Abstract

We present some basic concepts of a theory of modified gravity, inspired by the gauge theories, where the commutator algebra of covariant derivative gives us an added term with respect to the General Relativity, which represents the interaction of gravity with a substratum. New spherically symmetric solutions of this theory are obtained and can be viewed as solutions that reproduce the mass, the charge, the cosmological constant, and the Rindler acceleration, without coupling with the matter content, that is, in the vacuum.

1. Introduction

Through the need for modeling the phenomena and symmetries that the nature can present, the mathematical tools more used were the group. A group obeys some specified algebraic properties that characterize it. Several theories of group that served to model the symmetries of various quantum phenomena are well known and are for great success [1]. The standard model of interactions is basically divided into the Electroweak Interactions model, developed by Abdus Salam, Sheldon Glashow, and Steven Weinberg [2], whose symmetries group is 𝑆𝑈(2)×𝑈(1), and the Strong Interactions characterized by the symmetries group 𝑆𝑈(3) [1]. Note that this model does not include the gravitational interaction.

Various models of grand unification emerged in order to provide the unification of the elementary interactions. Some of the most important were those of Weyl [3–6], Supergravity [7], Strings and Superstrings [8], and Loop Quantum Gravity [9]. In all these models the group theory has an important role; it is an ingredient of the description of the known revealed symmetries of the nature.

As in the case of quantum or microscopic phenomena, the comprehension of the content and the evolution of the universe has grown in recent years. With a great advance in measurements of the cosmological data, and the increase of cosmological and astrophysical measures, it has been obtained the acceleration and the strange content of the universe, which according to the General Relativity (GR) are the dark matter and the dark energy. A not-so-small class of modifications of the GR has emerged as an attempt of a new approach to explain consistently with more recent data, as WMAP [10] and type Ia Supernovae [11, 12]. The theories of modified gravity most widely used as alternatives that allow us to understand the analysis of the expansion phenomenon of the universe or the dark matter content are the TEVES [13], the 𝑓(𝑅) [14, 15], the 𝑓(𝐺) [16] (with 𝑓(𝑅,𝐺) [17] and 𝑓(𝑅,𝑇) [18]), and the 𝑓(𝑇) [19] (generalization of the Teleparallel Theory [20]).

Inspired by Lorentz-Yang-Mills gauge theory, whose symmetries group is 𝑆𝑂(3,1), Camenzind [21], in different order, constructs a modification of the GR in which the Lagrangian contains a quadratic term of the scalar curvature and the quadratic contraction of the Ricci tensor. Using the Levi-Civita connection and an analogy with Yang-Mills theory, he obtains the field equations in which the energy-momentum tensor is not conserved. The conserved quantity in his theory, due to the Bianchi identity, is the 4-current associated with the matter content, that is, the energy momentum tensor.

Similarly, later, inspired by the foundations of the gauge theory, Mignani et al. [22] apply a new methodology for the commutators algebra of covariant derivatives which is the construction of a modification of the GR and which leads to the GR in a particular case. In the same way as Camenzind [21], Mignani et al. found an imposition on the theory, that is, the 4-current acts as the source of the matter content and is conserved, while the energy-momentum tensor is not. Through an approximate analysis, they get for the static and spherically symmetric case an increase in mass of a Schwarzchild-type black hole, which is a correction with respect to the GR in the light deflection factor, which travels next to an intense gravitational field. This correction leads to a value closer to the measured light deflection by both the eclipse in Sobral in 1919 [23, 24], as the more recent measurements [24, 25]. They also obtain some results for the weak field approximation, in gravitational waves, and a time-dependent cosmological constant, which does not need to be entered as additional term, but being a direct consequence of the nonconservation of the energy-momentum tensor.

In this paper, we will explain more clearly some basic concepts of the theory constructed by Mignani et al. [22], through the original inspiration of the gauge theory and the commutator algebra of covariant derivatives [26], for a Riemannian spacetime, defined through the Levi-Civita connection. We reobtain the equations of motion of this theory, taking into account all original ingredients used by Mignani, getting a model that possesses an equation of motion different from that presented in [22]. We will analyse in this paper only the vacuum case in which the metric is static and possesses spherical symmetry.

This paper is organized as follows. In Section 2, we present some fundamental concepts for the introduction to the Algebraic Modified Gravity, divided in Section 2.1, in which we put out the elements of the GR, and the Section 2.2, in which we construct the theory. In Section 3, we will present some new solutions for the vacuum case, with a static and spherically symmetric metric, comparing them with the known solutions in the literature and drawing a preliminary analysis. The conclusion and perspective are presented in the Section 4.

2. Fundamental Concepts of the Theory

2.1. The General Relativity

Let us firstly rewrite some equations coming from the GR, to be used as the inspiration to the modification made by Mignani et al. [22]. The spacetime is considered to have the Riemannian manifold, in which the line element is given by 𝑑𝑆2=𝑔𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈,(1) where 𝑔𝜇𝜈 are the components of the metric. The signature of the metric is defined as (+,−,−,−). For making a parallel transport of the vector field 𝑉𝛼(𝑥𝛾), between two infinitesimally close points, we define the covariant derivative as being ∇𝜇𝑉𝛼=𝜕𝜇𝑉𝛼−Γ𝜆𝜇𝛼𝑉𝜆,(2) where Γ𝜆𝜇𝛼 are the components of the Levi-Civita connection. If we apply the commutator of the covariant derivatives ∇𝜇 and ∇𝜈, on a vector 𝑉𝛼, we get ∇𝜇,∇𝜈𝑉𝛼=−𝑅𝜆𝛼𝜇𝜈𝑉𝜆,(3) where we define the Riemann tensor components 𝑅𝜆𝛼𝜇𝜈=𝜕𝜇Γ𝜆𝜈𝛼−𝜕𝜈Γ𝜆𝜇𝛼−Γ𝜎𝜇𝛼Γ𝜆𝜈𝜎+Γ𝜎𝜈𝛼Γ𝜆𝜇𝜎.(4) Using two times the commutator of the covariant derivatives ∇𝜇, ∇𝛼, and ∇𝛽 and making the cyclic subscripts, we get the Jacobi identity ∇𝜇,∇𝛼,∇𝛽+∇𝛼,∇𝛽,∇𝜇+∇𝛽,∇𝜇,∇𝛼=0,(5) which in terms of the Riemann tensor in (3) is written as ∇𝜇𝑅𝜆𝜎𝛼𝛽+∇𝛼𝑅𝜆𝜎𝛽𝜇+∇𝛽𝑅𝜆𝜎𝜇𝛼=0.(6) This latter is called the first Bianchi identity. Contracting 𝜆 with 𝛼 in (6), one obtains ∇𝛼𝑅𝛼𝜎𝛽𝜇=∇𝛽𝑅𝜎𝜇−∇𝜇𝑅𝜎𝛽,(7) where 𝑅𝜎𝜇 are the components of the Ricci tensor. Contracting 𝜎 with 𝛽 leads to the second Bianchi identity ∇𝛼𝑅𝛼𝜇−12𝛿𝛼𝜇𝑅=0,(8) where 𝑅=𝑔𝜇𝜈𝑅𝜇𝜈 is the curvature scalar. Considering the Einstein equations of the GR 𝑅𝜎𝜇−12𝑔𝜎𝜇𝑅=𝜒𝑇𝜎𝜇(9) and inserting it in (7) one gets ∇𝛼𝑅𝛼𝜎𝛽𝜇=𝜒𝐽𝜎𝛽𝜇,(10) where 𝐽𝜎𝛽𝜇=∇𝛽𝑇𝜎𝜇−∇𝜇𝑇𝜎𝛽−12𝑔𝜎𝜇∇𝛽𝑇−𝑔𝜎𝛽∇𝜇𝑇,(11) and 𝑇 is the trace of the energy-momentum tensor. Equations (10) are called the old quasi-Maxwellian gravitational equations [27]. Hence, all this is well known and established in the GR. In the next section, we will introduce the fundamental concepts of the theory constructed by Mignani et al. [22].

2.2. The Modified Algebraic Gravity

A modification of the GR made by Camenzind [21] was constructed as an analogy to the Yang-Mills theory, with the symmetry group 𝑆𝑂(3,1). Similarly, inspired by the foundations of the gauge theory, Mignani et al. [22] construct a modified theory of gravity, in which the algebra of the covariant derivatives operators adds a term to the equations of motion. In order to present some basic concepts of the theory, we will start applying the commutator of ∇𝜇 with the commutator of ∇𝛼 with ∇𝛽, to a vector field 𝐾𝜈, and we obtain ∇𝜇,∇𝛼,∇𝛽𝐾𝜈=∇𝜇∇𝛼,∇𝛽𝐾𝜈−∇𝛼,∇𝛽∇𝜇𝐾𝜈∇=−𝜇𝑅𝜆𝜈𝛼𝛽𝐾𝜆+𝑅𝜆𝜇𝛼𝛽∇𝜆𝐾𝜈,(12) then, ∇𝜇,∇𝛼,∇𝛽𝐾𝜈=∇𝜇𝑅𝜆𝜈𝛽𝛼𝐾𝜆+𝑅𝜆𝜇𝛼𝛽∇𝜆𝐾𝜈.(13) This relationship is demonstrated by Mignani et al. [28] through the commutative diagrams, as used in category theory. Using the matter source 𝐽𝜎𝛽𝜇 in (11), we can regain the Einstein equations of GR, for the particular case where the covariant derivative of the vector field 𝐾𝜈 vanishes identically. But firstly, we obtain the general equation of motion for this theory.

Considering as matter source, the same as that of the GR, the 𝐽𝜎𝛽𝜇 in (11), we can match the expression (13), with the new source condition ∇𝜇,∇𝛼,∇𝛽𝐾𝜈=𝜒𝐽𝜇𝛼𝛽𝐾𝜈,(14) which lead to 𝜒𝐽𝜇𝛼𝛽𝐾𝜈=∇𝜇𝑅𝜆𝜈𝛽𝛼𝐾𝜆+𝑅𝜆𝜇𝛼𝛽∇𝜆𝐾𝜈.(15) Contracting 𝜇 with 𝛼 and 𝛽 with 𝜈, we obtain the following general equation of motion: ∇𝜇𝑅𝜇𝜈𝑇+𝜒𝜇𝜈+12𝑔𝜇𝜈𝑇𝐾𝜈+𝑅𝜇𝜈∇𝜇𝐾𝜈=0.(16) In the particular case where ∇𝜇𝐾𝜈=0, considering the validity of Einstein’s equations (9), we reobtain the second Bianchi identity (8) for a nonzero vector field 𝐾𝜈. These motion equations are not stemming directly from a Lagrangian or a variational principle, as the case of the Rastall theory [29]. All the matter fields, as scalar, vector, or spinorial fields, are contained in the energy-momentum tensor, which gives us the interaction between the matter and the geometry, as in RG.

Equation (16) can be interpreted as the equation of motion of a theory which is viewed as a generalization of the GR, coming from the foundations of gauge theory, in which the added term with respect to the GR comes from the covariant derivatives commutator. Indeed, we call this theory the Algebraic Modified Gravity. The vector field 𝐾𝜈(𝑥𝜎) can be interpreted as a field that describes the interaction with a substratum [22]. As in gauge theories, the vector field 𝐾𝜈 can be a gauge field of a certain local (as 𝑆𝑈(2) gauge group, e.g.) or global (as 𝑆𝑈(3)) gauge group [30]. In the case used, the second term of the sum in (16) is the main coupling term between the substratum and the gravity, in such a way that if this term vanishes, the GR is recovered. The crucial point of the difference with the GR here, is the consideration of the source of the Riemann tensor in (10) for the RG, and being now a source of another operator, like an eigenvalue equation of the double commutator of covariant derivatives in (14).

3. New Solutions in Modified Algebraic Gravity

To begin with an application of the theory, let us take the case in which the matter content is identically zero, 𝑇𝜇𝜈=0 which implies that 𝐽𝛼𝜇𝜈=0. Equation (16) for this case is written as ∇𝜇𝑅𝜇𝜈𝐾𝜈+𝑅𝜇𝜈∇𝜇𝐾𝜈=0.(17)

As a possible simplification, we can take the eigenvalue condition [22]: ∇𝜇𝐾𝜈=𝛾𝜇𝐾𝜈,(18) which in (17) yields ∇𝜇𝑅𝜇𝜈+𝛾𝜇𝑅𝜇𝜈𝐾𝜈=0.(19) As the vector field is nonzero in general, we have ∇𝜇𝑅𝜇𝜈+𝛾𝜇𝑅𝜇𝜈=0.(20) We can also see that in the particular case where the eigenvalues 𝛾𝜇=0 (∇𝜇𝐾𝜈=0), the GR is recovered in the vacuum, which is the second identity Bianchi (8), for 𝑅=0. For a first approach, we chose a static spacetime with a spherical symmetry. The line element (1) of this spacetime can be written as 𝑑𝑆2=𝑒𝜈(𝑟)𝑑𝑡2−𝑒𝜆(𝑟)𝑑𝑟2−𝑟2𝑑𝜃2+sin2(𝜃)𝑑𝜙2.(21) The components of the inverse metric are given by 𝑔00=𝑒−𝜈(𝑟),𝑔11=−𝑒−𝜆(𝑟),𝑔22=−𝑟−2,𝑔33=−𝑟−2sin−2(𝜃).(22) The nonzero components of the Levi-Civita connection are Γ010=12𝜈(𝑟),Γ100=12𝜈(𝑟)𝑒𝜈(𝑟)−𝜆(𝑟),Γ111=12𝜆(𝑟),Γ122=Γ133=−𝑟𝑒−𝜆(𝑟),Γ221=Γ331=𝑟−1,Γ233=−sin𝜃cos𝜃,Γ332=cot𝜃,(23) where denotes the derivative with respect to the radial coordinate 𝑟. The nonzero components of the Ricci tensor are 𝑅00=14𝑒𝜈(𝑟)−𝜆(𝑟)2𝜈𝜈(𝑟)+(𝑟)2−𝜈(𝑟)𝜆4(𝑟)+𝑟𝜈,𝑅(𝑟)111=−42𝜈𝜈(𝑟)+(𝑟)2−𝜈(𝑟)𝜆4(𝑟)−𝑟𝜆,𝑅(𝑟)22=12𝑒−𝜆(𝑟)𝑟𝜆(𝑟)+2𝑒𝜆(𝑟)−𝑟𝜈,𝑅(𝑟)−233=sin2𝜃𝑅22.(24) We chose a more particular case, where there is only one nonzero eigenvalue of (18), 𝛾1(𝑟), which depends only on the radial coordinate 𝑟. We can rewrite (20) as 𝑔𝜇𝜎∇𝜎𝑅𝜇𝜈+𝛾𝜇𝑅𝜇𝜈=0,(25) where 𝛾𝜇=𝑔𝜇𝜎𝛾𝜎. Using 𝜈=1, (25) becomes 𝑔11𝜕1𝑅11−𝑅11𝑔00Γ100+2𝑔11Γ111+𝑔22Γ122+𝑔33Γ133−𝛾1−𝑔00Γ001𝑅00−𝑔22Γ221𝑅22−𝑔33Γ331𝑅33=0.(26) Substituting (22), (23), and (24) in (26), and after some algebraic manipulation, we obtain the following differential equation: −8+8𝑒𝜆(𝑟)+2𝑟3𝜈(𝑟)−4𝑟𝜈(𝑟)+𝑟2𝜆(𝑟)24+𝑟𝜈(𝑟)−4𝑟2𝜆(𝑟)−𝑟3𝜈(𝑟)𝜆(𝑟)+4𝑟2𝜈(𝑟)+2𝑟3𝜈(𝑟)𝜈(𝑟)−𝑟2𝜆4𝜈𝜈(𝑟)+𝑟(𝑟)2+3𝑟𝜈(𝑟)+𝑒𝜆(𝑟)𝑟2𝛾1𝜆(𝑟)(𝑟)4+𝑟𝜈𝜈(𝑟)−𝑟(𝑟)2+2𝜈(𝑟)=0.(27) This is a nonlinear differential equation of the third order, but it can be simplified to obtain exact solutions, as we will see later. Choosing the system of quasi-global coordinates where the metric functions satisfy 𝜈(𝑟)=−𝜆(𝑟)(𝑔11=−𝑔00), and making the redefinition []𝜈(𝑟)=−𝜆(𝑟)=ln1+𝑓(𝑟),(28) the differential equation (27), after some algebraic manipulations, becomes 𝑟3𝑓(𝑟)−4𝑓(𝑟)−2𝑟𝑓(𝑟)1+𝑟𝛾1(𝑟)1+𝑓(𝑟)+𝑟2𝑓(𝑟)4−𝑟𝛾1(𝑟)1+𝑓(𝑟)=0.(29) Remembering that 𝛾1(𝑟)=𝑔11𝛾1(𝑟)=−(1+𝑓(𝑟))𝛾1(𝑟), (29) leads to 𝑟3𝑓(𝑟)−4𝑓(𝑟)−2𝑟𝑓(𝑟)1−𝑟𝛾1(𝑟)+𝑟2𝑓(𝑟)4+𝑟𝛾1(𝑟)=0.(30) This differential equation is now linear and of the third order. For each type of 𝛾1(𝑟), we get a possible solution of this equation. We will show here some interesting cases.

(1) The first model we will take is that of GR, that is, 𝛾1(𝑟)=0. Equation (30) becomes 𝑟3𝑓(𝑟)−4𝑓(𝑟)−2𝑟𝑓(𝑟)+4𝑟2𝑓(𝑟)=0.(31) The general solution for this equation is given by 𝑐𝑓(𝑟)=1𝑟+𝑐2𝑟2+𝑐3𝑟2,(32) where 𝑐1, 𝑐2, and 𝑐3 are real constants. When we choose 𝑐1=−2𝑀, 𝑐2=𝑞2, and 𝑐3=Λ/3, with 𝑀 being the total mass, 𝑞 the charge, and Λ the cosmological constant, we have the solution of Reissner-Nordstrom-(anti)-de Sitter (RN-(A)dS). Equation (31) is obtained directly from the second Bianchi identity (8) for the vacuum (𝑅=0). Hence, (32) is solution of the second Bianchi identity (8) for the vacuum, that is, a particular solution of GR.

(2) Now let us take the model where 𝛾1(𝑟)=−𝐴, with 𝐴 being a real constant. Equation (30) becomes 𝑟3𝑓(𝑟)−4𝑓(𝑟)+𝑟2𝑓(𝑟)(4−𝐴𝑟)−2𝑟𝑓(𝑟)(1+𝐴𝑟)=0.(33) A particular solution of this equation in order to eliminate uninteresting terms, is given by 𝑐𝑓(𝑟)=2𝐴2+𝑐1𝑟−6𝑐2𝐴4𝑟2+4𝑐2𝐴3𝑟ln(𝑟),(34) where 𝑐1 and 𝑐2 are real constants. This solution is not asymptotically Minkowskian for 𝑐2≠0. When 𝑐1=−2𝑀 and 𝑐2=−𝑞2𝐴4/6, the solution resembles that of RN with a complicated logarithmic term. This term appears in the solutions of (1+2)-Dimensional Teleparallel Gravity [31] and Symmetric Teleparallel Gravity [32]. Note also that this sort of term has been obtained, in 𝑓(𝑅) theory, in the limit of the fourth post-Newtonian order correction [33, Equation (16.100)]. The signature of the metric in the asymptotic limit depends on the constants 𝑐2 and 𝐴. When 𝑐2=−𝑞2𝐴4/6and √−1<(|𝑞|𝐴/6)<1, the asymptotic signature is (+,−,−,−), and otherwise (−,+,−,−).

(3) When 𝛾1(𝑟)=−𝐴𝑟, with 𝐴 being a real constant, (30) becomes 𝑟3𝑓(𝑟)−4𝑓(𝑟)+𝑟2𝑓(𝑟)4−𝐴𝑟2−2𝑟𝑓(𝑟)1+𝐴𝑟2=0.(35) A particular solution of (35) is given by 𝑐𝑓(𝑟)=−2𝐴+𝑐1√2𝐴𝑟+2𝑐2𝐴2𝑟2,(36) where 𝑐1 and 𝑐2 are real constants. In the same way as for the previous solution, this solution is not asymptotically Minkowskian for 𝑐2≠0. When 𝑐1√=−2𝑀/2𝑎and 𝑐2=(𝑞𝐴)2/2, the solution is similar to that of RN. The signature of the metric in the asymptotic limit depends on the constant 𝑐2 and 𝐴. When 𝑐2=(𝑞𝐴)2/2and √−1<(|𝑞|𝐴/2)<1, the asymptotic signature is (+,−,−,−), otherwise is (−,+,−,−).

(4) When 𝛾1(𝑟)=−𝐴/𝑟, with 𝐴 being a real constant, (30) becomes 𝑟3𝑓(𝑟)−4𝑓(𝑟)+𝑟2𝑓(𝑟)(4−𝐴)−2𝑟𝑓(𝑟)(1+𝐴)=0.(37) The general solution of (37) is given by 𝑐𝑓(𝑟)=1𝑟+𝑐2𝑟√(1/2)(𝐴−𝐴2+16)+𝑐3𝑟√(1/2)(𝐴+𝐴2+16),(38) where 𝑐1, 𝑐2, and 𝑐3 are real constants. This solution is a generalization of the solution of RN-(A)dS. When 𝐴=0 (𝛾1(𝑟)=0), the GR is recovered, with the solution of RN-(A)dS in (32). The exponent 𝑟 of the constants terms 𝑐2 and 𝑐3 can take real values in general. Similar solutions arise in modified gravity, as in the case of 𝑓(𝑅) theory with 𝑅 being the scalar curvature. The powers of the radial coordinate 𝑟 in the Clifton paper [34] depend on the value of 𝛿, which is power of 𝑅 (𝑓(𝑅)=𝑅1+𝛿). Note that the solution proposed here is more general in the sense that power of 𝑟 is general and also there exists a mass term of order 1/𝑟.

(5) When 𝛾1(𝑟)=−𝐴/𝑟2, with 𝐴 a real constant, (30) reads 𝑟3𝑓(𝑟)−4𝑓(𝑟)+𝑟𝑓(𝑟)(4𝑟−𝐴)−2𝑓(𝑟)(𝑟+𝐴)=0.(39) The general solution of (39) is given by 𝐴𝑓(𝑟)=2𝑐2𝐴2+1−4𝐴+6𝑟𝑐1−𝐴2−2𝐴+2𝐴2𝑐−4𝐴+62−2𝐴𝑐2𝐴2−4𝐴+6𝑟+2𝑐2𝐴2𝑟−4𝐴+62+𝑐3𝑟2𝐴4𝐴2𝑒−4𝐴+6−𝐴/𝑟,(40) where 𝑐1, 𝑐2, and 𝑐3 are real constants. As in the two previous solutions, this solution is not asymptotically Minkowskian for 𝑐2≠0 and 𝑐3≠0. When 𝑐1=−2𝑀+(Λ/6)(𝐴2−2𝐴+2) and 𝑐2=(Λ/6)(𝐴2−2𝐴+2), the solution is similar with that of S-(A)dS, with a linear term and other exponential one that differentiates these solutions. An interesting term is the linear one, in (40), which is connected with the Rindler acceleration, which is more appreciable for large distances [35]. This Rindler acceleration 𝑎 with 𝑎=−𝐴𝑐2/(𝐴2−4𝐴+6) is in the direction of the source when 𝑎>0 and forth from the source when 𝑎<0. In fact, this term appears firstly in the solution of a theory whose Lagrangian has the quadratic contraction of the Weyl tensor, or a quadratic term of the scalar curvature 𝑅, added to a quadratic contraction term of the Ricci tensor. This solution was first obtained by Mannheim and Kazanas [36], where, according to their parameters, 𝛾=−𝐴𝑘, 𝛽=−(𝐴/6), 𝑘=2𝑐2/(𝐴2−4𝐴+6), and 𝑐3=0. They showed that, in this case, the solution interpolates the Schwarzschild’s one and also that of Robertson-Walker. This is not viewed directly as in [36] with 𝛽=0, which in our case leads to 𝐴=0. In our case, we must first assume that the constant term in (40) is much smaller than unity, which is acceptable due to the fact that it is proportional to the cosmological constant Λ in a particular case. Then, we choose 𝑐1=(Λ/6)(𝐴2−2𝐴+2), such that the mass is zero, and then it can be compared with the case of [36]. Just as in [35], the linear term in 𝑟 starts to dominate for large distances. Note that our solution is more general because of the exponential term. This linear term also has been obtained for the second-order post-Newtonian corrections in 𝑓(𝑅) theory [33, Equation (16.87)]. Also, note that the exponential term, of the form of Yukawa correction, has been found from the second-order post-Newtonian corrections in the 𝑓(𝑅) theory [33, Equation (16.74)]. However, it is important to mention that we have here the inverse of 𝑟 in the exponential argument, and then the correction would be observed in a cosmological constant term, proportional to 𝑟2, which is not the case of the 𝑓(𝑅) theory where the correction appears in a mass term (proportional to 𝑟−1).

(6) We could have several cases for the function 𝛾1(𝑟)=−𝐴𝑟𝑛, with an integer 𝑛, but all provide us very long solutions of little interest, which are not necessary to be shown here. We only put out these possibilities. A simple example is when 𝑛=−3, that is, 𝛾1(𝑟)=−𝐴/𝑟3. The general solution of (30) is 𝑓(𝑟)=−𝐴𝑐2+1𝐴−2𝑟𝑐1−2−3𝐴𝑐3(𝐴−2)2+2𝑐2𝑟3(𝐴−2)2−𝑐36(𝐴−2)𝐴2𝑟𝑟2𝑟2𝑒−𝐴−(𝐴/2𝑟2)−𝐴3/2√2𝜋𝐸𝑟𝐴21𝑟,(41) where √𝐸𝑟(𝑥)=(2/∫𝑒𝜋)−𝑥2𝑑𝑥. Using the particular case where 𝑐3=0, this solution is similar to that of S-(A)dS, for 𝑐1=−2𝑀+(Λ/6)(2−3𝐴) and 𝑐2=(Λ/2)(𝐴−2).

4. Conclusion

We present some basic concepts of the theory called Modified Algebraic Gravity through known elements in the GR and taking as inspiration the foundations of gauge theory. Our goal in revisiting this theory is to investigate the reconstruction of some black holes solutions, well known and already obtained from the GR and the most usual modified theories of gravitation, as 𝑓(𝑅) and 𝑓(𝑇). We obtained the equations of motion for this theory and chose as a first application the static vacuum with spherical symmetry. The equations for this particular case are nonlinear differential equations of the third order, coming from the Bianchi identities coupled with a term proportional to the Ricci tensor. Considering the covariant derivative of the gauge field 𝐾𝜈, which represents the interaction of gravity with the substratum, providing an equation as that of eigenvalue, we might rewrite the equation of motion as a linear differential equation of the third order. In such a way, we have shown some consistency with the well-established results, and then getting a trustable direct analogy with the usual modified theories, allowing us to obtain new results.

We divided the solutions obtained in this paper into six cases. The first is when the eigenvalue 𝛾1(𝑟)=0, and we reobtained the solution of RN-(A)dS of the GR. The second is when 𝛾1(𝑟)=−𝐴, with 𝐴∈ℛ, and we get a solution that generalizes the RN by a logarithm term, and the possibilities of asymptotic signature, 𝑟→+∞, and (+,−,−,−) and (−,+,−,−). The third is when 𝛾1(𝑟)=−𝐴𝑟, which results in a solution that resembles that of RN, with the possibility of asymptotic signature (+,−,−,−) and (−,+,−,−). The fourth case is when 𝛾1(𝑟)=−𝐴/𝑟, which results in a solution that generalizes the RN-(A)dS in any real power of radial coordinate 𝑟. This solution generalizes the Clifton [34] by a mass term proportional to 1/𝑟. The fifth case is when 𝛾1(𝑟)=−𝐴/𝑟2, which results into a solution that generalizes the S-(A) dS for two terms, one linear in 𝑟, which represents the Rindler acceleration, as in [35], and another which presents an exponential form. The solution, in a particular case, can interpolate the Schwarzschild and Robertson-Walker ones, as shown in [36]. We still have several possibilities for solutions for 𝛾1(𝑟)=−𝐴𝑟𝑛, with 𝑛 an integer, but which are very long and are not explicitly presented here. We only put out the case where 𝑛=−3, the sixth case, as the simplest example, resulting into a solution similar to that of S-(A) dS, for 𝑐3=0 in (41).

Then, we wrap up that, with this theory, we were able to reproduce partially and even fully, at least in the vacuum, some results previously obtained by the above usual modified theories of gravitation. In such a way, we believe that, with this theory, further investigations could lead to more information which are not clear until now about the dark sector of the universe. Note that this kind of work, in investigating the dark sector of the universe, has previously been made by Licata and Sakaji [26]. However, there are still many things to do in this way. Also, there is still a wide range of possible solutions, both static and cosmological to this theory. These aspects will be undertaken in our future works, as well as the introduction of a more complex structure for the field gauge 𝐾𝜈 in (16), as 𝑈(1) or 𝑆𝑈(2) gauge groups, for testing with vehemence this theory and trying to make some explanations to some points that continue being a challenge until now about the dark sector of the universe, and also obtaining new results in local gravitation and astrophysics, consistent with the observational data.

Acknowledgments

M. H. Daouda thanks CNPq/TWAS for financial support. M. E. Rodrigues thanks UFES for the hospitality during the development of this work. M. J. S. Houndjo thanks CNPq for partial financial support.