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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.06706 |
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| _version_ | 1866914281809772544 |
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| author | Hurtado, Roger Anderson |
| author_facet | Hurtado, Roger Anderson |
| contents | In this work, we linearize the field equations of $f(R)$ gravity using the Starobinsky model, $R+R^2/(6m^2)$, and examine the modifications to General Relativity. We derive an equation for the trace, $T$, of the energy-momentum tensor, which we then decompose using an auxiliary field. This field satisfies the wave equation with $T$ as its source, while simultaneously acting as an effective source for the classical deviation, $\bar h$, governed by the Klein-Gordon equation. The fields were expressed in terms of Green's functions, whose symmetry properties facilitated the solution of the trace equation. Then $\bar h_{μν}$ was determined in terms of a modified or effective matter-energy distribution. From this, the effective energy density was obtained as the usual energy density $T_{00}$, plus a perturbative correction proportional to $m^{-2}$, involving the Laplacian of the integral of $T$, weighted by the retarded propagator of the Klein-Gordon equation. Finally, we numerically computed the perturbative term in a binary star system, evaluating it as a function of $m$ and spatial position near the stars. In all cases, the results illustrate how the gravitational influence of the stars diminishes with distance. Additionally, the perturbation decreases as $m$ increases, consistently recovering the relativistic limit. These results highlight the role of modified gravity corrections in the vicinity of compact objects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06706 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linearized Gravity in the Starobinsky Model: Perturbative Deviations from General Relativity Hurtado, Roger Anderson General Relativity and Quantum Cosmology High Energy Astrophysical Phenomena In this work, we linearize the field equations of $f(R)$ gravity using the Starobinsky model, $R+R^2/(6m^2)$, and examine the modifications to General Relativity. We derive an equation for the trace, $T$, of the energy-momentum tensor, which we then decompose using an auxiliary field. This field satisfies the wave equation with $T$ as its source, while simultaneously acting as an effective source for the classical deviation, $\bar h$, governed by the Klein-Gordon equation. The fields were expressed in terms of Green's functions, whose symmetry properties facilitated the solution of the trace equation. Then $\bar h_{μν}$ was determined in terms of a modified or effective matter-energy distribution. From this, the effective energy density was obtained as the usual energy density $T_{00}$, plus a perturbative correction proportional to $m^{-2}$, involving the Laplacian of the integral of $T$, weighted by the retarded propagator of the Klein-Gordon equation. Finally, we numerically computed the perturbative term in a binary star system, evaluating it as a function of $m$ and spatial position near the stars. In all cases, the results illustrate how the gravitational influence of the stars diminishes with distance. Additionally, the perturbation decreases as $m$ increases, consistently recovering the relativistic limit. These results highlight the role of modified gravity corrections in the vicinity of compact objects. |
| title | Linearized Gravity in the Starobinsky Model: Perturbative Deviations from General Relativity |
| topic | General Relativity and Quantum Cosmology High Energy Astrophysical Phenomena |
| url | https://arxiv.org/abs/2411.06706 |