Spherically symmetric configurations in the quadratic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity

We study spherically symmetric configurations of the quadratic <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>f</a:mi><a:mo stretchy="false">(</a:mo><a:mi>R</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:math> gravity [<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>f</e:mi><e:mo stretchy="false">(</e:mo><e:mi>R</e:mi><e:mo stretchy="false">)</e:mo><e:mo>=</e:mo><e:mi>R</e:mi><e:mo>−</e:mo><e:msup><e:mrow><e:mi>R</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msup><e:mo>/</e:mo><e:mn>6</e:mn><e:msup><e:mrow><e:mi>μ</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msup></e:mrow></e:math>]. In the case of a purely gravitational system, we have fully investigated the global qualitative behavior of all static solutions satisfying the conditions of asymptotic flatness. These solutions are proved to be regular everywhere except for a naked singularity at the center; they are uniquely determined by the total mass <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi mathvariant="fraktur">M</i:mi></i:math> and the “scalar charge” <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>Q</l:mi></l:math> characterizing the strength of the scalaron field at spatial infinity. The case <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>Q</n:mi><n:mo>=</n:mo><n:mn>0</n:mn></n:math> yields the Schwarzschild solution, but an arbitrarily small <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>Q</p:mi><p:mo>≠</p:mo><p:mn>0</p:mn></p:math> leads to the appearance of a central naked singularity having a significant effect on the neighboring region, even when the space-time metric in the outer region is practically insensitive to the scalaron field. Approximation procedures are developed to derive asymptotic relations near the naked singularity and at spatial infinity, and the leading terms of the solutions are presented. We have investigated the linear stability of the static solutions with respect to radial perturbations satisfying the null Dirichlet boundary condition at the center and numerically estimate the range of parameters corresponding to stable/unstable configurations. In particular, the configurations with sufficiently small <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>Q</r:mi></r:math> turn out to be linearly unstable. Published by the American Physical Society 2024

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