We develop a novel model for Cosmological Hyperfluids, that is fluids with intrinsic hypermomentum that induce spacetime torsion and non-metricity. Imposing the Cosmological Principle to Metric-Affine Spaces, we present the most general covariant form of the hypermomentum tensor in an FLRW Universe along with its conservation laws and therefore construct a novel hyperfluid model for Cosmological purposes. Extending the previous model of the unconstrained hyperfluid in a Cosmological setting we establish the conservation laws for energy-momentum and hypermomentum and therefore provide the complete Cosmological setup to study non-Riemannian effects in Cosmology. With the help of this we find the forms of torsion and non-metricity that were earlier reported in the literature and also obtain the most general form of the Friedmann equations with torsion and non-metricity. We also discuss some applications of our model, make contact with the known results in the literature and point to future directions.
We set the foundation and formulate the Perfect (Ideal) Hyperfluid. The latter represents the natural generalization of the usual perfect fluid structure where now the microscopic characteristics of matter (spin, shear, dilation) are also taken into account, sourcing a non-Riemannian arena (i.e spacetime torsion and non-metricity) for Metric-Affine Gravity. We derive the energy tensors of this Hyperfluid structure and subsequently present the conservation laws obeyed by them. Finally, we consider a Cosmological application of this Perfect Hyperfluid and classify some possible forms of this flu…
We consider a generic Metric-Affine Cosmological setup and classify some particularly interesting specific cases of Perfect Hyperfluids. In particular, we present the form of conservation laws for the cases of pure spin, pure dilation and pure shear fluids. We also develop the concept of an incompressible hyperfluid and pay special attention to the case of a hypermomentum preserving hyperfluid. We also give a specific example on the emergence of the spin, dilation and shear currents through matter-connection couplings. In addition, starting from the generalized acceleration equation for the sc…
We study the potential effects of spacetime non-metricity in cosmology. In the spirit of Einstein-Cartan gravity, but with non-metricity replacing torsion, we consider the Einstein-Hilbert action and assume zero torsion. Adopting certain hyperfluid models, with non-vanishing hypermomentum that can source spacetime non-metricity, we add a matter component into the action and derive the field equations, along with the conservation laws. Applying our formulae to cosmology, we generalize the Friedmann and the Raychaudhuri equations in the presence of non-metricity. Our results show that, in a numb…
We investigate the cosmological aspects of the most general parity preserving Metric-Affine Gravity theory quadratic in torsion and non-metricity in the presence of a cosmological hyperfluid. The equations of motion are obtained by varying the action with respect to the metric and the independent affine connection. Subsequently, considering a Friedmann-Lemaître-Robertson-Walker background, we derive the most general form of the modified Friedmann equations for the full quadratic theory. We then focus on a characteristic sub-case involving only two quadratic contributions given in terms of tors…
We introduce and develop the 1+3 covariant approach to relativity and cosmology to spacetimes of arbitrary dimensions that have nonzero torsion and do not satisfy the metricity condition. Focusing on timelike observers, we identify and discuss the main differences between their kinematics and those of their counterparts living in standard Riemannian spacetimes. At the centre of our analysis lies the Raychaudhuri equation, which is the fundamental formula monitoring the convergence/divergence, namely the collapse/expansion, of timelike worldline congruences. To the best of our knowledge, we pro…
Starting from the generalized Raychaudhuri equation with torsion and non-metricity, and considering an FLRW spacetime we derive the most general form of acceleration equation in the presence of torsion and non-metricity. That is we derive the cosmic acceleration equation when the nonRiemannian degrees of freedom are also taken into account. We then discuss some conditions under which torsion and non-metricity accelerate/decelerate the expansion rate of the Universe.
Torsion and nonmetricity are inherent ingredients in modifications of Eintein's gravity that are based on affine spacetime geometries. In the context of pure f(R) gravity we discuss here, in some detail, the relatively unnoticed duality between torsion and nonmetricity. In particular we show that for R2 gravity torsion and nonmetricity are related by projective transformations. Since the latter correspond simply to redefining the affine parameters of autoparallels, we conclude that torsion and nonmetricity are physically equivalent properties of spacetime. As a simple example we show that both…
We study the inclusion of fermionic degrees of freedom and ${\cal N}=1$ supersymmetry in 4-dimensional manifolds with arbitrary torsion and non-metricity tensors deforming the connection. We inspect the closure of local diffeomorphism, Lorentz and supersymmetric transformations on the vielbein and gravitino at leading order in fermions. The closure of a pair of supersymmetric transformations acting on the vielbein restricts the form of the spin connection. The dependency of the latter on the fundamental fields is exclusively given by the vielbein, up to Lorentz and/or diffeomorphisms brackets,…
For both f(R) theories of gravity with an independent symmetric connection (no torsion), usually referred to as Palatini f(R) gravity theories, and for f(R) theories of gravity with torsion but no non-metricity, called U4 theories, it has been shown that the independent connection can actually be eliminated algebraically, as long as this connection does not couple to matter. Remarkably, the outcome in both case is the same theory, which is dynamically equivalent with an ω_0=-3/2 Brans--Dicke theory. It is shown here that even for the most general case of an independent connection with both non…
We derive the generalized Gauss-Codazzi-Mainardi (GCM) equation for a general affine connection with torsion and non-metricity. Moreover, we show that the metric compatibility and torsionless condition of a connection on a manifold are inherited to the connection of its hypersurface. As a physical application to these results, we derive the (3+1)-Einstein Field Equation (EFE) for a special case of Metric-Affine f(R)-gravity when f(R)=R, the Metric-Affine General Relativity (MAGR). Motivated by the concept of geometrodynamics, we introduce additional variables on the hypersurface as a consequen…
We study the renormalization of theories of gravity with an arbitrary (torsionful and non-metric) connection. The class of actions we consider is of the Palatini type, including the most general terms with up to two derivatives of the metric, but no derivatives of the connection. It contains 19 independent parameters. We calculate the one loop beta functions of these parameters and find their fixed points. The Holst subspace is discussed in some detail and found not to be stable under renormalization. Some possible implications for ultraviolet and infrared gravity are discussed.
A Cartan-geometrical perspective on torsion and non-metricity
Scuola Superiore Meridionale（南方高等学院）；INFN– Sezione di Napoli（那不勒斯意大利国家核物理研究所）；University of Tartu（塔尔图大学）；Institute of Theoretical Physics and Astrophysics, University of Gdańsk（格但斯克大学理论物理与天体物理研究所）
This Thesis is devoted to the study of Metric-Affine Theories of Gravity and Applications to Cosmology. The thesis is organized as follows. In the first Chapter we define the various geometrical quantities that characterize a non-Riemannian geometry. In the second Chapter we explore the MAG model building. In Chapter 3 we use a well known procedure to excite torsional degrees of freedom by coupling surface terms to scalars. Then, in Chapter 4 which seems to be the most important Chapter of the thesis, at least with regards to its use in applications, we present a step by step way to solve for …
I investigate the general extension of Einstein's gravity by considering the third rank non-metricity tensor and the torsion tensor. The minimal coupling to Dirac fields faces an ambiguity coming from a severe arbitrariness of the Fock-Ivanenko coefficients. This arbitrariness is fed in part by the covariant derivative of Dirac matrices, which is not completely determined as well. It is remarkable that this feature is not exclusive to the non-metricity case: it happens also for gravity with torsion alone. Nevertheless, theory in vacuum is well defined and non-trivial, where torsion is the sour…
We deal with the question of what it means to define a minimal coupling prescription in presence of torsion and/or non-metricity, carefully explaining while the naive substitution $\partial\to\na$ introduces extra couplings between the matter fields and the connection that can be regarded as non-minimal in presence of torsion and/or non-metricity. We will also investigate whether minimal coupling prescriptions at the level of the action (MCPL) or at the level of field equations (MCPF) lead to different dynamics. To that end, we will first write the Euler-Lagrange equations for matter fields in…
Relaxing the Riemannian condition to incorporate geometric quantities such as torsion and non-metricity may allow to explore new physics associated with defects in a hypothetical space-time microstructure. Here we show that non-metricity produces observable effects in quantum fields in the form of 4-fermion contact interactions, thereby allowing us to constrain the scale of non-metricity to be greater than 1 TeV by using results on Bhabha scattering. Our analysis is carried out in the framework of a wide class of theories of gravity in the metric-affine approach. The bound obtained represents …
We extend the results of antecedent literature on quadratic Metric-Affine Gravity by studying a new quadratic gravity action in vacuum which, besides the usual (non-Riemannian) Einstein-Hilbert contribution, involves all the parity even quadratic terms in torsion and non-metricity plus a Lagrangian that is quadratic in the field-strengths of the torsion and non-metricity vector traces. The theory result to be equivalent, on-shell, to a Vector-Tensor theory. We also discuss the sub-cases in which the contribution to the Lagrangian quadratic in the field-strengths of the torsion and non-metricit…
In this article, we consider a special case of Metric-Affine f(R)-gravity for f(R) = R, i.e. the Metric-Affine General Relativity (MAGR). As a companion to the first article in the series, we perform the (3+1) decomposition to the hypermomentum equation, obtained from the minimization of the MAGR action S [g, ω] with respect to the connection ω. Moreover, we show that the hypermomentum tensor H could be constructed completely from 10 hypersurfaces variables that arise from its dilation, shear, and rotational (spin) parts. The (3+1) hypermomentum equations consists of 1 scalar, 3 vector, 3 matr…
We discuss the general structure of metric geometries, and how metricity implies the complete antisymmetry of Cartan tensor; an application in the frame of Lie group theory is given. Interpretations of the completely antisymmetric torsion in physical models are reviewed.
A Weyl structure is usually defined by an equivalence class of pairs $({\bf g}, \boldsymbolω)$ related by Weyl transformations, which preserve the relation $\nabla {\bf g}=\boldsymbolω\otimes{\bf g}$, where ${\bf g}$ and $\boldsymbolω$ denote the metric tensor and a 1-form field. An equivalent way of defining such a structure is as an equivalence class of conformally related metrics with a unique affine connection $Γ_{\boldsymbolω}$, which is invariant under Weyl transformations. In a standard Weyl structure, this unique connection is assumed to be torsion-free and have vectorial non-metricity…
General Relativity enjoys the freedom of different geometrical interpretations in terms of curvature, torsion or non-metricity. Within this geometrical trinity, a simpler geometrical formulation of General Relativity manifests itself in the latter, where gravity is entirely attributed to non-metricity. In this Letter, we consider non-linear extensions of Coincident General Relativity $f(\mathring{\mathbb{Q}})$ for phenomenological applications on both cosmological as well as galactic scales. The theory not only delivers dark energy on large scales but also recovers MOND on galactic scales, tog…
We derive the full set of field equations for the Metric-Affine version of the Myrzakulov gravity model and also extend this family of theories to a broader one. More specifically, we consider theories whose gravitational Lagrangian is given by $F(R,T,Q, {\cal T},{\cal D})$ where $T$, $Q$ are the torsion and non-metricity scalars, ${\cal T}$ is the trace of the energy-momentum tensor and ${\cal D}$ the divergence of the dilation current. We then consider the linear case of the aforementioned theory and assuming a cosmological setup we obtain the modified Friedmann equations. In addition, focus…
We consider the most general 11 parameter parity even quadratic Metric-Affine Theory whose action consists of the usual Einstein-Hilbert plus the 11 quadratic terms in torsion, non-metricity as well as their mixing. By following a certain procedure and using a simple trick we are able to find the unique solution of the affine connection in terms of an arbitrary hypermomentum. Given a fairly general non-degeneracy condition our result provides the exact form of the affine connection for all types of matter. Subsequently we compute the forms of torsion and non-metricity in terms of their sources…
After recalling the differential geometry of non-metric connections in the formalism of differential forms, we introduce the idea of a Non-Metricity (NM) connection, whose connection $1$--forms coincides with the non-metricity $1$--forms for a class of cobase fields. Then we formulate a theory of gravitation (equivalent to General Relativity (GR)) which admits a geometrical interpretation in a flat torsionless space where the gravitational field is completely manifest in the non-metricity of a NM connection. We define and then apply the non-metricity gauge to a gravitational Lagrangian density…
General Relativity and its higher derivative extensions have symmetric teleparallel reformulations in terms of the non-metricity tensor within a torsion-free and flat geometry. These notes present a derivation of the exact propagator for the most general infinite-derivative, even-parity and generally covariant theory in the symmetric teleparallel spacetime. The action made up of the non-metricity tensor and its contractions is decomposed into terms involving the metric and a gauge vector field and is found to complement the previously known non-local ghost- and singularity-free theories.
Connections with (skew-symmetric) torsion on non-symmetric Riemannian manifold satisfying the Einstein metricity condition (NGT with torsion) are considered. It is shown that an almost Hermitian manifold is an NGT with torsion if and only if it is a Nearly Kähler manifold. In the case of an almost contact metric manifold the NGT with torsion spaces are characterized and a possibly new class of almost contact metric manifolds is extracted. Similar considerations lead to a definition of a particular classes of almost para-Hermitian and almost paracontact metric manifolds. The conditions are give…
We study a large family of metric-affine theories with a projective symmetry, including non-minimally coupled matter fields which respect this invariance. The symmetry is straightforwardly realised by imposing that the connection only enters through the symmetric part of the Ricci tensor, even in the matter sector. We leave the connection completely free (including torsion) and obtain its general solution as the Levi-Civita connection of an auxiliary metric, showing that the torsion only appears as a projective mode. This result justifies the widely used condition of setting vanishing torsion …
We study the variational principle over an Hilbert-Einstein like action for an extended geometry taking into account torsion and non-metricity. By extending the semi-Riemannian geometry, we obtain an effective energy-momentum tensor which can be interpreted as physical sources. As an application we develop a new manner to obtain the gravitational wave equations on a Weyl-integrable manifold taking into account the non-metricity and non-trivial boundary conditions on the minimization of the action, which can be identified as possible sources for the cosmological constant and provides two differ…
We analyse the Einstein-Cartan gravity in its standard form cal-R = R + cal-K^2, where cal-R and R are the Ricci scalar curvatures in the Einstein-Cartan and Einstein gravity, respectively, and cal-K^2 is the quadratic contribution of torsion in terms of the contorsion tensor cal-K. We treat torsion as an external (or a background) field and show that the contribution of torsion to the Einstein equations can be interpreted in terms of the torsion energy-momentum tensor, local conservation of which in a curved spacetime with an arbitrary metric or an arbitrary gravitational field demands a prop…
Gravity is attributed to the spacetime curvature in classical General Relativity (GR). But, other equivalent formulation or representations of GR, such as torsion or non-metricity have altered the perception. We consider the Weyl-type $f(Q, T)$ gravity, where $Q$ represents the non-metricity and $T$ is the trace of energy momentum temsor, in which the vector field $ω_μ$ determines the non-metricity $Q_{μνα}$ of the spacetime. In this work, we employ the well-motivated $f(Q, T)= αQ+ \fracβ{6k^{2}} T$, where $α$ and $β$ are the model parameters. Furthermore, we assume that the universe is domina…
On Gauge Gravity's 'Hypermomentum Challenge' to the Geometric Trinity
Lichtenberg Group for History and Philosophy of Physics, Institute of Philosophy, University of Bonn（波恩大学哲学研究所利希滕贝格历史与物理哲学小组）；Argelander Institute for Astrophysics, University of Bonn（波恩大学阿根兰德天体物理研究所）
We consider the most general Quadratic Metric-Affine Gravity setup in the presence of generic matter sources with non-vanishing hypermomentum. The gravitational action consists of all $17$ quadratic invariants (both parity even and odd) in torsion and non-metricity as well as their mixings, along with the terms that are linear in the curvature namely the Ricci scalar and the totally antisymmetric Riemann piece. Adding also a matter sector to the latter we first obtain the field equations for the generalized quadratic Theory. Then, using a recent Theorem, we successfully find the exact form of …
Geometrical analysis of a new type of Unified Field Theoretical models follow the guidelines of previous works of the authors is presented. These new unified theoretical models are characterized by an underlying hypercomplex structure, zero non-metricity and the geometrical action is determined fundamentally by the curvature provenient of the breaking of symmetry of a group manifold in higher dimensions. This mechanism of Cartan-MacDowell-Mansouri type, permits us to construct geometrical actions of determinantal type leading a non topological physical Lagrangian due the splitting of a reducti…
We present the Dirac equation in a geometry with torsion and non-metricity balancing generality and simplicity as much as possible. In doing so, we use the vielbein formalism and the Clifford algebra. We also use an index-free formalism which allows us to construct objects that are totally invariant. It turns out that the previous apparatuses not only make possible a simple deduction of the Dirac equation but also allow us to exhibit some details that is generally obscure in the literature.
Symmetric teleparallel gravity (STG) offers an interesting avenue to formulate a theory of gravitation that relies neither on curvature nor torsion but only on non-metricity Q. Given the growing number of observations of gravitational waves (GWs) and their use to explore gravitational theories, in this work we investigate the GWs in various extensions of STG, focusing on their speed and polarization. For the simple STG, for theories that arise from the generalized irreducible decomposition of STG, and for f(Q) gravity, we obtain the same speed and polarizations with general relativity. For sca…
