Test-field limit of metric nonlinear gravity theories

In the framework of alternative metric gravity theories, it has been shown by several authors that a generic Lagrangian depending on the Riemann tensor describes a theory with 8 degrees of freedom (which reduce to 3 for f(R) Lagrangians depending only on the curvature scalar). This result is often related to a reformulation of the fourth-order equations for the metric into a set of second-order equations for a multiplet of fields, including – besides the metric – a massive scalar field and a massive spin-2 field (the latter being usually regarded as a ghost): this is commonly assumed to represent the particle spectrum of the theory. In this article we investigate an issue which does not seem to have been addressed so far: in ordinary general-relativistic field theories, all fundamental fields (i.e. fields with definite spin and mass) reduce to test fields in some appropriate limit of the model, where they cease to act as sources for the metric curvature. In this limit, each of the fundamental fields can be excited from its ground state independently from the others (which does not happen, instead, as long as the fields are coupled through the gravitational interaction). We thus address the following question: does higher-derivative gravity admit a test-field limit for its fundamental fields? It is easy to show that for a generic f(R) theory (carrying 3 d.o.f.) the test-field limit does exist; then, we consider the case of Lagrangians depending on the full Ricci tensor, relying on a previous analysis published several years ago. We show that, already for a quadratic Lagrangian, the constraint binding together the scalar field and the massive spin-2 field does not disappear in the limit where they should be expected to act as test fields. A proper test-field limit exists only for a particular choice of the coefficients in the Lagrangian, which cause the scalar field to disappear (so that the resulting model has only 7 d.o.f.). We finally consider the possible addition of an arbitrary function of the quadratic invariant of the Weyl tensor, \(C^{\alpha }{}_{\beta \mu \nu }C_{\alpha }{}^{\beta \mu \nu }\), showing that the appearance of the Weyl tensor does not add physical degrees of freedom (in accordance to the known results for Lagrangians depending on the full Riemann tensor) and the resulting model with 8 d.o.f. still lacks a proper test-field limit: the differential constraints between the fundamental fields do not cancel out when gravitational interaction is suppressed. We argue that the lack of a test-field limit for the dynamics of the fundamental fields may constitute a serious drawback of the full 8 d.o.f. higher-order gravity models, which is not encountered in the restricted 7 d.o.f. or 3 d.o.f. cases.