**1. Introduction**

One of the outstanding problems in physics is to account for the apparent dark energy and dark matter in the universe since it accounts for roughly 95% of the total matter in the universe. Reviews of the dark matter and dark energy cosmological problem, and the models that have been introduced to account for it, include those of Peebles and Ratra [1], Sahni [2], Copeland, Sami, and Tsujikawa [3], Frieman, Turner, and Huterer [4], Amendola and Tsujikawa [5], Li, Li, Wang, and Wang [6], and Arun, Gudennavar, and Sivaram [7]. We will not survey the literature here as these reviews do an excellent job of that. As is often the case, we use dimensions where the speed of light *c* is 1; we use the Einstein summation convention where sums over repeated indices are assumed, and a comma in front of a lower index such as *f*,*<sup>i</sup>* denotes differentiation of *f* with respect to *x i* .

Maybe the most favored model is the ΛCDM model. Here, Λ is Einstein's cosmological constant, giving rise to dark energy with *p* = −*µ*0, and CDM is cold dark matter introduced to give the observed ratio of pressure to total mass density, which is about −0.8. Constraints on dark matter and dark energy properties are imposed by the results of the DES collaboration [8,9]. Gravitational lensing measurements [10] give a Hubble constant that is consistent with long-period Cepheid measurements in the large Magellanic cloud [11], but both strongly indicate significant discrepancies with the ΛCDM model. Experimental tests of the strong equivalence principle [12] provide further evidence casting doubt on the model in favor of modified gravity theories. Alternatively, there may be late time dark matter creation [13].

The relativistic model we introduce here has no adjustable parameters and incorporates a torsion vector field. It is perhaps the simplest gravitational model involving torsion; yet, we believe it could explain the dark energy and dark mass in the universe. If the simplicity of the underlying equations is to be a guiding principle in physics, then these equations surely meet that principle. Of course, our equations still need to be compatible with both existing and future experimental observations, both qualitatively and quantitatively, and this remains to be seen. It is to be stressed that our equations govern the curvature of empty space and do not fully determine the interaction between matter and

**Citation:** Milton, G.W. A Possible Explanation of Dark Matter and Dark Energy Involving a Vector Torsion Field. *Universe* **2022**, *8*, 298. https://doi.org/10.3390/ universe8060298

Academic Editors: Fabbri Luca and Matteo Luca Ruggiero

Received: 26 March 2022 Accepted: 16 May 2022 Published: 25 May 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the curvature. We believe the simpler problem of obtaining the equations for empty space should be addressed first, as a stepping stone towards a more general theory where matter is included. The main demands that drive our formulation of the equations are:


It may be argued that these should not be assumed a priori, but that convincing physical arguments should be presented as well. On the other hand, Einstein's equations for empty space can be obtained from the first three of these requirements without any necessity to introduce physical considerations. Indeed, as is well known, it is natural that the Ricci tensor (with the possible addition of the cosmological constant term) is zero in empty space as this provides 10 equations for the 10 metric elements, with the 4 functions associated with freedom in the choice of coordinates being compensated by the 4 Bianchi identities. Only when matter is present is physics needed to determine the full Einstein equations, as embodied in the constraints that the equations reduce to Newton's gravitational equations when the spacetime curvature is small and that small test particles follow geodesics. Since we do not consider the full interaction of matter and curvature, we cannot claim that small test particles will still follow geodesics: that would be a natural demand to be required of a more general theory.

Despite the simplicity of our underlying equations the resultant dynamics of the torsion vector field, even in the weak field approximation, is enormously complicated, suggesting the torsion vector field has some sort of turbulent behavior. This is the main novel feature of our theory: the suggestion that torsion may induce intrinsic inhomogeneity on many length scales, even in the absence of matter. This goes further than the idea that space is inhomogeneous on the Planck length scale and is also a feature of anti-de Sitter spacetime [14]. Other work shows that the inhomogeneities of matter in the universe may account for the perceived acceleration of the universe without any need to introduce a negative cosmological constant (Λ < 0) (see [15] and the references therein).

Numerical simulations of the torsion field behavior will almost certainly be necessary to test the theory and assess its compatibility with astronomical and cosmological observations. The equations can be reinterpreted as a model using the Einstein gravitational equations where spacetime has regions filled with a perfect fluid with negative energy (pressure) and positive mass density, other regions containing an anisotropic substance, which in the local rest frame (where the momentum is zero) has negative mass density and a uniaxial stress tensor, and possibly other "luminal" regions where there is no natural local "rest frame". We emphasize, though, that all three regions are manifestations of the torsion vector field, and the three regions accordingly correspond to regions where the vector field points inside, outside, or on the boundary of the light cone. Our theory predicts that dark energy and dark matter (which are both manifestations of the torsion field) interact and exchange energy. Other models where dark energy and dark matter interact were reviewed by Wang, Abdalla, Atrio-Barandela, and Pavón [16] (see also the more recent work of Borges and Wands [17]).

It has been noted before by De Sabbata and Sivaram [18] that torsion provides a natural framework for negative mass, as has been suggested to occur in the early universe. Cosmological models with negative mass have been studied by Ray, Khlopov, Ghosh, and Mukhopadhyay [19] and by Famaey and McGaugh [20] and yield promising explanations for the acceleration of the expansion rate of the universe.

In addition to the cosmological dark mass problem, there is also the dark mass problem, which is associated with the observations of higher-than-expected rotational velocities

of stars far from the galactic center. One empirically motivated model that successfully accounts for this is Modified Newtonian Dynamics (MOND), first introduced by Milgrom [21]. He suggested that Newton's law, where the gravitational force is proportional to the acceleration, be replaced at low accelerations, below a critical acceleration *<sup>a</sup>*<sup>0</sup> <sup>=</sup> <sup>≈</sup>1.2 <sup>×</sup> <sup>10</sup>−<sup>10</sup> ms−<sup>2</sup> , by one where the force is proportional to the square of the acceleration; see Figure 1. Later, this idea motivated a relativistic theory developed by Bekenstein [22] and generalized by Skordis [23]. One prediction of MOND, later verified, was that there should be a universal relation between the rotation speeds of stars in the outermost parts of a galaxy and the total mass, not dark mass, of the galaxy; see the book of Merritt [24] for further discussion on this point. In particular, on the basis of this, it seems unlikely that unseen particles will provide the explanation for the galactic missing mass problem. Other reviews of MOND, including these and other relativistic extensions and their implications for cosmology, have been given by Famaey and McGaugh [20], Merritt [24], and Milgrom [25]. It is not yet clear whether the torsion field model developed here will be successful in explaining the galactic dark mass problem, though the success of Farnes [26] in explaining the flattening of rotation curves by introducing negative mass suggests that it might meet with success on this front.

**Figure 1.** Figure, courtesy of M. Milgrom, taken from http://www.scholarpedia.org/article/The\_ MOND\_paradigm\_of\_modified\_dynamics, (accessed on 31 March 2020) showing its predictions, which are consistent with experimental observations. Plotted is the acceleration as a function of the distance from an isolated mass *<sup>M</sup>*, for a star with *<sup>M</sup>* <sup>=</sup> *<sup>M</sup>*<sup>⊙</sup> (red), a globular cluster with *<sup>M</sup>* <sup>=</sup> <sup>10</sup>5*M*<sup>⊙</sup> (blue), a galaxy with *<sup>M</sup>* <sup>=</sup> <sup>3</sup> <sup>×</sup> <sup>10</sup>10*M*<sup>⊙</sup> (green), and a galaxy cluster with *<sup>M</sup>* <sup>=</sup> <sup>3</sup> <sup>×</sup> <sup>10</sup>13*M*<sup>⊙</sup> (magenta), in which *M*<sup>⊙</sup> represents one solar mass.

Torsion is the antisymmetric part of the affine connection. The affine connection determines how vectors change under parallel displacements. Cartan introduced torsion and applied it to develop generalizations of Einstein's gravitational equations. His work dates back to the early 1920s; see [27] and the references therein (translated in [28]). A brief introduction to torsion is in the classic book on gravitation by Misner, Thorne, and Wheeler [29]. More extensive reviews of general relativistic models that include torsion, with further developments, include those of Hehl, von der Heyde, and Kerlick [30], De Sabbata and Sivaram [18], Hehl, McCrea, Mielke, and Ne'eman [31], Shapiro [32], Ortín [33], Trautman [34], Poplawski [35], and Fabbri [36]. Interestingly, Jose Beltrán Jiméneza, Lavinia Heisenberg, and Tomi S. Koivisto have recently shown [37] that Einstein's gravitational equations can be reformulated in terms of the torsion alone, eliminating the metric.

Typically, general relativistic models with torsion have been introduced to allow for the intrinsic spin of matter and are quite complicated. By contrast, our focus here is on developing a simple model that may account for the dark mass and dark energy in the universe.

Ivanov and Wellenzohn suggested that the Einstein–Cartan theory may account for dark energy [38]. Another gravitational model that incorporates the same torsion vector field we use, as well as additional fields and a fifth dimension, was developed by Sengupta [39], who suggests it may solve both the cosmological and galactic dark matter problem. Other models incorporating torsion, quite different from the one explored here, that may explain the accelerated expansion of the universe have been developed by Watanabe and Hayashi [40], Minkevich [41], de Berredo-Peixoto and de Freitas [42], Belyaev, Thomas, and Shapiro [43], and Vasak, Kirsch, and Struckmeier [44].

The analysis in the following sections is more or less standard, though equivalent formulations are clearly possible according to one's mathematical taste. The key step to arriving at our equations is simply to postulate that geodesics and autoparallels coincide. There is nothing difficult in the analysis leading to our equations governing the spacetime curvature.

### **2. Metric and Affinities**

The functions *guv* of the metric field describe with respect to the arbitrarily chosen system of co-ordinates the metrical relations of the spacetime continuum:

$$ds^2 = g\_{\mu\nu} d\mathfrak{x}^{\mu} d\mathfrak{x}^{\nu}.\tag{1}$$

Here, we will assume that the *guv* are real and symmetric in the indices *u* and *v*, and thus, (1) provides the defining equation for *guv* with respect to a given coordinate system.

Now, consider the affinity Γ *i st*, which determines a vector after parallel displacement. To a real contravariant vector **A** with components *A <sup>i</sup>* at a point *P* with coordinates *x t* , we correlate a vector **A** + *δ***A** with components *A <sup>i</sup>* + *δA <sup>i</sup>* at the infinitesimally close point with coordinates *x <sup>t</sup>* + *δx <sup>t</sup>* by

$$
\delta A^i = -\Gamma^i\_{st} A^s \delta \mathbf{x}^t. \tag{2}
$$

Since the magnitude of **A** in parallel displacement does not change to first order in that displacement, we obtain

$$0 = \delta[g\_{\mu\upsilon}A^{\mu}A^{\upsilon}] = \frac{dg\_{\mu\upsilon}}{d\mathfrak{x}^{a}}A^{\mu}A^{\upsilon}d\mathfrak{x}^{a} + g\_{\mu\upsilon}A^{\mu}(\delta A^{\upsilon}) + g\_{\mu\upsilon}A^{\upsilon}(\delta A^{\mu}),\tag{3}$$

and so, using (2), we obtain

$$
\delta g\_{\nu\upsilon\mu} - \mathcal{g}\_{\nu\beta} \Gamma^{\beta}\_{\upsilon\mu} - \mathcal{g}\_{\upsilon\beta} \Gamma^{\beta}\_{\mu\alpha} = 0,\tag{4}
$$

where the comma denotes partial differentiation. Now, by considering this equation together with the two equations

$$g\_{\upsilon\alpha,\mu} - g\_{\upsilon\beta}\Gamma^{\beta}\_{\alpha\mu} - g\_{\alpha\beta}\Gamma^{\beta}\_{\upsilon\mu} = 0,\tag{5}$$

$$g\_{\mu\nu,\upsilon} - g\_{a\beta} \Gamma^{\beta}\_{\upsilon\upsilon} - g\_{u\beta} \Gamma^{\beta}\_{\alpha\upsilon} = 0,\tag{6}$$

that are obtained by a cyclic interchange of indices and by subtracting (4) from the sum of (5) and (6), we obtain

$$
\hat{\Gamma}[\boldsymbol{u}\,\boldsymbol{v},\boldsymbol{u}] + \mathcal{g}\_{\upsilon\beta}\hat{\Gamma}^{\beta}\_{\mu\alpha} + \mathcal{g}\_{\imath\beta}\hat{\Gamma}^{\beta}\_{\upsilon\alpha} - \mathcal{g}\_{a\beta}\hat{\Gamma}^{\beta}\_{\mu\upsilon} = \mathcal{g}\_{a\beta}\Gamma^{\beta}\_{\upsilon\upsilon} \tag{7}
$$

where [*u v*, *α*] is the Christoffel symbol of the first kind, given by

$$[\mu\ v,\mathfrak{a}] = \frac{1}{2}(g\_{\text{au},\text{v}} + g\_{\text{av},\text{u}} - g\_{\text{uv},\text{a}}), \quad \hat{\Gamma}^{\beta}\_{\text{ij}} = \frac{1}{2}(\Gamma^{\beta}\_{\text{ij}} - \Gamma^{\beta}\_{\text{ji}}).\tag{8}$$

The antisymmetric part of the affinity <sup>Γ</sup><sup>b</sup> *β ij*, in contrast to Γ *β ji*, is a tensor—Cartan's torsion tensor.

### **3. Equating Geodesics with Autoparallels**

Geodesics are trajectories **x**(*s*), which we chose to parametrize by the distance *s* along them, that have an extremal distance between two points. Since they clearly only depend on the metric, they satisfy the standard formula:

$$\frac{d^2\mathbf{x}^\mu}{ds^2} + g^{\mu r}[\alpha \not\rh, r] \frac{d\mathbf{x}^\alpha}{ds} \frac{d\mathbf{x}^\beta}{ds} = 0. \tag{9}$$

Alternatively, we may consider an autoparallel constructed in such a way that successive elements arise from each other by parallel displacements. An element is the vector *d***x**/*ds*, and under parallel displacement, its components transform as

$$
\delta \left( \frac{d\mathbf{x}^{\mu}}{ds} \right) = -\Gamma^{\mu}\_{a\beta} \frac{d\mathbf{x}^{a}}{ds} \delta \mathbf{x}^{\beta}. \tag{10}
$$

The left-hand side is to be replaced by (*d* 2*x <sup>µ</sup>*/*ds*<sup>2</sup> )*δs*, giving

$$\frac{d^2\mathbf{x}^\mu}{ds^2} + \Gamma^\mu\_{\alpha\beta}\frac{d\mathbf{x}^\alpha}{ds}\frac{d\mathbf{x}^\beta}{ds} = \mathbf{0}.\tag{11}$$

We postulate that geodesics coincide with autoparallels, thus giving

$$\left\{\Gamma^{\mu}\_{a\beta} - g^{\mu r}[a \; \beta \; r] \right\} \frac{d\mathbf{x}^{a}}{ds} \frac{d\mathbf{x}^{\beta}}{ds} = \mathbf{0},\tag{12}$$

or equivalently,

$$\left\{ \frac{1}{2} (\Gamma^{\mu}\_{\alpha\beta} + \Gamma^{\mu}\_{\beta\alpha}) - g^{\mu r} [\alpha \ \beta \ \jmath r] \right\} \frac{d\mathfrak{x}^{\alpha}}{ds} \frac{d\mathfrak{x}^{\beta}}{ds} = 0. \tag{13}$$

This postulate is fundamental to the theory. While it is absent of any physical justification, aside from removing possible ambiguity in the path that test particles are required to follow in a more general theory, it is essential to keep the governing equations as simple as possible. This is our motivation for this constraint.

As (13) holds for all *dxα*/*ds* and *dxβ*/*ds*, we obtain

$$\overline{\Gamma}^{\mu}\_{\mathfrak{a}\mathfrak{F}} \equiv \frac{1}{2} (\Gamma^{\mu}\_{\mathfrak{a}\mathfrak{F}} + \Gamma^{\mu}\_{\mathfrak{f}\mathfrak{a}}) = \mathcal{g}^{\mu r} [\mathfrak{a}\,\mathfrak{F}, r]. \tag{14}$$

Multiplying both sides by *gµ<sup>s</sup>* and summing over *µ* gives

$$\mathbf{g}\_{\mu\mathbf{s}}\Gamma^{\mu}\_{\alpha\beta} + \mathbf{g}\_{\mu\mathbf{s}}\Gamma^{\mu}\_{\beta\alpha} = \mathbf{2}[\mathbf{a}\ \boldsymbol{\beta}, \mathbf{s}].\tag{15}$$

Combining this with (7) then yields

$$\mathcal{S}\_{a\mathcal{B}\mu} \equiv \mathcal{g}\_{ar}\widehat{\Gamma}^{r}\_{\mathcal{\beta}\mu} = -\mathcal{g}\_{\mathcal{\beta}r}\widehat{\Gamma}^{r}\_{a\mu} = -\mathcal{S}\_{\mathcal{\beta}a\mu} = \mathcal{S}\_{\mathcal{\beta}\mu a}.\tag{16}$$

Therefore, *Sαβµ* is antisymmetric with respect to the interchange of any pair of its three indices, and this implies (see, for example, the text below Equation (2.16) in [30]) that

$$
\widehat{\Gamma}^{i}\_{jk} = \mathcal{g}^{ir} e\_{rjk\ell} \mathcal{U}^{\ell} \,. \tag{17}
$$

for some contravariant vector density **U**, where, as is standard, *erjk*<sup>ℓ</sup> is the Levi-Civita tensor density, with *e*<sup>1234</sup> = 1 and antisymmetric with respect to the interchange of any pair of indices. **U** is known as the axial part of the torsion [30]. A parallel derivation

of the complete antisymmetry of torsion is in the review of Fabbri [36]. Combining (17) with (14) gives

$$
\Gamma^{\mu}\_{a\beta} = \overline{\Gamma}^{\mu}\_{a\beta} + \mathcal{g}^{\mu r} e\_{ra\beta\ell} \mathcal{U}^{\ell}. \tag{18}
$$
