Abstract
Carrying out an analysis of the constraints and their linearizations on a spacelike hypersurface, we show that topologically massive gravity has a linearization instability at the chiral gravity limit about $AdS_3$. We also calculate the symplectic structure for all the known perturbative modes (including the log-mode) for the linearized field equations and find it to be degenerate (non-invertible) hence these modes do not approximate exact solutions and so do not belong to the linearized phase space of the theory. Naive perturbation theory fails: the linearized field equations are necessary but not sufficient in finding viable linearized solutions. This has important consequences for both classical and possible quantum versions of the theory.
Highlights
Quantum gravity is elusive not mainly because we lack computational tools, but because we do not know what to compute and so how to define the theory for a generic spacetime
One possible exception and a promising path is the case of asymptotically anti–de Sitter (AdS) spacetimes for which a dual quantum conformal field theory that lives on the boundary of a bulk spacetime with gravity would amount to a definition of quantum gravity
Linearized solutions by definition satisfy the linearized equations but this is not sufficient; they should satisfy a quadratic constraint to represent linearized versions of exact solutions. This deep result comes from the Bianchi identities and their linearizations and it is connected to the conserved quantities
Summary
Quantum gravity is elusive not mainly because we lack computational tools, but because we do not know what to compute and so how to define the theory for a generic spacetime. This obstruction to a viable classical and perhaps quantum theory was observed to disappear in an important work [5], where it was realized that at a “chiral point” defined by a tuned topological mass in terms of the AdS radius, one of the Virasoro algebras has a vanishing central charge (and so admits a trivial unitary representation) and the other has a positive nonzero central charge with unitary nontrivial representations, the theory has a positive energy black hole and zero energy bulk gravitons This tuned version of TMG, called “chiral gravity,” seems to be a viable candidate for a well-behaved classical and quantum gravity.
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