Θ-twisted Gravity
نویسنده
چکیده
We describe a theory of gravitation on canonical noncommutative spacetimes. The construction is based on θ-twisted General Coordinate Transformations and Local Lorentz Invariance. 1. It can be suspected from various arguments, e.g. related with short-distance divergences in quantum field theory and singularities in General Relativity, that the classical concept of spacetime continuum breaks down at small scales and must be replaced by some sort of ’quantum’ spacetime with an intrinsic ’fundamental’ lenght scale. In recent years, there were several attempts to formulate a theory of gravitation on “quantized” spacetimes [1]-[5]. The simplest such a spacetime is the spacetime with canonical noncommutativity [x ⋆, x ] = ilθ , (1) where l is a ’fundamental’ length of noncommutativity and θ = −θ are real constants. The ⋆-algebra of coordinates (1) explicitly violates the basic symmetry of Einstein’s General Relativity under the General Coordinate Transformations (GCT), leaving only its subgroup of (symplectic) volume-preserving diffeomorphisms. Noncommutative theory of gravity based on this residual symmetry has been constructed in [3], . An important development in studies of (flat) noncommutative spacetimes was the identification of θ-twisted Poncare group as a symmetry of noncommutative spacetimes [6],[7] (see also [8]). θ-twisted diffeomorphisms has been introduced soon afterwards, and the corresponding theory of gravitation has bee constructed in [1],[2]. In these works the Local Lorentz Invariance (LLI) is left unspecified since the authors essentially work with the second order formalism of General Relativity. However, LLI has to be consistently incorporated into a theory if one needs to couple fermions to gravity. Furthermore, the noncommutative metric tensor in [1],[2] is defined as a symmetrized ⋆-product of vierbeins, and this already requires specification of LLI, irrespective to the problem with fermions. In this paper we construct noncommutative theory of gravitation based on θ-twisted symmetries of General Relativity, GCT and LLI, working in the first order formalism. 2. An element of symmetry group of ordinary General Relativity can be represented as:
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