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Plot of σ(η) as given by equation (136) with both sign choices. Note that the dashed line corresponds to a contracting universe.
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Cosmologies based on General Relativity encompassing an anti-symmetric connection (torsion) can display nice desirable features as the absence of the initial singularity and the possibility of inflation in the early stage of the universe. After briefly reviewing the standard approach to the cosmology with torsion, we generalize it to demonstrate th...
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Citations
... Now let us apply these techniques to the possibility of a torsional version of Inflation. (See for example [41,65,66] for torsional cosmology. It is interesting to note that the history of torsional cosmology goes back surprisingly far, back, in fact, to the work of Friedmann and Schouten in 1924: see [52].) ...
We study the intrinsic and extrinsic torsions (defined by analogy with the intrinsic and extrinsic curvatures) of the spatial sections of torsional spacetimes. We consider two possibilities. First, that the intrinsic torsion might prove to be directly observable. Second, that it is not observable, having been ‘inflated away’ in the early Universe. We argue that, even in this second case, the extrinsic torsion may grow during the inflationary era and be non-negligible at reheating and thereafter. Even if the spatial intrinsic curvature and torsion are too small to be detected directly, then, the extrinsic torsion might not be. We point out that, if its presence is not recognised, the extrinsic torsion could lead to anomalies in the theoretical estimate of the Hubble parameter—a result with obvious potential applications. We stress that extrinsic torsion is by far the most natural way to produce such anomalies, simply because it mixes naturally with the Hubble parameter; that is, the second fundamental form of a spacelike section depends on a sum of two terms, one determined by the Hubble parameter, the other by the extrinsic torsion.
... which shows that the Ricci tensor is in general not symmetric, like in conventional Riemann-Cartan spacetime [74,77]. By computing the trace of R ab via the pseudoinverse metric, we obtain R :=g ab R ab . ...
... Remarkably, the complex Schwarzschild geometry satisfies the relations (76). As a consequence, the connection is given by formula (77), which, contrary to the general case, is compatible with the metric. The Ricci tensor (67) can be easily constructed through the simplified expression (78), and it is found to be zero, like for the real-valued solutions pertaining to the Euclidean and Lorentzian sections of the complexified spacetime (see Sect. 2). ...
The Moore–Penrose algorithm provides a generalized notion of an inverse, applicable to degenerate matrices. In this paper, we introduce a covariant extension of the Moore–Penrose method that permits to deal with general relativity involving complex non-invertible metrics. Unlike the standard technique, this approach guarantees the uniqueness of the pseudoinverse metric through the fulfillment of a set of covariant relations, and it allows for the proper definition of a covariant derivative operator and curvature-related tensors. Remarkably, the degenerate nature of the metric can be given a geometrical representation in terms of a torsion tensor, which vanishes only in special cases. Applications of the new scheme to complex black hole geometries and cosmological models are also investigated, and a generalized concept of geodesics that exploits the notion of autoparallel and extremal curves is presented. Relevance of our findings to quantum gravity and quantum cosmology is finally discussed.
... Another open question is how Elko affects torsion in space-time. In Einstein-Cartan gravity, which is the gauge theory of the Poincaré group [73,74], the spin density of matter can cause torsion [75,76]. Since the Dirac and Elko field have different spin density tensors, it can be expected that the two fields will induce different torsion effects. ...
Elko is a massive spin-half field of mass dimension one. Elko differs from the Dirac and Majorana fermions because it furnishes the irreducible representation of the extended Poincaré group with a two-fold Wigner degeneracy where the particle and anti-particle states both have four degrees of freedom. Elko has a renormalizable quartic self interaction which makes it a candidate for self-interacting dark matter. We study Elko in the spatially flat FLRW space-time and find exact solutions in the de Sitter space. Furthermore, we study its quantization under de Sitter space evolution. By choosing the appropriate solutions and phases, the fields satisfy the canonical anti-commutation relations and have the correct time evolutions in the flat space limit.
... Cubero and Poplawski [14] find that fermion spin in Einstein-Cartan gravity leads to a nonsingular big bounce instead of a big bang, with specific conditions for closed universes. Medina and Nowakowski [15] demonstrate that General Relativity with torsion can avoid initial singularities and enable early inflation, with various theories allowing accelerated expansion. Numerous researchers such as Tolman [16], Vaidya [17], Trautman [18], Nduka [19,20], Singh and Yadav [21], Singh and Griffiths [22], Singh et al. [23], Singh and Prasad [24], Arkuszewski et al. [25], Raychaudhuri and Banerji [26], Bedran and Som [27], Yadav and Prasad [28], Katkar [29,30], Katkar and Patil [31], Katkar and Phadatare [32][33][34][35] have investigated different aspects of solutions within the Einstein-Cartan frame-work. ...
By harnessing the power of differential forms, especially suited for a non-Riemannian space-time of Einstein–Cartan theory, we have successfully solved the field equations for a Weyssenhoff fluid-a fascinating source of gravitation and spin. Our innovative approach is based on a simple yet elegant equation of state, with two distinct cases: (i) ρ = p , and (ii) ρ = 3 p , and an exponential relationship e μ = e n ν . We explore the fascinating physical and geometrical properties of these ground-breaking solutions, uncovering new insights into the mysteries of gravitation and the behavior of matter under extreme conditions.
... which shows that the Ricci tensor is in general not symmetric, like in conventional Riemann-Cartan spacetime [74,77]. By computing the trace of R ab via the pseudoinverse metric, we obtain R :=g ab R ab . ...
The Moore-Penrose algorithm provides a generalized notion of an inverse, applicable to degenerate matrices. In this paper, we introduce a covariant extension of the Moore-Penrose method that permits to deal with general relativity involving complex non-invertible metrics. Unlike the standard technique, this approach guarantees the uniqueness of the pseudoinverse metric through the fulfillment of a set of covariant relations, and it allows for the proper definition of a covariant derivative operator and curvature-related tensors. Remarkably, the degenerate nature of the metric can be given a geometrical representation in terms of a torsion tensor, which vanishes only in special cases. Applications of the new scheme to complex black hole geometries and cosmological models are also investigated, and a generalized concept of geodesics that exploits the notion of autoparallel and extremal curves is presented. Relevance of our findings to quantum gravity and quantum cosmology is finally discussed.
... (See for example [37,56,57] for torsional cosmology.) ...
We study the intrinsic and extrinsic torsions (defined by analogy with the intrinsic and extrinsic curvatures) of the spatial sections of torsional spacetimes. We consider two possibilities. First, that the intrinsic torsion might prove to be directly observable. Second, that it is not observable, having been ``inflated away'' in the early Universe. We argue that, even in this second case, the extrinsic torsion may grow during the inflationary era and be non-negligible at reheating and thereafter. Even if the spatial intrinsic curvature and torsion are too small to be detected directly, then, the extrinsic torsion might not be. We point out that, if its presence is not recognised, the extrinsic torsion could lead to anomalies in the theoretical estimate of the Hubble parameter, a result with obvious potential applications. We stress that extrinsic torsion is by far the most natural way to produce such anomalies, simply because it mixes naturally with the Hubble parameter.
... The first attempt to describe a perfect fluid endowed with intrinsic spin was made by Weyssenhoff and Raabe [43]. Such construction allows a Lagrangian description [44][45][46][47][48] and has been considered extensively in the Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology for EC gravity [49][50][51][52][53][54][55][56][57] including recent attempts to reduce the Hubble tension [58,59]. The type of spin density with microscopic origin has been also used to generate the inflationary expansion of the early universe within the framework of EC theory [60,61], although such models require extreme fine-tuning to achieve the required number of e-folds [60]. ...
... Next, we will assume that w and w ℎ are constant. This is surely a strong hypothesis for w ℎ considering its definition (59b) combined with (57). However, we saw in the previous section V A 1 that a simple hypothesis about the relation between spin and matter density does indeed give a constant w ℎ . ...
... In the context of Einstein-Cartan theory which has only torsion and no nonmetricity, a popular model for the source of torsion is the Weyssenhoff fluid [43] (see also [44][45][46][47][48]), whereby the spin part of hypermomentum is linked to the quantum mechanical intrinsic spin density of a fermionic fluid. In an unpolarized case the quantum fluctuations of the spin orientation are random, and the average of the spin density will vanish, but the terms that are quadratic in spin have a nonvanishing average which can be related to the energy density as [55,[57][58][59][60] ...
In metric-affine gravity, both the gravitational and matter actions depend not just on the metric, but also on the independent affine connection. Thus matter can be modeled as a hyperfluid, characterized by both the energy-momentum and hypermomentum tensors. The latter is defined as the variation of the matter action with respect to the connection and it encodes extra (micro)properties of particles. For a homogeneous and isotropic universe, it was recently shown that the generic cosmological hypermomentum possesses five degrees of freedom: one in dilation, two in shear, and two in spin part. The aim of the current work is to present the first systematic study of the implications of this perfect hyperfluid on the universe with Friedmann-Lema\^itre-Robertson-Walker metric. We adopt a simple model with non-Riemannian Einstein-Hilbert gravitational action plus arbitrary hyperfluid matter, and solve analytically the cosmological equations for single and multiple component hypermomentum contributions using different assumptions about the equation of state. It is remarkable, that in a number of cases the forms of the time evolution of the Hubble function and energy density still coincide with their general relativity counterparts, only the respective indexes and start to differ due to the hypermomentum corrections. The results and insights we obtained are very general and can assist in constructing interesting models to resolve the issues in standard cosmology.
... Cartan's ideas were forgotten later on, but they were reconsidered and extended in the 1960s, leading to the formulation of the so-called Einstein-Cartan theory (see [25] for a detailed review of the early developments in this field), in which the spin of matter couples to a non-Riemannian geometric structure, namely, the torsion tensor. For discussions on the cosmological applications and mathematical foundations of the Einstein-Cartan theory, see [26][27][28]. ...
... The possible presence of torsion in the Universe has been extensively investigated in various theoretical frameworks. In particular, the Einstein-Cartan-type theories [25][26][27] have attracted significant attention. In the Einstein-Cartan theory, torsion is directly related to a physical property of matter, with the torsion proportional to the spin tensor. ...
We present a review of the Semi-Symmetric Metric Gravity (SSMG) theory, representing a geometric extension of standard general relativity, based on a connection introduced by Friedmann and Schouten in 1924. The semi-symmetric connection is a connection that generalizes the Levi-Civita one by allowing for the presence of a simple form of the torsion, described in terms of a torsion vector. The Einstein field equations are postulated to have the same form as in standard general relativity, thus relating the Einstein tensor constructed with the help of the semi-symmetric connection, with the energy–momentum tensor. The inclusion of the torsion contributions in the field equations has intriguing cosmological implications, particularly during the late-time evolution of the Universe. Presumably, these effects also dominate under high-energy conditions, and thus SSMG could potentially address unresolved issues in general relativity and cosmology, such as the initial singularity, inflation, or the ⁷Li problem of the Big-Bang Nucleosynthesis. The explicit presence of torsion in the field equations leads to the non-conservation of the energy–momentum tensor, which can be interpreted within the irreversible thermodynamics of open systems as describing particle creation processes. We also review in detail the cosmological applications of the theory, and investigate the statistical tests for several models, by constraining the model parameters via comparison with several observational datasets.
... As opposed to other matter models (e.g., spin fluids [13]), in kinetic theory the constraint (5) does not automatically entail evolution equations on the matter fields; on the contrary, it complicates the task of deriving admissible kinetic models. In this paper we introduce an evolution equation on the kinetic density f (x, u, s) of spin particles which is compatible with (5) and which generalizes the well-known Vlasov model for spinless particles [1,17]. ...
... The purpose of this section is to recall a few important definitions and formulas in Riemann-Cartan's geometry. More details can be found e.g. in [11,13,18,21]. ...
A kinetic model for the dynamics of collisionless spin neutral particles in a spacetime with torsion is proposed. The fundamental matter field is the kinetic density f(x,u,s) of particles with four-velocity u and four-spin s. The stress-energy tensor and the spin current of the particles distribution are defined as suitable integral moments of f in the (u,s) variables. By requiring compatibility with the contracted Bianchi identity in Einstein-Cartan theory, we derive a transport equation on the kinetic density f that generalizes the well-known Vlasov equation for spinless particles. The total number of particles in the new model is not conserved. To restore this important property we assume the existence in spacetime of a second species of particles with the same mass and spin magnitude. The Vlasov equation on the kinetic density of the new particles is derived by requiring that the sum of total numbers of particles of the two species should be conserved.
... Torsion tensor 1 is defined as the anti-symmetric part of the connection [83] ...
... For the remainder of this section we adopt the notation of Ref. [83]. Since the above mentioned connection (2) is not necessarily symmetric, the Riemann tensor in ECT will have additional terms compared to GR 2 ...
... The two field equations in Einstein-Cartan gravity are obtained by variation of the Lagrangian with respect to the metric and connection fields. One of the field equations gives an algebraic relation between the torsion and contorsion and their source, the spin tensor of the matter fields τ ανν as [83] S αν ...
We study the evolution of scalar and tensor cosmological perturbations in the framework of the Einstein–Cartan theory of gravity. The value of the gravitational slip parameter which is defined as the ratio of the two scalar potentials in the Newtonian gauge, can be used to determine whether or not the gravity is modified. We calculate the value of slip parameter in the Einstein–Cartan cosmology and show that it falls within the observed range. We also discuss the evolution of the cosmic gravitational waves as another measure of the modification of gravity.
