发表机构
Indian Institute of Engineering Science and Technology, Shibpur（印度工程科学和技术学院（希布尔））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于Abel变换的逆透镜形式，从光偏折角直接重建静态球对称时空度规，无需预设度规，并在太阳偏折案例中恢复Schwarzschild解，为观测与时空几何建立新桥梁。
AI 中文摘要
我们发展了一种逆透镜形式，用于直接从光的引力偏折角重建静态球对称时空的度规函数。通过将逆问题表述为Abel变换，我们推导出一个积分-微分关系，将可观测的偏折轮廓与潜在的时空几何联系起来。作为一致性检验，我们考虑了零偏折情形，\\(\alpha(b)=0\\)，并恢复出\\(A(r)=1\\)，对应于平坦的闵可夫斯基时空。然后，我们将该形式应用于太阳对光的引力偏折，使用观测驱动的领头阶表达式\\(\alpha(b)=\frac{2(1+\gamma)GM_{\odot}}{c^{2}b}\\)。所得度规函数在弱场极限\\(r\gg M\\)下恢复为Schwarzschild形式。我们进一步考虑太阳偏折角的高阶修正，并在领头阶近似之外重建相应的度规。有趣的是，在所得弱场展开中，\\((M/r)^2\\)项消失，与从领头阶偏折角重建的度规相比，提供了改进的近似。我们的结果表明，引力透镜观测可以提供一条直接途径来重建潜在的时空几何，而无需先验地假设特定度规。因此，该形式为连接观测光传播与时空几何提供了一个新颖的框架。
英文摘要
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.
