A derivation of Weyl gravity

TL;DR

This paper derives Weyl gravity as the unique consistent deformation of linearized conformal gravity in four dimensions, and explores the possibility of interactions between multiple Weyl gravitons.

Contribution

The paper provides a cohomological derivation of Weyl gravity and analyzes the conditions under which cross-interactions between different Weyl gravitons are possible.

Findings

Weyl action is the only consistent deformation with up to four derivatives.

Cross-interactions between different Weyl gravitons are impossible under certain conditions.

Cross-couplings are possible if different free limits are allowed.

Abstract

In this paper, two things are done. (i) Using cohomological techniques, we explore the consistent deformations of linearized conformal gravity in 4 dimensions. We show that the only possibility involving no more than 4 derivatives of the metric (i.e., terms of the form $\partial^4 g_{\mu \nu}$, $\partial^3 g_{\mu \nu} \partial g_{\alpha \beta}$, $\partial^2 g_{\mu \nu} \partial^2g_{\alpha \beta}$, $\partial^2 g_{\mu \nu} \partial g_{\alpha \beta} \partial g_{\rho \sigma}$ or $\partial g_{\mu \nu} \partial g_{\alpha \beta} \partial g_{\rho \sigma} \partial g_{\gamma \delta}$ with coefficients that involve undifferentiated metric components - or terms with less derivatives) is given by the Weyl action $\int d^4x \sqrt{-g} W_{\a\b\g\d} W^{\a\b\g\d}$, in much the same way as the Einstein-Hilbert action describes the only consistent manner to make a Pauli-Fierz massless spin-2 field self-interact with no more than 2 derivatives. No a priori requirement of invariance under diffeomorphisms is imposed: this follows automatically from consistency. (ii) We then turn to "multi-Weyl graviton" theories. We show the impossibility to introduce cross-interactions between the different types of Weyl gravitons if one requests that the action reduces, in the free limit, to a sum of linearized Weyl actions. However, if different free limits are authorized, cross-couplings become possible. An explicit example is given. We discuss also how the results extend to other spacetime dimensions.