Aspects of the polynomial affine model of gravity in three dimensions

The polynomial affine gravity is a model that is built up without the explicit use of a metric tensor field. In this article we reformulate the three-dimensional model and, given the decomposition of the affine connection, we analyse the consistently truncated sectors. Using the cosmological ansatz for the connection, we scan the cosmological solutions on the truncated sectors. We discuss the emergence of different kinds of metrics.

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

The present paper is intended to show the close interrelationship between non-linear models of solids, produced with additive manufacturing, and models of solids with distributed defects. The common feature of these models is the incompatibility of local deformations. Meanwhile, in contrast with the conventional statement of the problems for solids with defects, the distribution for incompatible local deformations in additively created deformable body is not known a priori, and can be found from the solution of the specific evolutionary problem. The statement of the problem is related to the mechanical and physical peculiarities of the additive process. The specific character of incompatible deformations, evolved in additive manufactured solids, could be completely characterized within a differential-geometric approach by specific affine connection. This approach results in a global definition of the unstressed reference shape in non-Euclidean space. The paper is focused on such a formalism. One more common factor is the dataset which yields a full description of the response of a hyperelastic solid with distributed defects and a similar dataset for the additively manufactured one. In both cases, one can define a triple: elastic potential, gauged at stress-free state, and reference shape, and some specific field of incompatible relaxing distortion, related to the given stressed shape. Optionally, the last element of the triple may be replaced by some geometrical characteristics of the non-Euclidean reference shape, such as torsion, curvature, or, equivalently, as the density of defects. All the mentioned conformities are illustrated in the paper with a non-linear problem for a hyperelastic hollow ball.

Affine connection representation of gauge fields

There are two ways to unify gravitational field and gauge field. One is to represent gravitational field asprincipal bundle connection, and the other is to represent gauge field as affine connection. Poincaré gauge theoryand metric-affine gauge theory adopt the first approach. This paper adopts the second. In this approach:(i) Gauge field and gravitational field can both be represented by affine connection; they can be described by aunified spatial frame.(ii) Time can be regarded as the total metric with respect to all dimensions of internal coordinate space andexternal coordinate space. On-shell can be regarded as gradient direction. Quantum theory can be regarded as ageometric theory of distribution of gradient directions. Hence, gauge theory, gravitational theory, and quantumtheory all reflect intrinsic geometric properties of manifold.(iii) Coupling constants, chiral asymmetry, PMNS mixing and CKM mixing arise spontaneously as geometricproperties in affine connection representation, so they are not necessary to be regarded as direct postulates in theLagrangian anymore.(iv) The unification theory of gauge fields that are represented by affine connection can avoid the problem thata proton decays into a lepton in theories such as SU(5).(v) There exists a geometric interpretation to the color confinement of quarks.In the affine connection representation, we can get better interpretations to the above physical properties,therefore, to represent gauge fields by affine connection is probably a necessary step towards the ultimate theory ofphysics.

The affine connection

The connection and the covariant derivative are treated. Connection coefficients are introduced in their role of expressing the change in the coordinate basis vectors between neighbouring points. The covariant derivative of a vector is then defined. Next we relate the connection to the metric, and obtain the Levi-Civita connection. The logic concerning what is defined and what is derived is explained carefuly. The notion of a derivative along a curve is defined. The emphasis through is on clarity and avoiding confusions arising from the plethora of concepts and symbols.

Metric-Affine Myrzakulov Gravity Theories

In this paper, we review the so-called Myrzakulov Gravity models (MG-N, with N = I, II, …, VIII) and derive their respective metric-affine generalizations (MAMG-N), discussing also their particular sub-cases. The field equations of the theories are obtained by regarding the metric tensor and the general affine connection as independent variables. We then focus on the case in which the function characterizing the aforementioned metric-affine models is linear and consider a Friedmann-Lemaître–Robertson–Walker background to study cosmological aspects and applications. Historical motivation for this research is thoroughly reviewed and specific physical motivations are provided for the aforementioned family of alternative theories of gravity.

Affine connection representation of gauge fields

There are two ways to unify gravitational field and gauge field. One is to represent gravitational field as principal bundle connection, and the other is to represent gauge field as affine connection. Poincar{\'{e}} gauge theory and metric-affine gauge theory adopt the first approach. This paper adopts the second. In this approach:(i) Gauge field and gravitational field can both be represented by affine connection; they can be described by a unified spatial frame.(ii) Time can be regarded as the total metric with respect to all dimensions of internal coordinate space and external coordinate space. On-shell can be regarded as gradient direction. Quantum theory can be regarded as a geometric theory of distribution of gradient directions. Hence, gauge theory, gravitational theory and quantum theory obtain the same geometric foundation and a unified description of evolution.(iii) Coupling constants, chiral asymmetry, PMNS mixing and CKM mixing arise spontaneously as geometric properties in affine connection representation, so they are not necessary to be regarded as direct postulates in the Lagrangian anymore.(iv) The unification theory of gauge fields that are represented by affine connection can avoid the problem that a proton decays into a lepton in theories such as $SU(5)$.(v) There exists a geometric interpretation to the color confinement of quarks.In the affine connection representation, we can get better interpretations to the above physical properties, therefore, to represent gauge fields by affine connection is probably a necessary step towards the ultimate theory of physics.

Affine connection representation of gauge fields

There are two ways to unify gravitational field and gauge field. One is to represent gravitational field as principal bundle connection, and the other is to represent gauge field as affine connection. Poincaré gauge theory and metric-affine gauge theory adopt the first approach. This paper adopts the second. In this approach: (i) Gauge field and gravitational field can both be represented by affine connection; they can be described by a unified spatial frame. (ii) Time can be regarded as the total metric with respect to all dimensions of internal coordinate space and external coordinate space. On-shell can be regarded as gradient direction. Quantum theory can be regarded as a geometric theory of distribution of gradient directions. Hence, gauge theory, gravitational theory and quantum theory obtain the same geometric foundation and a unified description of evolution. (iii) Coupling constants, chiral asymmetry, PMNS mixing and CKM mixing can appear spontaneously as geometric properties in affine connection representation. It is remarkable that PMNS mixing and CKM mixing can be interpreted as a spontaneous geometric consequence of symmetry reduction, so they are not necessary to be regarded as direct postulates in the Lagrangian anymore. (iv) The unification theory of gauge fields that are represented by affine connection can avoid the problem that a proton decays into a lepton in theories such as SU(5). (v) There exists a possible geometric interpretation to the color confinement of quarks. In the affine connection representation, we can get better interpretations of the above physical properties, therefore, to represent gauge fields by affine connection is probably a necessary step towards the ultimate theory of physics.

Minimum Time Problem Controlled by Affine Connection

Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented.

Dark matter from a dark connection

In the first part of this note, we observe that a non-Riemannian piece in the affine connection (a “dark connection”) leads to an algebraically determined, conserved, symmetric 2-tensor in the Einstein field equations that is a natural dark matter candidate. The only other effect it has, is through its coupling to standard model fermions via covariant derivatives. If the local dark matter density is the result of a background classical dark connection, these Yukawa-like mass corrections are minuscule ([Formula: see text] for terrestrial fermions) and none of the tests of general relativity or the equivalence principle are affected. In the second part of the note, we give dynamics to the dark connection and show how it can be re-interpreted in terms of conventional dark matter particles. The simplest way to do this is to treat it as a composite field involving scalars or vectors. The (pseudo-)scalar model naturally has a perturbative shift-symmetry and leads to versions of the Fuzzy Dark Matter (FDM) scenario that has recently become popular (e.g. arXiv:1610.08297) as an alternative to WIMPs. A vector model with a [Formula: see text]-parity falls into the Planckian Interacting Dark Matter (PIDM) paradigm, introduced in arXiv:1511.03278. It is possible to construct versions of these theories that yield the correct relic density, fit with inflation and are falsifiable in the next round of CMB experiments. Our work is an explicit demonstration that the meaningful distinction is not between gravity modification and dark matter, but between theories with extra fields and those without.