Beyond Einstein: A Polynomial Affine Model of Gravity

We show that the effective field equations for a recently formulated polynomial affine model of gravity, in the sector of a torsion-free connection, accept general Einstein manifolds---with or without cosmological constant---as solutions. Moreover, the effective field equations are partially those obtained from a gravitational Yang-Mills theory known as the Stephenson-Kilmister-Yang (SKY) theory. Additionally, we find a generalisation of a minimally coupled massless scalar field in general relativity within a"minimally"coupled scalar field in this affine model. Finally, we present the road map to finding general solutions to the effective field equations with either isotropic or cosmologic (i.e., homogeneous and isotropic) symmetry.


Introduction
During the last century (approximately), we reached a high level of understanding of the four fundamental interactions, i.e., Electromagnetic, Weak, Strong and Gravitational. However, our understanding splits into two streams: The first describes three kinds of interactions and includes them into a single model, called standard model of particle physics; while the second covers only the gravitational interactions.
The interactions within the standard model of particles are described by connection fields, modelled by gauge theories, and their quantization procedure is successfully applied. On the other hand, the gravitational interaction, as formulated by Einstein [1] and Hilbert [2], is described by the metric field, whose model does not fit into the category of gauge theory, and its quantization procedure is not yet well-defined [3][4][5][6][7][8][9] (see Ref. [10] for a historical review).
The above suggests that some of the theoretical problems encountered when trying to quantize the gravitational interactions, 1 are due to fact that it is formulated as a field theory for the metric, and not as a theory for a connection. Therefore, there have been several attempts of describing the gravitational interaction from an affine view point, by using only the connection as fundamental field of the model [11,12], but these descriptions were not very successful. In addition, Cartan's proposal of considering the same field equations (or action) than Einstein-Hilbert, but with more general connections, see Refs. [13][14][15][16], was left aside because the field equations, in pure gravity, impose the vanishing torsion.
Before continuing our arguments for the necessity of considering affine generalisations of General Relativity, we shall briefly remind some basic concepts in geometry. In the antiquity, the plane geometry was built essentially with the aid of a (straight) rule and a compass. Counter-intuitively, the compass was used to measure distances, while the rule was used to define parallelism. In modern differential geometry language, the object that allows us to measure distances is the metric, (g µν ), while the one associated with the concept of parallelism is the connection (Γ λ µν ).
Although the concepts of distance and parallelism are independent, there exists a (unique) particular case in which the concepts relate with each other, and thus one needs just of the metric-while the connection is a potential for the metric. This particular case is known as Riemannian geometry, and General Relativity stands on such particular construction.
In the following sections we will analyse a recently proposed model called Polynomial Affine Gravity [29,30], which is built up with an affine connection as sole field, and under the premise of preserving the whole group of diffeomorphisms.

Polynomial affine gravity: the model
The connection is the field that allows us to define the notion of parallelism as follows. Given a connectionΓ µ ρσ , one defines a covariant derivative, (∇Γ), such that, if the directional derivative of a geometrical object, V, along a vector (X) vanishes, one says that the object is parallel transported along (the integral curve defined by) the vector, The affine connection accepts a decomposition on irreducible components aŝ where Γ µ ρσ =Γ µ (ρσ) is symmetric in the lower indices, A µ is a vector field corresponding to the trace of torsion, and T µ,λκ is a Curtright-like field [47], 2 satisfying T κ,µν = −T κ,νµ and ǫ λκµν T κ,µν = 0. 3 Using the above decomposition, we need to build the most general action preserving diffeomorphisms. In order to guarantee the correct transformation of the Lagrangian density, the geometrical objects used to write down the action are a Curtright (T µ,νλ ), a vector (A µ ), the covariant derivative defined with the Levi-Civita connection (∇ µ ), both Levi-Civita tensors (ǫ µνλρ and ǫ µνλρ ), and the Riemannian curvature (R µν λ ρ ). Since the Riemannian curvature is defined as the commutator of the covariant derivative, it is not an independent field, so it will be left out of the analysis, and only five ingredients remain. In Ref. [30], a method of dimensional analysis was introduced to ensure that all possible terms were taken into account, and the general action-up 2 Notice that the Curtright-like field is defined as the quasi-Hodge dual of the traceless part of the torsion. 3 Since no metric is present, the epsilon symbols are not related by raising (lowering) their indices, but instead we demand that to boundary and topological terms-is Despite the complex structure of the action, it shows very interesting features: (a) The structure is rigid (it does not accept extra terms), which forbid the appearance of counter-terms, if one would like to quantise the model; (b) All coupling constants are dimensionless, which might be a hint of conformal invariance of the model; (c) The action turns out to be power-counting renormalizable, which does not guarantee renormalizability, but is a nice feature; (d) The structure of the model yields no three-point graviton vertices, which might allow to overcome the no-go theorems found in Refs. [48,49].

Limit of vanishing torsion
We now want to restrict ourselves to the limit of vanishing torsion, which simplify the comparison between our model and General Relativity. the vanishing torsion limit-equivalent to take T λ,µν → 0 and A µ → 0-cannot be taken at the action level, but in the field equations, and the limit is a consistent truncation of the whole field equations [30].
The only nontrivial field equation after the limit will be the one for the Curtright-like field, T ν,µρ , with κ a constant related with the original couplings of the model. These field equations are simpler if one restricts to connections compatible with a volume form, also known as equi-affine [32,50,51], which assures that the Ricci tensor of the connection is symmetric, and the contraction of the last indices vanishes, thus the equations are Equation (4) is a generalization of Einstein's field equation in vacuum. This can be seem as follows: All Einstein manifolds posses Ricci tensor proportional to the metric, R µν ∝ g µν , the metricity condition thus ensures that every vacuum solution to the Einstein's equations solves the (simplified) field equations of our model.
Moreover, the Eq. (4) is related through the second Bianchi identity to the harmonic curvature condition [52], The Eqs. (4) and (5) accept a geometrical interpretation equivalent to that of the field equations of a pure Yang-Mills theory, which in the language of differential forms are where F = DA is the field strength 2-form (the curvature 2-form of the connection in the principal bundle, see for example Ref. [53,54]), and the operator ⋆ denotes the Hodge star. Now, these Yang-Mills field equations are obtained from the variation of the action functional and the Jacobi identity for the covariant derivative. Similarly, Eq. (4) (equivalently Eq. (5)) can be obtained from an effective gravitational Yang-Mills functional action [55][56][57], where R ∈ Ω 2 (M, T * M ⊗ T M) is the curvature two-form, the operator ⋆ denotes the Hodge star, and the trace is taken on the bundle indices (see Ref. [53]). The gravitational model described by the action in Eq. (8) is called Stephenson-Kilmister-Yang (or SKY for short), and its physical interpretation relies-as in General Relativity-in the fact that the metric is the fundamental field for describing the gravitational interaction. In that case, the field equations are third order partial differential equations, and there are several undesirable behaviours due to this characteristic of the equations. However, in our model the field mediating the gravitational interaction is the connection and, therefore, the Eqs. (4) and (5) are second order field equations for the components of the connection.
It is worth noticing that, according to the arguments in Refs. [48,49], the SKY theory is not renormalizable. However, it is possible that the polynomial affine gravity could be renormalizable, in the sense that SKY is an effective describtion for the torsionless limit of polynomial affine gravity.

Polynomial affine gravity coupled to a scalar field
In the standard formulation of physical theories (even in flat spacetimes), the metric is a required ingredient. The metric and its inverse define an homomorphism between the tangent and cotangent bundle, allowing to build the kinetic energy term in the action Therefore, the inclusion of matter within models with no necessity of a metric, is a nontrivial task.
Inspired in the method of dimensional analysis introduced in Ref. [30], we attempt to couple a scalar field to the polynomial affine gravity by defining the most general, symmetric 2 0 -tensor density, g µν , built with the available fields, and use it to build Lagrangian densities for the matter content. It can be shown that such a density is given by with α, β and γ arbitrary coefficients. Therefore, the action defined by the "kinetic term" is Remarkably, it induces a nontrivial contribution to the field equations once we restrict to the torsionless sector. The field nontrivial equations, when the scalar field is turned on, are 4 4 We have fixed the coefficient C1 = 1.
which for equi-affine connections simplifies to Equation (11) can be integrated once, and the solution takes the familiar form, where the integration (covariantly) constant tensor, which is invertible and symmetric, has been suggestively denoted by Λg µν . The above equation can be written in the more conventional form Thus, we have been able to recover a set of equations, similar to those of General Relativity coupled with a free, massless scalar field, but with an arbitrary rank two, symmetric, covariantly constant tensor playing the role of a metric. Notice that, from the action in Eq. (10) it is not possible to obtain the field equation for the scalar φ, when the vanishing torsion limit is taken. Nonetheless, the second Bianchi identity in Eq. (4) imposes This condition is, in the sense argued in Ref. [58], the equation of motion for the scalar field.

Finding symmetric ansätze
The usual procedure for solving the Einstein's equation is to propose an ansatz for the metric. That ansatz must be compatible with the symmetries we would like to respect in the problem. The formal study of the symmetries of the fields is accomplish via the Lie derivative (for reviews, see Refs. [54,[59][60][61]). Below, we use of the Lie derivative for obtaining ansätze for either the metric or the connection. The form of the Lie derivative for tensors is well-known, but the Lie derivative for a connection is not. Thus, for the sake of completeness we remind the readers that where ξ is the vector defining the symmetry flow. We shall restrict ourselves to the isotropic (spherically symmetric) and, homogeneous and isotropic (cosmological symmetry). In the tedious task of calculating the Lie derivative of different objects, we have used the mathematical software SAGE together with its differential geometry package SageManifolds [62,63].

Isotropic ansätze
The two-dimensional sphere, S 2 , is a maximally symmetric space whose Killing vectors generate an SO(3) symmetry group. The se vectors can be expressed in spherical coordinates as

Isotropic (covariant) two-tensor
Let us start by finding the most general isotropic, four-dimensional, covariant rank two-tensor. We shall obtain a generalisation of the famous ansatz for the Schwarzschild metric.
We start from a general rank two tensor, i.e., the sixteen components of the tensor depend on all the coordinates. Then, the Lie derivative of the metric along the vector J 3 in Eq. (15) yields which vanishes only if none of the components of the metric depend on the ϕ coordinate. The Lie derivative along the other two generators of the angular momentum yield a nontrivial set of differential equations (not shown here) whose solution fixes a tensor of the form, This result was found by A. Papapetrou [64]. Notice that there are six functions of the coordinates t and r, while the θ dependence is fixed by the symmetry. Two out of the six functions vanish whenever one restrict to symmetric tensors, i.e. g(X, Y ) = g(Y, X), such as the metric tensor. The form of the symmetric, covariant, rank two tensor is which, under a redefinition of the radial and temporal coordinates, takes the standard form, It is worth noticing that Eq. (19) is the most general spherical ansatz is one wants to solve Einstein's field equations. The static condition is only assured by the Birkhoff theorem [65][66][67][68], once the field equations are given.

Isotropic affine connection
The strategy used in the previous section can be repeated for an affine connection, and we shall end up with the most general isotropic (affine) connection. For the sake of simplicity, we do not include the differential equations obtained from the calculation of the Lie derivative. 5 As before, the Lie derivative along the generators of the spherical symmetry fix the angular dependence of the connection's components. The nonvanishing components of an isotropicΓ a bc , arê This general isotropic connection, depends on twenty functions of the coordinates t and r, and has nonvanishing torsion and non-metricity. Therefore, for our purposes within this paper, we can restrict even further to a torsion-free connection, whose components are The last connection depends on twelve functions of t and r, and these are the functions to be fixed by solving the field equations (4). Using the Eq. (20), we calculated the Ricci tensor and noticed that it is not symmetric, because the connection is not equi-affine. In order for the Ricci to be symmetric, the following conditions must hold, and There are several solutions to these conditions, and each of them could (in principle) provides a solution to the field equations (4). It is worth mentioning that if we do not demand the connection to be equi-affine, the field equations to solve would be Eq. (3), which have and extra term which cannot be obtained from the Yang-Mills-like effective action in Eq. (8).

Isotropic and homogeneous (covariant) two-tensor
In order to find the most general isotropic and homogeneous, covariant two-tensor, we can start from the result in Eq. (17), and impose now the symmetries from the extra generators. The equations are 8 £ P 3 T tt : 1 − κr 2 cos (θ) ∂T 00 ∂r = 0, £ P 3 T tr : κrT 01 + κr 2 − 1 ∂T 01 ∂r = 0, The above equations are solved for a tensor of the form, Under a redefinition of the "time" coordinate, the tensor is nothing but the standard ansatz for the Friedman-Robertson-Walker metric.

Isotropic And homogeneous affine connection
Without further details, we present the nonvanishing components of an isotropic and homogeneous affine connection. It is given by, which is determined by five independent functions, but this affine connection still possess torsion. The imposition of vanishing torsion kills two of the above functions, and the remaining components of the connection are 6 Toward the solution of the field equations

Cosmological solutions
Using the connection in Eq. (29), the Ricci is calculated, and yields Notice that the Ricci tensor is symmetric, determined by only three independent functions and, as expected by the symmetry, just two of their components behave differently. A first kind of solutions can be found by solving the system of equations determined by vanishing Ricci. However, this strategy requires the fixing of one of the unknown functions. A solution inspired in the components of the connection for Friedmann-Robertson-Walker, gives A second class of solutions can be found by solving the parallel Ricci equation, ∇ λ R µν = 0, which surprisingly yield three independent field equations, However, the system of equations is complicated enough to avoid an analytic solution. 9 Despite the complication, we can try a couple of assumptions that simplify the system of equations, for example: • Again, inspired in the Friedmann-Robertson-Walker results we choose G 000 = 0, and solve for the other functions (see Ref. [70]).
• We could choose G 000 = G 101 , to eliminate the nonlinear terms in Eq. (32).
Finally, the third class of solutions are those of Eq. (4). The set of equations degenerate and yield a single independent field equation, Therefore, we need to set two out of the three unknown functions to be able of solving for the connection.

Conclusions
We have presented a short review of the polynomial affine gravity, whose field equations (in the torsion-free sector) generalises whose of the standard general relativity. In the mentioned approximation, the field equations coincide with (part of) those of a gravitational Yang-Mills theory of gravity, known in the literature as the SKY model. Among the features of the polynimial affine gravity, we highlighted the following: • Although our spacetime could be metric, the metric plays no role in the model building.
• The non-relativistic limit of the model yields a Keplerian potential, even with the contributions of torsion and non-metricity.
• In the torsion-free sector, all the vacuum solutions to the Einstein gravity are solutions of the polynomial affine gravity.
• Scalar matter can be coupled to the polynomial affine gravity through a symmetric, 2 0 -tensor density. The coupled field equations can be written in a similar form to those of general relativity, with the subtlety that where ever the metric appears in the Einstein's equations, we just need a covariantly constant, symmetric, • Although the model is built up without the necessity of a metric, one can still assume that the connection is a metric potential. Such consideration yields to obtain new solutions to the Eq. (4), which are not solutions of the standard Einstein's equations. 10 • We found the general ansatz for the connection, compatible with isotropic and cosmological symmetries. Additionally, we have sketched the road to solving the Eq. (4), for the cosmological connection ansatz. A thorough analysis will be presented in Ref. [70].
We would like to finish commenting about the necessity of coupling other form of matters and a formal counting of degrees of freedoms.