Deflection angle in the strong deflection limit: A perspective from local geometrical invariants and matter distributions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911452073295872 |
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| author | Igata, Takahisa |
| author_facet | Igata, Takahisa |
| contents | In static, spherically symmetric spacetimes, the deflection angle of photons in the strong deflection limit exhibits a logarithmic divergence. We introduce an analytical framework that clarifies the physical origin of this divergence by employing local, coordinate-invariant geometric quantities alongside the properties of the matter distribution. In contrast to conventional formulations -- where the divergence rate $\bar{a}$ is expressed via coordinate-dependent metric functions -- our approach relates $\bar{a}$ to the components of the Einstein tensor in an orthonormal basis adapted to the spacetime symmetry. By applying the Einstein equations, we derive the expression \begin{align*} \bar{a}=\frac{1}{\sqrt{1-8πR_{\mathrm{m}}^2\left(ρ_{\mathrm{m}}+Π_{\mathrm{m}}\right)}}, \end{align*} where $ρ_{\mathrm{m}}$ and $Π_{\mathrm{m}}$ denote the local energy density and tangential pressure evaluated at the photon sphere of areal radius $R_{\mathrm{m}}$. This result reveals that $\bar{a}$ is intrinsically governed by the local matter distribution, with the universal value $\bar{a}=1$ emerging when $ρ_{\mathrm{m}}+Π_{\mathrm{m}}=0$. Notably, this finding resolves the long-standing puzzle of obtaining $\bar{a}=1$ in a class of spacetimes supported by a massless scalar field. Furthermore, these local properties are reflected in the frequencies of quasinormal modes, suggesting a profound connection between strong gravitational lensing and the dynamical response of gravitational wave signals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02320 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deflection angle in the strong deflection limit: A perspective from local geometrical invariants and matter distributions Igata, Takahisa General Relativity and Quantum Cosmology High Energy Astrophysical Phenomena High Energy Physics - Theory In static, spherically symmetric spacetimes, the deflection angle of photons in the strong deflection limit exhibits a logarithmic divergence. We introduce an analytical framework that clarifies the physical origin of this divergence by employing local, coordinate-invariant geometric quantities alongside the properties of the matter distribution. In contrast to conventional formulations -- where the divergence rate $\bar{a}$ is expressed via coordinate-dependent metric functions -- our approach relates $\bar{a}$ to the components of the Einstein tensor in an orthonormal basis adapted to the spacetime symmetry. By applying the Einstein equations, we derive the expression \begin{align*} \bar{a}=\frac{1}{\sqrt{1-8πR_{\mathrm{m}}^2\left(ρ_{\mathrm{m}}+Π_{\mathrm{m}}\right)}}, \end{align*} where $ρ_{\mathrm{m}}$ and $Π_{\mathrm{m}}$ denote the local energy density and tangential pressure evaluated at the photon sphere of areal radius $R_{\mathrm{m}}$. This result reveals that $\bar{a}$ is intrinsically governed by the local matter distribution, with the universal value $\bar{a}=1$ emerging when $ρ_{\mathrm{m}}+Π_{\mathrm{m}}=0$. Notably, this finding resolves the long-standing puzzle of obtaining $\bar{a}=1$ in a class of spacetimes supported by a massless scalar field. Furthermore, these local properties are reflected in the frequencies of quasinormal modes, suggesting a profound connection between strong gravitational lensing and the dynamical response of gravitational wave signals. |
| title | Deflection angle in the strong deflection limit: A perspective from local geometrical invariants and matter distributions |
| topic | General Relativity and Quantum Cosmology High Energy Astrophysical Phenomena High Energy Physics - Theory |
| url | https://arxiv.org/abs/2503.02320 |