Topological regularization and self-duality in four-dimensional anti–de Sitter gravity

Abstract

It is shown that the addition of a topological invariant (Gauss-Bonnet term) to the anti-de Sitter (AdS) gravity action in four dimensions recovers the standard regularization given by holographic renormalization procedure. This crucial step makes possible the inclusion of an odd parity invariant (Pontryagin term) whose coupling is fixed by demanding an asymptotic (anti) self-dual condition on the Weyl tensor. This argument allows to find the dual point of the theory where the holographic stress tensor is related to the boundary Cotton tensor as T i j = ±(ℓ 2 /8πG)C i j , which has been observed in recent literature in solitonic solutions and hydrodynamic models.A general procedure to generate the counterterm series for AdS gravity in any even dimension from the corresponding Euler term is also briefly discussed.

Magnetic mass in 4D AdS gravity

“…Furthermore, the connection to Holographic Renormalization [11] in the context of AdS/CFT correspondence was shown in Refs. [12,13], as the asymptotic expansion of the extrinsic curvature reproduces the standard counterterm series [14,15].…”

“…The appearance of the Weyl tensor in the surface term coming from the variation of the total action (2.1) reflects the link to Conformal Mass definition in AAdS gravity [19]. Indeed, upon suitable expansion of the tensors involved, one can prove that the physical information on the conformal boundary is encoded in the electric part of the Weyl tensor [12,20].…”

“…The Weyl tensor is the only combination between the Riemann and Ricci tensors that has a suitable asymptotic behavior. A formal proof of the finiteness of the action, however, requires precise fall-off conditions in the metric, valid for any AAdS spacetime [12]. The appearance of the Weyl tensor in the surface term coming from the variation of the total action (2.1) reflects the link to Conformal Mass definition in AAdS gravity [19].…”

Holographic perfect fluidity, Cotton energy-momentum duality and transport properties

“…When mapped into Lorentzian signature, the self-duality of the Weyl tensor hints at a certain duality between the Cotton-York tensor and the energy-momentum tensor of the boundary geometry [5][6][7][8][9]. This duality implies that the Cotton-York tensor of the boundary geometry is proportional to the energy-momentum tensor, and we call geometries with such property as perfect-Cotton geometries.…”

Non-geodesic motion in f(G) gravity with non-minimal coupling

“…It is well known that adding the Gauss-Bonnet curvature G to the usual scalar curvature in Einstein-Hilbert action does not alter the equations of motion of general relativity due to the fact that it is a topological invariant in four dimensions, even though it may have some other non-trivial contributions [2,3]. However this term can be incorporated into space-time dynamics in several ways, namely by considering higher dimensions, e.g as in [4] or via non-minimal coupling to some scalar field in arbitrary space-time dimensions, which is also called modified Gauss-Bonnet gravity, as in string gravity, see e.g [5,6].…”