General approach to the Lagrangian ambiguity in $f(R, T)$ gravity

The $f(R,T)$ gravity is a theory whose gravitational action depends arbitrarily on the Ricci scalar, $R$, and the trace of the stress-energy tensor, $T$; its field equations also depend on matter Lagrangian, $\mathcal{L}_{m}$. In the modified theories of gravity where field equations depend on Lagrangian, there is no uniqueness on the Lagrangian definition and the dynamics of the gravitational and matter fields can be different depending on the choice performed. In this letter, we have eliminated the $\mathcal{L}_{m}$ dependence from $f(R,T)$ gravity field equations by generalizing the approach of Moraes [Eur. Phys. J. C 79(8), 674 (2019)]. We also propose a general approach where we argue that the trace of the energy-momentum tensor must be an"unknown"variable of the field equations. The trace can only depend on fundamental constants and few inputs from the standard model. We show that our proposal resolves two limitations: First the energy-momentum tensor of the $f(R,T)$ gravity is not the perfect fluid one; second, the Lagrangian is not well-defined.


Introduction
General Relativity (GR) is one of the cornerstones of modern physics being stated as the standard model of gravitation and cosmology. However, in the last years, shortcomings came out in the Einstein's theory and the investigations whether GR is the fundamental theory capable of explaining the gravitational interaction in different regimes initiated.
Combined data from Cosmic Microwave Background Radiation (CMB) [2] and from Baryonic Acoustic Oscillations (BAO), indicate that the Universe is spatially flat, it is in accelerated expansion [3,4], and it is composed of 96% of unknown matter-energy, commonly known as dark matter a e-mail: geanderson.araujo.carvalho@gmail.com b e-mail: rocha.pereira.flavia@gmail.com c e-mail: ronaldo.lobato@tamuc.edu and dark energy respectively. It is widely accepted that the reason for the present accelerated expansion phase of the Universe is due dark energy [5][6][7], while that an invisible matter (or dark matter) accounts for the galaxies' rotation curves flatness [8,9].
To overcome this situation different researchers came up with more sophisticated gravity theories by modifying the Einstein-Hilbert action, which gave arise a new avenue known as modified or extended theories of gravity. The extended theories of gravity have born out as an opportunity to solve problems which are still without explanation within GR framework. The f (R) theory of gravity is one of the most well studied modified theory, and consists of choosing a more general action to replace the Einstein-Hilbert one, assuming that the gravitational action is an arbitrary function of the Ricci scalar R as discussed in Refs. [10,11].
In this letter we are particularly interested in the f (R, T ) theory of gravity that is a generalization of f (R) type theories of gravity. The f (R, T ) gravity, proposed by Harko et al. [12], consists of choosing a gravitational action as an arbitrary function of the Ricci scalar and also the trace of the energy-momentum tensor T . Moraes [1] has used f (R, T ) = R + f (T ) to calculate the trace of the f (R, T ) gravity field equations. In this case the author describes only a minimal coupling between the Ricci tensor and an arbitrary function of the energy-momentum tensor, i.e., a specific model. Here, we are going further in calculating the trace of the f (R, T ) gravity field equations and deriving a new field equation for the theory that does not depend on the matter Lagrangian. In our approach matter and curvature can have a more complex coupling, i.e., it is a general approach. As pointed by [13], a more rich phenomenology could arise from a non-minimal geometry-matter coupling, what is within the motivations behind the present letter.
This new general approach eliminates the Lagrangian ambiguity choice. We argue that the trace of the energy-arXiv:2008.13326v1 [gr-qc] 31 Aug 2020 momentum tensor is the macroscopic description of the more fundamental gravity structure, i.e., it is the quantity that encodes the degree of freedom of the matter to the scalar curvature. We show that our propose resolves two limitations: the Lagrangian choice, and the fact that the energy-momentum tensor cannot be the perfect fluid. From this novel approach we consider that the trace of the energy-momentum is an "unknown" variable, and thus, the trace of the field equations can be exploited to eliminate it.
This letter is organized as follows: Section 2 presents an basic overview on general properties of the f (R, T ) gravity, in Section 3 we derive the traceless field equations for a generic f (R, T ) functional, in Section 4 we present a consistent approach for the Langrangian ambiguity choice and in Section 5 we conclude and discuss possible applications of the theory presented here.
The f (R, T ) gravity is derived by adopting the following gravitational action [12] where f (R, T ) is a generic function of the Ricci scalar R, and to the trace T of the energy-momentum tensor T µν . L represents the matter Lagrangian density. Natural units are adopted and metric signature -2. By variation of the action (1) with respect to the metric tensor g µν , one obtains the field equations of the f (R, T ) gravity theory as follows where 2 is the D'Alambertian operator, R µν is the Ricci tensor and ∇ µ represents the covariant derivative associated with the Levi-Civita connection of g µν . For sake of simplic- Taking the trace of (2) we obtain Combining (2) with (3) yields The covariant derivative of the stress-energy tensor is given by where L can be eliminated from Eq. (3). As one can see, the four-divergence is non-null and in a traceless formulation of the field equations, the f (R, T ) shares a similarity with the unimodular gravity as will see ahead.
3.1 f (R, T ) gravity and unimodular gravity, connection through energy-momentum violation Trying to deal with elementary particles in a geometrical framework, Einstein proposed [14,15] in 1919 a trace-free field equation The formulation derived from the Einstein-Hilbert was persuaded in order to have an understanding in the righthand side of the field equations of General Relativity. The gravitational field equations involve only traceless parts of the Riemann/energy-momentum tensor.
Nowadays, this formulation was reborn as "unimodular gravity", due to a fixation on the metric determinantdetg µν ≡ g = 1, and it is applied to solve the problem of the discrepancy between the vacuum energy density and the observed value of the cosmological constant [16][17][18][19][20].
In Eq. (6), the Bianchi identity still holds for the Einstein tensor, ∇ µ G µν = 0, but the vanishing of the four-divergence of energy-momentum tensor, ∇ µ T µν = 0, is not a geometrical consequence. As have been shown, the difference between the field equations in unimodular and in GR is a scalar stress 1/4(T + R/8π)g µν [16].
The field equations are derived by restricting the variations preserving the volume form. These restrictions lead to violations of the energy-momentum conservation. For a conservative case, the condition must be satisfied and it leads to GR with cosmological constant, i.e., dark energy.
In the case of f (R, T ) gravity, which is a theory with a presence of coupling in the gravitational field, the nonvanishing of the energy-momentum tensor, Eq. (5), arises without restrictions in the variations and it is associated with particle creation in a quantum level, being plausible that gravitational field theories intrinsically contain effective particle creation in a phenomenological description [21]. Particle creation is a feature in quantum field theories described in curved spacetime and in noncommutative quantum field theories, which is field theory in a noncommutative spacetime and can be interpreted as a low energy limit of a quantum gravity theory. As we stated in a previous work [22], the energy nonconservation in a four dimensional spacetime can be related to a noncommutative compact extra dimension with circular topology. In this regard, a letter by Josset, Perez & Sudarsky [23] considered the unimodular gravity with violation of the conservation of energy-momentum, investigating sources of nonconservation in quantum mechanics. In a first scenario studied by them, is evoked a Markovian equation (used to describe creation and evolution of black holes) of the density matrixρ. This leads to a non-constant average energy E ≡ Tr[ρ,Ĥ]. In the second scenario the nonconservation arises naturally from quantum gravity. In a more recently letter [24], exploring this second case, they showed that the nonconservation arises from the discreteness at the Planck level, similar to our line of thought [22]. They have shown that these quantum phenomena is relevant in a cosmological scale, i.e., the underline granularity of the spacetime would lead to the emergence of an effective dark energy. The relevance of the discreteness arises by the interaction of the gravity with scale-invariance-breaking fields (massive fields could interact with quantum gravity structure and exchange energy with it). The quantity that would describe macroscopically the phenomenon is the trace of the energy-momentum tensor T , which for a perfect fluid is given by T = ρ − 3p, the trace characterizes the breaking of the conformal and scale invariance [25], and it is related to the scalar curvature, therefore captured geometrically by scalar curvature R. A non-vanishing of trace leads to a trace anomaly [26,27].
As we can see, the trace is an important ingredient in the quantum and gravitational level description, and it is intrinsically associated with energy violations. We will use it in a more consistent approach to the Lagrangian problem in f (R, T ) gravity in the next section.

A more consistent approach to the Lagrangian ambiguity choice
In this section we present a new approach to the Lagrangian ambiguity problem in f (R, T ) gravity. Our approach consists of considering the trace of the energy-momentum tensor as a variable of the field equations.
Taking the most general definition [13] of the energymomentum given by and calculating the trace we obtain that Then the field equations become In this way, the field equations become independent of the matter Lagrangian. In a flat spacetime limit, the trace is free of anomaly, however, considering a coupling, we can have trace anomaly, i.e., corrections terms to energy-momentum tensor, which would lead to phenomenological implications as pointed by Perez & Sudarsky [24]. Rewriting the energy-momentum tensor we have We can also calculate the four divergence of the energymomentum tensor by replacing (9) into Eq. (5). One must realize that, from now on, field equations depend only on energy-momentum tensor and its trace. In previous works in f (R, T ) gravity the trace of the energy-momentum tensor depends on matter Lagrangian, being not well-defined. Assuming the trace to be an unknown entity, we can treat it as a variable of the f (R, T ) theory. To solve this issue one can take the trace of the field equations to obtain When taking the trace of the field equations one more equation is added to the problems to be solved. It is worth to quote that in this approach the trace, T , will have a similar role as the curvature scalar, R, in f (R) gravity theories. This approach has two major advances: it solves the Lagrangian choice problem; and it also respects the fact that in f (R, T ) gravity the energy-momentum tensor cannot be the one for perfect fluids. As the energy-momentum tensor is not welldefined as in GR our proposed approach solves this issue by coupling the trace of the field equation to themselves.
Moraes [1] has provided a solution for the Lagrangian choice problem by deriving a field equation for the f (R, T ) gravity that does not depend on the matter Lagrangian. However, he considered the specific case f (R, T ) = R + f (T ). The case studied by Moraes is an advance on f (R, T ) gravity research field, in the sense that now researchers have the possibility to study f (R, T ) gravity with no need for choosing a specific matter Lagrangian, thus, working on a general basis. In addition to Moraes' approach, we consider in this letter a generalization of his seminal idea. Here, we work with no specific case, so, the f (R, T ) functional remains as arbitrary as possible. This study was inspired by the work of Fisher & Carlson [13], where they studied the on-shell Lagrangian problem in f (R, T ) gravity. In their work, they suggest that only cross terms between matter and geometry could survive as a theory which brings new insights for the gravitational theory. Our work here is then presented as a possible way to eliminate the matter Lagrangian 1 as a variable of the field equations. This is done for any f (R, T ) functional, and hence it is also valid for cross terms between matter and geometry.
Another way to remove the matter Lagrangian form field equations is also presented here. Our approach was again motivated by the work of Fisher and & Carlson [13], in the sense that they have shown that the energy-momentum tensor cannot be given by the perfect fluid definition. In this letter, we take the general definition of the energy-momentum tensor to remove the dependence on matter Lagrangian of the field equations. We also argued that trace of energymomentum becomes an unknown variable that can be obtained from the trace of the field equations. Hence, this approach unfold two problems of the f (R, T ) gravity, which are the Lagrangian choice one and the energy-momentum tensor that becomes not well-defined. So, works on f (R, T ) = R+λ T gravity, where λ is a constant, could be reformulated by using our approach. In this regard, new results could be useful to constraint the value of λ in comparison with results of the previous works. 1 for perfect fluids Forthcoming applications of our approaches are encouraged including the study of flat rotation curves of galaxies, cosmological models, astrophysical systems and so on.