − g µν T (cid:19) . (90)In addition, by taking the divergence of the above equation and using the Bianchiidentities, R ,ν πG (cid:18) T µν ; µ − T ; ν (cid:19) . (91)It is worth noting that there is an extra constraint on the determinant of the metric,and therefore there are nine independent equations of motion, one less than GR. Then,energy–momentum conservation ad hoc imposed in this theory.Using the approach adopted in Ref. [91], it is possible to design a non-conservationunimodular theory. In Ref. [92], a constant curvature R =const. has been used to con-strain unimodular gravity. If this condition is applied to Equation (91), one immediatelyfinds T µν = T ; ν . (92)A consequence of this case is the fixing of the scaling law for the enthalpy of thesystem such that ρ + p = Ca − , (93)where C is a constant.It is also noted in Ref. [92] that other constraining conditions such as e.g., when thequantity √− g ( R +4Λ) is constant, also lead to nonconservative models with backgroundexpansions similar to the ΛCDM, but with distinct perturbative behavior.3 XI. ENERGY CONDITIONS WHEN T µν ; ν = 0. The interface between non-conservation in modified gravity theories and how energyconditions are employed is worth being highlighted. Most of the theories discussedin this work are considered nonconservative since new geometric terms appear on theleft-hand side of their field equations. Then, very often seen in the literature, in orderto maintain the application of the contracted Bianchi identities to the Einstein tensor G µν , all remaining new geometric contributions are sent to the right-hand side of fieldequations to compose an effective T effµν = ¯ T µν . While formally possible to impose inan ad hoc manner that such new ¯ T µν conserves i.e., ¯ T µν ; µ = 0, this procedure leads to adifferent interpretation e.g., on how the familiar energy conditions should apply to the¯ T µν components.Let us sketch now the basic idea behind the above argument. At the field equationslevel, one can formally recast most of the modified gravity theories in the form σ (Ψ i ) ( G µν + W µν ) = κT µν , (94)where the factor σ (Ψ i ) is a coupling to the gravity while Ψ i represents curvature invari-ants or other fields, like scalar ones. The symmetric tensor W µν stands for additionalgeometrical terms which may appear in specific theory under consideration. We wantto mention that, in Equation (94), the energy–momentum tensor T µν will be consideredas the one of a perfect fluid as defined in (3). Of course, Equation (94) does not en-compass all the possible alternatives to GR at the field equations level. However, mostof the main proposals in the market (including most of the theories discussed in thiswork) can be reshaped in this form. From the structure presented in (94), one identifiesthat GR is immediately recovered if σ (Ψ i ) = 1 and W µν = 0. Equation (94) can alsobe rewritten as a GR-like theory according to G µν = κ ¯ T µν = κσ T µν − W µν . (95)In principle, one can not postulate that the effective energy–momentum tensor isconserved i.e., ¯ T µν ; µ = 0. However, this is indirectly achieved due to the Bianchiidentities. Then, what is the meaning of the former statement ( ¯ T µν ; µ =0 )?4This issue has been extensively discussed in Refs. [93, 94] (see also [95]), and webriefly review it now since this topic is critical for the discussion of the viability ofmodified gravity models.Firstly, let diffeomorphism invariance take place in the perfect fluid matter actioni.e., T µν ; µ = 0. On the other hand, contracted Bianchi identities assure that G µν ; µ = 0.Then, if both conditions take place, there appears a new constraint equation derivedfrom (95). It reads W µν ; µ = − κσ T µν σ ; µ . (96)However, if one evades the diffeomorphism invariance of the matter action, thephysical meaning of the above constraint can be translated to the notion of a non-conservation, i.e., the mere application of the Bianchi identities to the field Equation(95) yields to T µν ; ν = ( σW µν ) ; ν + (cid:16) σ ; ν σ (cid:17) [ T µν − ( σW µν )] , (97)implying a different geodesic structure for the matter fields.As an example of how delicate this issue is, let us focus on the weak energy conditionas a simple instance of a situation in which care must be taken when comparing thestructure of modified gravity theories with GR. In GR, the weak energy condition isusually interpreted as the one that guarantees positiveness of energy density in a locallyinertial frame i.e., ρ >
0. Actually, in a coordinate-independent way, the weak energycondition reads T µν U µ U ν > , (98)where U µ is any timelike vector.It is a common practice in the literature to replace T µν in the above inequality by¯ T µν i.e., one writes down an effective energy density ¯ ρ which depends on the curvatureterms and interprets ¯ ρ > W , one can find situations unacceptable situations where ¯ ρ > ρ , derived from the microphysical description of the matter fields, is negative. XII. CONCLUSIONS
Modified gravity theories represent a fruitful way to study the astronomical obser-vations behind the dark matter/energy phenomena. While the direct searches for thecosmic dark components do not deliver positive results proving the existence of exoticmatter fields in nature, there is room for investigations aiming to explain the observedastrophysical/cosmological data via the introduction of geometrical features beyondGR. In order to promote departures from GR, one has to either abandon one (or more)of the pillars over which GR has been built up, or add new fields. A modificationthat leads to a new modified gravitational theory can be seen as more or less radicaldepending on how strong the assumptions it is based on are.In this contribution, we have discussed some of the attempts found in literatureto explore alternative gravity theories in which the null covariant divergence of theenergy–momentum tensor is not achieved. Such non-conservation appears in a modi-fied gravitational theory by different methods. As reviewed in this work, this occurseither by imposing in an ad hoc manner the non-vanishing of the covariant deriva-tive of the energy–momentum tensor or obtaining this feature as a consequence froma first principle construction. Though we have probably failed to mention all existingproposals, the main ideas that have motivated current research in this field have beendiscussed here. In addition, interesting proposals in which there appears violation ofthe energy–momentum conservation include e.g., cosmological diffusion effects [96–98]and other physical mechanisms [99–102].We have also revisited in Section XI how subtle the direct application is of well-established results of the general relativistic framework to the case of nonconservativetheories. In particular, the application of energy conditions when T µν ; ν = 0 requires acareful treatment not usually seen in the literature.While one can not find conclusive evidence that nonconservative theories of gravity6should be ruled out as viable alternatives, they will stay on the market and be a matterof intense investigation as seen currently in the literature. Acknowledgments
We thank three anonymous reviewers for provided helpful comments on earlier draftsof the manuscript. We acknowledge discussions with Jose Beltr´an Jimenez, J´ulio Fabris,Oliver Piattella and Saulo Carneiro. This research was partially funded by CAPES,CNPq, FAPES and Proppi/UFOP. [1] Brading, K. A Note on General Relativity, Energy Conservation, and Noether’s Theo-rems.
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