[Paper Review] The Dynamics of Supertranslations and Superrotations in 2+1 Dimensions

S. Carlip|arXiv (Cornell University)|Aug 17, 2016

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper demonstrates that in 2+1-dimensional asymptotically flat gravity, supertranslations and superrotations—once considered mere symmetries—acquire dynamical degrees of freedom through a boundary term in the Einstein-Hilbert action. The induced dynamics for these Goldstone-like fields is described by a chiral Liouville theory, closely related to the Schwarzian action in near-AdS2 gravity, revealing a conformal field theory structure at null infinity that generalizes the Brown-Henneaux result to flat spacetime.

ABSTRACT

Supertranslations, and at least in 2+1 dimensions superrotations, are asymptotic symmetries of the metric in asymptotically flat spacetimes. They are not, however, symmetries of the boundary term of the Einstein-Hilbert action, which therefore induces an action for the Goldstone-like fields that parametrize these symmetries. I show that in 2+1 dimensions, this action is closely related to a chiral Liouville action, as well as the "Schwarzian" action that appears in two-dimensional near-AdS physics.

Motivation & Objective

• To understand the dynamical origin of supertranslation and superrotation degrees of freedom in 2+1-dimensional asymptotically flat gravity.
• To show that the boundary term in the Einstein-Hilbert action induces a non-trivial action for asymptotic symmetries that are not symmetries of the boundary term itself.
• To establish a connection between the induced dynamics of superrotations and chiral Liouville theory, as well as the Schwarzian action in near-AdS2 gravity.
• To clarify the role of the BMS3 group and its coadjoint orbit structure in the context of flat space holography.

Proposed method

• Use of Bondi coordinates in 2+1 dimensions to describe the metric near future null infinity 𝒮⁺, with asymptotic form ds² = -2dudr + gᵤᵤdu² + 2gᵤᵠdudϕ + r²e²ωdϕ².
• Imposition of the gauge V=1 and ω₁=0 to simplify the metric and field equations, allowing analysis of asymptotic symmetries.
• Derivation of the transformation laws for diffeomorphisms that preserve the asymptotic metric form, identifying u₀ and ϕ₀ as supertranslations and superrotations.
• Computation of the boundary variation of the Einstein-Hilbert action, showing that the boundary term induces a non-invariant action for the asymptotic fields.
• Identification of the induced boundary action as a chiral Liouville theory with central charge matching the BMS₃ algebra.
• Use of coadjoint orbit quantization of the Virasoro group to interpret the dynamics of the superrotation modes.

Experimental results

Research questions

• RQ1 How do supertranslations and superrotations in 2+1D asymptotically flat gravity acquire physical dynamics despite being asymptotic symmetries?
• RQ2 What is the precise form of the boundary action induced by the Einstein-Hilbert action at null infinity?
• RQ3 How is the induced dynamics related to chiral Liouville theory and the Schwarzian action found in near-AdS2 gravity?
• RQ4 Why does the superrotation charge Ξ not appear in the final boundary action, and what does this imply about the completeness of the dynamics?
• RQ5 Can the induced boundary theory fully capture the algebraic structure of the BMS₃ group?

Key findings

• The boundary term in the Einstein-Hilbert action induces a dynamical action for supertranslation and superrotation modes, making them physical degrees of freedom at null infinity.
• The induced action for superrotations is a chiral Liouville theory, which matches the structure of the Schwarzian action in 2D near-AdS gravity.
• The central charge of the chiral Liouville theory is consistent with the BMS₃ algebra, linking the dynamics to the coadjoint orbit quantization of the Virasoro group.
• The absence of the superrotation charge Ξ in the final action suggests that subleading-order effects may be missing, indicating a possible incompleteness in the current description.
• Allowing ω₁ ≠ 0 in the metric expansion does not alter the induced boundary action, as long as the modified gᵤᵤ⁽⁰⁾ is used, showing robustness of the result.