When minimal length uncertainty emerging from a generalized uncertainty principle (GUP) is thoughtfully implemented, it is of great interest to consider its impacts on gravitational Einstein field equations (gEFEs) and to try to assess consequential modifications in metric manifesting properties of quantum geometry due to quantum gravity. GUP takes into account the gravitational impacts on the noncommutation relations of length (distance) and momentum operators or time and energy operators and so on. On the other hand, gEFE relates classical geometry or general relativity gravity to the energy–momentum tensors, that is, proposing quantum equations of state. Despite the technical difficulties, we intend to insert GUP into the metric tensor so that the line element and the geodesic equation in flat and curved space are accordingly modified. The latter apparently encompasses acceleration, jerk, and snap (jounce) of a particle in the quasi‐quantized gravitational field. Finite higher orders of acceleration apparently manifest phenomena such as accelerating expansion and transitions between different radii of curvature and so on.

中文翻译：

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线元，度量张量和测地线方程最小长度离散化的结果

当深思熟虑地实现了从广义不确定性原理（GUP）产生的最小长度不确定性时，考虑其对重力爱因斯坦场方程（gEFE）的影响并尝试评估由于量子引力。GUP考虑了重力对长度（距离）和动量算符或时间和能量算符等非交换关系的影响。另一方面，gEFE与经典几何或广义相对论引力有关到能量动量张量，即提出了状态量子方程。尽管存在技术上的困难，我们还是打算将GUP插入度量张量中，以便对平面和弯曲空间中的线元素和测地线方程进行相应的修改。后者显然包含了准量化引力场中粒子的加速度，加速度和弹跳（跳动）。有限的高阶加速度显然表现出诸如加速膨胀和不同曲率半径之间的过渡等现象。