A tale of two theories of gravity in asymptotically Anti-de Sitter spacetime

TL;DR

This paper compares two BF formulations of gravity in asymptotically AdS spacetimes, showing that the MacDowell-Mansouri approach yields finite charges and does not require counterterms, unlike the Plebanski formulation.

Contribution

It provides an explicit comparison of asymptotic charges in two BF formulations of AdS gravity, demonstrating the finiteness of charges in MacDowell-Mansouri theory.

Findings

Charges are divergent in Plebanski theory.

Charges are finite in MacDowell-Mansouri theory.

MacDowell-Mansouri theory does not require counterterms.

Abstract

We consider two BF formulations of the theory of gravity with a negative cosmological constant, of Plebanski and of MacDowell-Mansouri. Both give the standard Einstein equations in the bulk but differ in expressions of edge charges. We compute the asymptotic charges explicitly in both theories for AdS-Schwarzschild, AdS-Kerr, and AdS-Taub--NUT solutions. We find that while in the case of the Plebanski theory the charges are divergent, they are finite for MacDowell-Mansouri theory. Furthermore, we show that in the case of the arbitrary asymptotically AdS spacetimes, MacDowell--Mansouri asymptotic charges, action, and symplectic form are all finite. Therefore MacDowell-Mansouri theory of gravity in asymptotically AdS spaces does not need any counterterms.