A derivation of Weyl gravity

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 ${\partial }^4{g}_{\mu \nu }$, ${\partial }^3{g}_{\mu \nu }\partial g_{\alpha \beta }$, ${\partial }^2{g}_{\mu \nu }{\partial }^2{g}_{\alpha \beta }$, ${\partial }^2{g}_{\mu \nu }\partial g_{\alpha \beta }\partial g_{\rho \sigma }$ 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.

This work was presented in part at the meeting of the Deutsche Physikalische Gesellschaft held in Bonn, 26-29 March 2001. To appear in Annalen der Physik

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