The expansion of both sides of Einstein’s field equations in the weak-field approximation, up to terms of order 1/c4 is derived. This new approach leads to an extended form of gravitomagnetism (GEM) properly named as Beyond Gravitomagnetism (BGEM). The metric of BGEM includes a quadratic term in the gravitoelectric potential n the time and also space metric functions in contrast with first post-Newtonian 1PN approximation where the quadratic term appears only in the time metric function. This nonlinear term does not appear in conventional GEM, but is essential in achieving the exact value of Mercury’s perihelion advance as we explicitly show. The new BGEM metric is also applied to the classical problem of light deflection by the Sun, but the contribution of the new nonlinear terms produce higher-order terms in this problem and can be neglected, giving the correct result obtained already in the Lense–Thirring (GEM) approximation. The BGEM approximation also provides new terms that depend on the dynamics of the system, which may bring new insights into galactic and stellar physics.

中文翻译：

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超越引力磁学，应用于水星近日点推进和光的弯曲

爱因斯坦场方程两边在弱场近似中的展开，直到阶项1/C4是派生的。这种新方法导致了一种扩展形式的引力磁（GEM），被恰当地命名为超越引力磁（BGEM）。BGEM 的度量包括重力电势中的二次项 n 时间和空间度量函数，与第一个后牛顿1PN 近似，其中二次项仅出现在时间度量函数中。这个非线性项没有出现在传统的 GEM 中，但对于实现我们明确展示的水星近日点推进的确切值至关重要。新的 BGEM 度量也适用于太阳光偏转的经典问题，但新非线性项的贡献在该问题中产生了高阶项，可以忽略不计，给出了 Lense-Thirring 中已经获得的正确结果(GEM) 近似。BGEM 近似还提供了依赖于系统动力学的新术语，这可能会给银河系和恒星物理学带来新的见解。