100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Gravitational Billiards, Dualities and Hidden Symmetries 43and hyperbolic Kac Moody algebras on the other — is most remarkableand surely has some deep significance.2. Known Duality SymmetriesWe first review the two types of duality symmetries of Einstein’s theory thathave been known for a long time. The first concerns the linearized versionof Einstein’s equations and works in any space-time dimension. The secondis an example of a non-linear duality, which works only for the specialclass of solutions admitting two commuting Killing vectors (axisymmetricstationary and colliding plane wave solutions). This second duality is moresubtle, not only in that it is non-linear, but in that it is linked to theappearance of an infinite dimensional symmetry.2.1. Linearized dualityThe duality invariance of the linearized Einstein equations generalizes thewell known duality invariance of electromagnetism in four spacetime dimensions.Recall that Maxwell’s equations in vacuo∂ µ F µν = 0 , ∂ [µ F νρ] = 0 (1)are invariant under U(1) rotations of the complex field strengthwith the dual (‘magnetic’) field strengthF µν := F µν + i ˜F µν (2)˜F µν := 1 2 ɛ µνρσF ρσ (3)The action of this symmetry can be extended to the combined electromagneticcharge q = e + ig, where e is the electric, and g is the magneticcharge. The partner of the one-form electric potential A µ is a dual magneticone-form potential à µ , obeying˜F µν := ∂ µ à ν − ∂ ν à µ (4)Observe that this dual potential can only be defined on-shell, when F µνobeys its equation of motion, which is equivalent to the Bianchi identityfor ˜F µν . Consequently, the U(1) duality transformations constitute an onshellsymmetry because they are valid only at the level of the equations ofmotion. The two potentials A µ and õ are obviously non-local functionsof one another. Under their exchange, the equations of motion and the