We present a class of first-order perturbative spacetime solutions within the framework of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$F(R) = R + \alpha R^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>α</mi> <msup> <mi>R</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> gravity, coupled with a phenomenological hybrid equation of state <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p = w\rho + \beta \rho ^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>w</mi> <mi>ρ</mi> <mo>+</mo> <mi>β</mi> <msup> <mi>ρ</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. Using a covariant (1+1+2) approach and a perturbative expansion in the parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, we obtain metric solutions truncated at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {O}(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for static spherically symmetric configurations. These approximate solutions are designed to isolate and study the specific effects of quadratic gravitational corrections on spacetime geometry, separate from the complexities of realistic nuclear matter. The solutions reveal key theoretical features: (i) a characteristic deformation of the spacetime metric governed by the parameter <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma = (w + 2\beta \rho _c)/(1 + w + \beta \rho _c)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo>+</mo> <mn>2</mn> <mi>β</mi> <msub> <mi>ρ</mi> <mi>c</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>w</mi> <mo>+</mo> <mi>β</mi> <msub> <mi>ρ</mi> <mi>c</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, (ii) internal curvature transitions when <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$1-3w+6\beta \rho _c < 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>-</mo> <mn>3</mn> <mi>w</mi> <mo>+</mo> <mn>6</mn> <mi>β</mi> <msub> <mi>ρ</mi> <mi>c</mi> </msub> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and (iii) modified stability criteria within this theoretical framework. Our work provides a mathematically tractable perturbative model for studying how strong-field F(R) gravity interacts with dense matter, offering closed-form expressions at first order that illuminate the functional dependence of spacetime geometry on model parameter. These perturbative results provide a new class of benchmark solutions for testing modified gravity theories in the strong-field regime, complementing numerical studies that use microphysical equations of state.

Perturbative Spacetime Structures in $F(R) = R + \alpha R^2$ Gravity with a Hybrid Equation of State

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发表日期：

2026年7月14日

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Adnan Malik,
Jamshed Khan,
Ahdab K. Althukair,
Uzma Gul,
Sumaira Saleem Akhtar

摘要

We present a class of first-order perturbative spacetime solutions within the framework of $F(R) = R + \alpha R^2$ $F (R) = R + α R^{2}$ gravity, coupled with a phenomenological hybrid equation of state $p = w\rho + \beta \rho ^2$ $p = w ρ + β ρ^{2}$ . Using a covariant (1+1+2) approach and a perturbative expansion in the parameter $\alpha $ $α$ , we obtain metric solutions truncated at $\mathcal {O}(\alpha )$ $O (α)$ for static spherically symmetric configurations. These approximate solutions are designed to isolate and study the specific effects of quadratic gravitational corrections on spacetime geometry, separate from the complexities of realistic nuclear matter. The solutions reveal key theoretical features: (i) a characteristic deformation of the spacetime metric governed by the parameter $\gamma = (w + 2\beta \rho _c)/(1 + w + \beta \rho _c)$ $γ = (w + 2 β ρ_{c}) / (1 + w + β ρ_{c})$ , (ii) internal curvature transitions when $1-3w+6\beta \rho _c < 0$ $1 - 3 w + 6 β ρ_{c} < 0$ , and (iii) modified stability criteria within this theoretical framework. Our work provides a mathematically tractable perturbative model for studying how strong-field F(R) gravity interacts with dense matter, offering closed-form expressions at first order that illuminate the functional dependence of spacetime geometry on model parameter. These perturbative results provide a new class of benchmark solutions for testing modified gravity theories in the strong-field regime, complementing numerical studies that use microphysical equations of state.

Perturbative Spacetime Structures in \(F(R) = R + \alpha R^2\) Gravity with a Hybrid Equation of State

摘要