Weak field limit and gravitational waves in higher-order gravity

We derive the weak field limit for a gravitational Lagrangian density [Formula: see text], where higher-order derivative terms in the Ricci scalar [Formula: see text] are taken into account. The interest for this kind of effective theories comes out from the consideration of the infrared and ultraviolet behaviors of gravitational field and, in general, from the formulation of quantum field theory in curved spacetimes. Here, we obtain solutions in weak field regime both in vacuum and in the presence of matter and derive gravitational waves considering the contribution of [Formula: see text] terms. By using a suitable set of coefficients [Formula: see text], it is possible to find up to [Formula: see text] normal modes of oscillation with six polarization states with helicity 0 or 2. Here [Formula: see text] is the higher-order term in the [Formula: see text] operator appearing in the gravitational Lagrangian. More specifically: the mode [Formula: see text], with [Formula: see text], has transverse polarizations [Formula: see text] and [Formula: see text] with helicity 2; the [Formula: see text] modes [Formula: see text], with [Formula: see text], have transverse polarizations [Formula: see text] and non-transverse ones [Formula: see text], [Formula: see text], [Formula: see text] with helicity 0.

Higher-order gravity and the classical equivalence principle

As is well known, the deflection of any particle by a gravitational field within the context of Einstein’s general relativity — which is a geometrical theory — is, of course, nondispersive. Nevertheless, as we shall show in this paper, the mentioned result will change totally if the bending is analyzed — at the tree level — in the framework of higher-order gravity. Indeed, to first order, the deflection angle corresponding to the scattering of different quantum particles by the gravitational field mentioned above is not only spin dependent, it is also dispersive (energy-dependent). Consequently, it violates the classical equivalence principle (universality of free fall, or equality of inertial and gravitational masses) which is a nonlocal principle. However, contrary to popular belief, it is in agreement with the weak equivalence principle which is nothing but a statement about purely local effects. It is worthy of note that the weak equivalence principle encompasses the classical equivalence principle locally. We also show that the claim that there exists an incompatibility between quantum mechanics and the weak equivalence principle, is incorrect.

The regular state in higher order gravity

We consider the higher-order gravity theory derived from the quadratic Lagrangian [Formula: see text] in vacuum as a first-order (ADM-type) system with constraints, and build time developments of solutions of an initial value formulation of the theory. We show that all such solutions, if analytic, contain the right number of free functions to qualify as general solutions of the theory. We further show that any regular analytic solution which satisfies the constraints and the evolution equations can be given in the form of an asymptotic formal power series expansion.