1305.2403 (Edwin Beggs et al.)

Gravity induced from quantum spacetime    [PDF]

Edwin Beggs, Shahn Majid

We show that tensoriality constraints in noncommutative Riemannian geometry in the 2-dimensional bicrossproduct model \lambda-Minkowski (or \kappa-Minkowski) spacetime algebra [x,t]=\lambda x drastically reduce the possible metrics g to a 2-parameter space with classical limit having Ricci=x^{-2}g and Einstein=0, i.e. a vacuum at the classical level, and corrections at order \lambda^2 in the noncommutative version. The noncommutative Riemannian geometry includes a second Levi-Civita connection with no classical limit, and we find the moduli space more generally with torsion. Our analysis also suggests a reduction of moduli in n-dimensions and we study the resulting classical geometry in n=4 in detail, identifying two 1-parameter subcases where the Einstein tensor matches that of a perfect fluid for (a) positive pressure, zero density and (b) negative pressure and positive density. The classical geometry is conformally flat and its geodesics motivate new coordinates which we extend to the quantum case as a new description of the \lambda-Minkowski spacetime model as a quadratic algebra.

View original: http://arxiv.org/abs/1305.2403