Abstract
We consider a recently proposed extension of the Bondi-Metzner-Sachs algebra to include arbitrary infinitesimal diffeomorphisms on a 2-sphere. To realize this extended algebra as asymptotic symmetries, we work with an extended class of spacetimes in which the unphysical metric at null infinity is not universal. We show that the symplectic current evaluated on these extended symmetries is divergent in the limit to null infinity. We also show that this divergence cannot be removed by a local and covariant redefinition of the symplectic current. This suggests that such an extended symmetry algebra cannot be realized as symmetries on the phase space of vacuum general relativity at null infinity, and that the corresponding asymptotic charges are ill-defined. However, a possible loophole in the argument is the possibility that symplectic current may not need to be covariant in order to have a covariant symplectic form. We also show that the extended algebra does not have a preferred subalgebra of translations and therefore does not admit a universal definition of Bondi 4-momentum.
Highlights
Gravitational phase space to give additional asymptotic symmetries
We show that the answer is no: the symplectic current of general relativity diverges in the limit to null infinity, in general, when one of the perturbations is generated by an extended BMS symmetry
Using the definition of asymptotically-flat spacetimes we showed how the BMS algebra b arises as the asymptotic symmetry algebra at null infinity, emphasizing the role of the smoothness and topological assumptions in the definition
Summary
We follow the conventions of Wald [32] throughout. Tensors on spacetime will be denoted by Latin indices a, b, c,. We will frequently use an index-free notation for differential forms and denote by a bold-face, e.g. ω ≡ ωabc is the 3-form symplectic current. Tensors on the physical spacetime will be denoted by a “tilde” while those on the conformal completion (unphysical spacetime) will not have a “tilde”, e.g. gab is the physical metric while gab is the unphysical metric in the conformal completion. Indices on unphysical, unbarred quantities will be raised and lowered with the unphysical metric, for example nana = gabnanb
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