Gravitation and Noncommutativity of Spacetime

It is generally believed that the
structure of spacetime as a classical manifold will cease to
exist at distances of the order of the Planck length. A
theory of gravitation on a noncommutative spacetime could be
compatible with quantum mechanics and capture the expected
nonlocality of physics at very small distances, while
reproducing Einstein's general relativity at long distances.
We consider gravitation as a gauge theory of the Poincar
symmetry and aim at generalizing this point of view to
noncommutative spacetimes. The important role of the
Poincar symmetry in relativistic physics is reviewed. The
formulation of quantum field and gauge theories on
noncommutative spacetime is presented, emphasizing the
recently discovered twisted Poincar symmetry of these
theories. Some formulations of gravitation on noncommutative
spacetimes proposed in the literature are also discussed.
Finally, we study the possibility to formulate the twisted
Poincar symmetry as a gauge symmetry of gravitational
theory on a noncommutative spacetime. This book is intended
for researchers and students who are interested in
gravitation and elementary particle physics.