Noncommutative geometry and gravity

2008

Using noncommutative deformed canonical commutation relations, a model of gravity is constructed and a schwarchild like static solutions are obtained. As a consequence, the Newtonian potential is modified and it is shown to have a form similar to the one postulated by Fishbach et al. to explain the proposed fifth force. More interesting is the form of the gravitational acceleration expression proposed in the modified Newtonian dynamics theories (MOND) which is obtained explicitly in our model without any ad hoc asymptions.

Communications in Mathematical Physics, 1993

We study general relativity in the framework of non-commutative differential geometry. In particular, we introduce a gravity action for a space-time which is the product of a four dimensional manifold by a two-point space. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.

2003

In this paper, starting from the common foundation of Connes' noncommutative geometry (NCG) [1,2,3,4], various possible alternatives in the formulation of a theory of gravity in noncommutative spacetime are discussed in detail. The diversity in the final physical content of the theory is shown to the the consequence of the arbitrariness in each construction steps. As an alternative in the last step, when the staructure equations are to be solved, a minimal set of constraints on the torsion and connection is found to determine all the geometric notions in terms of metric. In the Connes-Lott model of noncommutative spacetime, in order to keep the full spectrum of the discretized Kaluza-Klein theory [5], it is necessary to include the torsion in the generalized Einstein-Hilbert-Cartan action.

Classical and Quantum Gravity, 2008

We study a noncommutative deformation of general relativity where the gravitational field is described by a matrix-valued symmetric twotensor field. The equations of motion are derived in the framework of this new theory by varying a diffeomorphisms and gauge invariant action constructed by using a matrix-valued scalar curvature. Interestingly the genuine noncommutative part of the dynamical equations is described only in terms of a particular tensor density that vanishes identically in the commutative limit. A noncommutative generalization of the energy-momentum tensor for the matter field is studied as well.

AIP Conference Proceedings, 2008

We briefly review ideas about "noncommutativity of space-time" and approaches toward a corresponding theory of gravity.

2009

We present a survey of the application of Cones' Non-Commutative Geometry to gravitation. Bases of the theory and Euclidian gravity models are reviewed. Then we discuss the problem of a Lorentzian generalization of the theory and review existing attempts of solution.

Physics Letters B, 2003

We develop a novel approach to gravity in which gravity is described by a matrixvalued symmetric two-tensor field and construct an invariant functional that reduces to the standard Einstein-Hilbert action in the commutative limit. We also introduce a gauge symmetry associated with the new degrees of freedom.

˜The œJournal of high energy physics/˜The œjournal of high energy physics, 2024

We use the framework of Hopf algebra and noncommutative differential geometry to build a noncommutative (NC) theory of gravity in a bottom-up approach. Noncommutativity is introduced via deformed Hopf algebra of diffeomorphisms by means of a Drinfeld twist. The final result of the construction is a general formalism for obtaining NC corrections to the classical theory of gravity for a wide class of deformations and a general background. This also includes a novel proposal for noncommutative Einstein manifold. Moreover, the general construction is applied to the case of a linearized gravitational perturbation theory to describe a NC deformation of the metric perturbations. We specifically present an example for the Schwarzschild background and axial perturbations, which gives rise to a generalization of the work by Regge and Wheeler. All calculations are performed up to first order in perturbation of the metric and noncommutativity parameter. The main result is the noncommutative Regge-Wheeler potential. Finally, we comment on some differences in properties between the Regge-Wheeler potential and its noncommutative counterpart.

Journal of Geometry and Physics, 1989

The structure of amanifold can be encoded in the commutative algebra of functions on the manifold it sell-this is usual-. In the case of a non com.mut.ative algebra thereis no underlying manifold and the usual concepts and tools of diffe.rential geometry (differentialforms, De Rham cohomology, vector bundles, connections, elliptic operators, index theory.. .) have to be generalized. This is the subject of non commutative differential geometry and is believed to be of fundamental importance in our understanding of quantum field theories. The presentpaper is an introduction for the non specialist and a review oftheprincipal results on the field.

Journal of Geometry and Physics, 1993

This is an introduction to the old and new concepts of non-commutative (N.C.) geometry. We review the ideas underlying N.C. measure and topology, N.C. differential calculus, N.C. connections on N.C. vector bundles, and N.C. Riemannian geometry by following A. Connes' point of view.