Emerging Newtonian potential in pure $R^2$ gravity on a de Sitter background

Abstract

In Fortsch. Phys. $\text{64}$, 176 (2016), Alvarez-Gaume et al established that pure $R^2$ theory propagates $\text{massless}$ spin-2 graviton on a de Sitter (dS) background but $\text{not}$ on a locally flat background. We build on this insight to derive a Newtonian limit for the theory. Unlike most previous works that linearized the metric around a locally flat background, we explicitly employ the dS background to start with. We directly solve the field equation of the action $(2\kappa {)}^{-1}\int d^{4}x\sqrt{-g}{R}^2$ coupled with the stress-energy tensor of normal matter in the form $T_{\mu \nu }=M{c}^2\delta (\vec{r}){\delta }_{\mu }^0{\delta }_{\nu }^0$. We obtain the following Schwarzschild-de Sitter metric $d{s}^2=-(1-\frac{\Lambda }{3}{r}^2-\frac{\kappa c^2}{48\pi \Lambda }\frac{M}{r})c^{2}d{t}^2+(1-\frac{\Lambda }{3}{r}^2-\frac{\kappa c^2}{48\pi \Lambda }\frac{M}{r}{)}^{-1}d{r}^2+r^{2}d{\Omega }^2$ which features a potential $V(r)=-\frac{\kappa c^4}{96\pi \Lambda }\frac{M}{r}$ with the correct Newtonian tail. The parameter $\Lambda$ plays a dual role: (i) it sets the scalar curvature for the background dS metric, and (ii) it partakes in the Newtonian potential $V(r)$. We reach two key findings. Firstly, the Newtonian limit only emerges owing to the de Sitter background. Most existing studies of the Newtonian limit in modified gravity chose to linearize the metric around a locally flat background. However, this is a $\text{false}$ vacuum to start with for pure $R^2$ gravity. These studies unknowingly omitted the information about $\Lambda$ of the de Sitter background, hence incapable of attaining a Newtonian behavior in pure $R^2$ gravity. Secondly, as $\Lambda$ appears in $V(r)$ in a $\text{singular}$ manner, viz. $V(r)\propto {\Lambda }^{-1}$, the Newtonian limit for pure $R^2$ gravity cannot be obtained by any perturbative approach treating $\Lambda$ as a small parameter.

Accepted to JHEP

References in corpus (13)

• Avoiding Dark Energy with 1/R Modifications of Gravity
• Black Holes in Higher-Derivative Gravity
• Spherically Symmetric Solutions to Fourth-Order Theories of Gravity
• New Exact Solutions of Quadratic Curvature Gravity
• The Newtonian limit of metric gravity theories with quadratic Lagrangians
• Restricted Weyl invariance in four-dimensional curved spacetime
• Black Holes in Higher Dimensional Gravity Theory with Quadratic in Curvature Corrections
• Traversable Morris-Thorne-Buchdahl wormholes in quadratic gravity
• Beyond Schwarzschild-de Sitter spacetimes: I. A new exhaustive class of metrics inspired by Buchdahl for pure $R^2$ gravity in a compact form
• Beyond Schwarzschild-de Sitter spacetimes: II. An exact non-Schwarzschild metric in pure $R^2$ gravity and new anomalous properties of $R^2$ spacetimes
• Scale Invariant Gravity and Black Hole Ringdown
• Physical black holes in fourth-order gravity
• Beyond Schwarzschild-de Sitter spacetimes: III. A perturbative vacuum with non-constant scalar curvature in $R+R^2$ gravity

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• Observational tests of asymptotically flat ${\mathcal{R}}^2$ spacetimes
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