Variational principle and 1-point functions in 3-dimensional flat space Einstein gravity

TL;DR

This work establishes a well-defined variational principle for three-dimensional flat-space Einstein gravity by adding one half of the Gibbons–Hawking–York boundary term. Using this action, the authors verify the 0-point function reproducing consistent flat-space thermodynamics for vacuum and flat space cosmologies, and compute 1-point functions that match canonical charges. The flat-space analysis is anchored in a careful translation of AdS3 boundary data and shows that the half-boundary term is crucial to obtain correct free energies and holographic stress responses. The results confirm the smooth AdS limit with vanishing cosmological constant and provide explicit expressions for mass and angular momentum in both vacuum and flat space cosmologies, reinforcing the holographic picture in flat space. Overall, the paper strengthens flat-space holography in 3D by deriving consistent variational principles and holographic data for the vacuum and flat space cosmologies.

Abstract

We provide a well-defined variational principle for 3-dimensional flat space Einstein gravity by adding one half of the Gibbons-Hawking-York boundary term to the bulk action. We check the 0-point function, recovering consistency with thermodynamics of flat space cosmologies. We then apply our result to calculate the 1-point functions in flat space Einstein gravity for the vacuum and all flat space cosmologies. The results are compatible with the ones for the zero mode charges obtained by canonical analysis.