广义相对论的一种非线性牛顿近似
A nonlinear Newtonian approximation to General Relativity
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- Università degli Studi dell’Insubria（因苏布里亚大学）
- INFN, Sezione di Milano（意大利国家核物理研究所米兰分部）
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中文总结 AI 辅助
提出一种不要求弱场条件的非线性牛顿近似，通过调和坐标保留约化爱因斯坦方程主部，在弱场恢复牛顿方程，并预言引力透镜偏离标准弱场描述。
中文摘要 AI 辅助
我们研究了一种非相对论性的广义相对论近似，该近似不要求度规系数是闵可夫斯基度规的弱扰动。相反，我们假设一个微分层级，其中包含度规二阶导数的曲率贡献占主导地位，而包含一阶导数二次项的贡献次之，同时保留的项所乘的依赖于度规的系数保持完全非线性。由于这种分离不是协变的，我们在调和坐标下表述该近似，在此坐标下它对应于保留约化爱因斯坦方程的主部。我们证明了截断的爱因斯坦张量满足缩并的比安基恒等式，精确到导数层级中忽略的阶数，从而确保在保留阶数上与能量-动量守恒相容。在弱场区域恢复标准牛顿方程，而在弱场区域之外，动力学保留了对时移和空间度规的非线性依赖。特别地，零测地线在领头阶探测空间几何，这表明引力透镜是可能产生可观测的偏离标准弱场描述的自然场景。我们用一个静态空间各向同性构型来说明该构造，并讨论了该近似的适用范围和局限性。
英文摘要
We investigate a nonrelativistic approximation to general relativity that does not require the metric coefficients to be weak perturbations of the Minkowski metric. Instead, we assume a differential hierarchy in which the contributions to the curvature containing second derivatives of the metric dominate over those quadratic in its first derivatives, while the metric-dependent coefficients multiplying the retained terms are kept fully nonlinear. Since this separation is not covariant, we formulate the approximation in harmonic coordinates, where it corresponds to retaining the principal part of the reduced Einstein equations. We show that the truncated Einstein tensor satisfies the contracted Bianchi identities up to the order neglected in the derivative hierarchy, ensuring compatibility with energy--momentum conservation at the retained order. The standard Newtonian equations are recovered in the weak-field regime, whereas outside it the dynamics retains a nonlinear dependence on the lapse and spatial metric. In particular, null geodesics probe the spatial geometry at leading order, suggesting gravitational lensing as a natural setting in which observable departures from the standard weak-field description may arise. We illustrate the construction with a static spatially isotropic configuration and discuss the scope and limitations of the approximation.
