Effective delta sources and regularity in higher-derivative and ghost-free gravity

Abstract

It is shown that polynomial gravity theories with more than four derivatives in each scalar and tensor sectors have a regular weak-field limit, without curvature singularities. This is achieved by proving that in these models the effect of the higher derivatives can be regarded as a complete regularization of the delta-source. We also show how this result implies that a wide class of non-local ghost-free gravities has a regular Newtonian limit too, and discuss the applicability of this approach to the case of weakly non-local models.1 We use the same sign conventions as [10]. Also, we set c = = 1. 2 That is, we shall consider non-local gravity models which are extensions of GR in the UV-limit, which means that for large momentum the propagator decays faster than in GR. Specifically, we require that f 0 (z) and f 2 (z) (defined in (3) and (4)) are constant or diverge at least linearly as z −→ ∞, and that f s (0) = 1. Owed to this improved behaviour in the UV, sometimes these models are called non-local HDG. The situation is quite different from non-local IR modifications of GR, such as those defined by form factors of the type F i ∝ ✷ −1 and F i ∝ ✷ −2 [24-26], or the logarithmic ones, F i ∝ ln ✷, which come from the integration of quantum matter fields in curved space-time [27][28][29][30][31].

Weak-field limit and regular solutions in polynomial higher-derivative gravities

“…In this spirit, one may be tempted to ask whether the condition ϕ 1 = ψ 1 = 0 is recurrent in higher-derivative gravity models. For example, there is a large class of nonlocal gravities that satisfy this condition when coupled to a δ-source [48,63,64] (see also [65] for more general non-local theories). On what concerns local models, the ones with only fourth derivatives do not satisfy this condition [52,66,67]; however, it holds for the sixth-order gravity with a pair of complex poles [14], and in [68] it was given general considerations supporting the conjecture that for theories with more than four derivatives one has ϕ 1 = ψ 1 = 0.…”

Horizonless ultracompact objects and dark matter in quadratic gravity

“…This can be checked directly by, e.g., computing the Ricci scalar R corresponding to (3.5) and noting that it diverges in the limit r → 0 (see e.g. [66]). It is important to note that this problem is not a sign of sickness of the theory but appears because a point mass is a singular source at the origin 11 .…”

Nonlinear stability in nonlocal gravity

“…Just as in general relativity [27], Schwarzschild spacetime is a stable solution in NLG at the linear level. Whether this solution also represents an actual, astrophysical black hole remains to be seen, since approximated solutions of the linearized equations of motion (EOM) look regular [28][29][30][31][32][33]. More recently, in [34] it was found that the NLG dynamics of small perturbations of Minkowski metric is the same as in Einstein gravity.…”