Abstract
We study asymptotically flat spacetimes in five spacetime dimensions by Hamiltonian methods, focusing on spatial infinity and keeping all asymptotically relevant nonlinearities in the transformation laws and in the charge generators. Precise boundary conditions that lead to a consistent variational principle are given. We show that the algebra of asymptotic symmetries, which had not been uncovered before, is a nonlinear deformation of the semidirect product of the Lorentz algebra by an Abelian algebra involving four independent (and not just one) arbitrary functions of the angles on the three-sphere at infinity, with nontrivial central charges. The nonlinearities occur in the Poisson brackets of the boost generators with themselves and with the other generators. They would be invisible in a linearized treatment of infinity.
Highlights
The structure of null infinity in five spacetime dimensions is rather complicated [1,2,3,4]. This is because the gravitational field decays with negative fractional powers of r as one tends to infinity along null curves, preventing a smooth conformal compactification of spacetime along the lines proposed by Penrose [5]
The gravitational field decays with a Coulombtype ∼r−Dþ3 behavior up to diffeomorphisms, where D is the number of spacetime dimensions
What makes spatial infinity simpler from this point of view is that its existence is not a dynamical question, contrary to the existence of a null infinity with given smoothness properties, which is a delicate dynamical question even in four spacetime dimensions [6,7]
Summary
Πij corresponds to the conjugate momentum of the four-dimensional spatial metric gij, while N and Ni stand for the lapse and shift functions, respectively, which we take to behave asymptotically as N → 1; Ni → 0. The surface integral on the three-sphere at spatial infinity B∞ coincides with the standard Arnowitt-Deser-Misner energy with this asymptotic behavior of the lapse and the shift
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