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 $α(b)=\frac{2(1+γ)GM_{\odot}}{c^{2}b}.$The resulting metric function is shown to recover the Schwarzschild form in the weak-field limit \(r\gg M\). 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.