An Abel-Inversion Formalism for Spacetime Metric Reconstruction from Light Deflection
Abstract
We develop an inverse lensing formalism for reconstructing the metric function of a static, spherically symmetric spacetime directly from the gravitational deflection angle of light. By formulating the inverse problem through an Abel transformation, we derive an integro-differential relation connecting the observable deflection profile to the underlying spacetime geometry. As a consistency check, we consider the case of vanishing deflection, α(b)=0, and recover A(r)=1, corresponding to flat Minkowski spacetime. We then apply the formalism to the gravitational deflection of light by the Sun using the observationally motivated leading-order expression α(b)=(2(1+γ)GM⊙)/c2b.The resulting metric function is shown to recover the Schwarzschild form in the weak-field limit r≫ M. We further consider the higher-order correction to the solar deflection angle and reconstruct the corresponding metric beyond the leading-order approximation. Interestingly, the (M/r)2 term vanishes in the resulting weak-field expansion, yielding an improved approximation compared with the metric reconstructed from the leading-order deflection angle. Our results demonstrate that gravitational lensing observations can provide a direct route to reconstructing the underlying spacetime geometry without assuming a specific metric a priori. This formalism therefore, offers a novel framework for connecting observational light propagation with spacetime geometry.
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BibTeX
@misc{ghosh2026abelinversion,
title = {{An Abel-Inversion Formalism for Spacetime Metric Reconstruction from Light Deflection}},
author = {Aniruddha Ghosh and Ujjal Debnath},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16844},
eprint = {2609.16844},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16844},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Abel-Inversion Formalism for Spacetime Metric Reconstruction from Light Deflection
AU - Aniruddha Ghosh
AU - Ujjal Debnath
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16844
UR - https://arxiv.org/abs/2609.16844
AB - We develop an inverse lensing formalism for reconstructing the metric function of a static, spherically symmetric spacetime directly from the gravitational deflection angle of light. By formulating the inverse problem through an Abel transformation, we derive an integro-differential relation connecting the observable deflection profile to the underlying spacetime geometry. As a consistency check, we consider the case of vanishing deflection, α(b)=0, and recover A(r)=1, corresponding to flat Minkowski spacetime. We then apply the formalism to the gravitational deflection of light by the Sun using the observationally motivated leading-order expression α(b)=(2(1+γ)GM_⊙)/c^2b.The resulting metric function is shown to recover the Schwarzschild form in the weak-field limit r≫ M. We further consider the higher-order correction to the solar deflection angle and reconstruct the corresponding metric beyond the leading-order approximation. Interestingly, the (M/r)^2 term vanishes in the resulting weak-field expansion, yielding an improved approximation compared with the metric reconstructed from the leading-order deflection angle. Our results demonstrate that gravitational lensing observations can provide a direct route to reconstructing the underlying spacetime geometry without assuming a specific metric a priori. This formalism therefore, offers a novel framework for connecting observational light propagation with spacetime geometry.
ER -