Arguing from his "hole" thought experiment, Einstein became convinced that, in cases in which the energy-momentum-tensor source vanishes in a spacetime hole, a solution to his general relativistic field equation cannot be uniquely determined by that source. After reviewing the definition of active diffeomorphisms, this paper uses them to outline a mathematical proof of Einstein's result. The relativistic field equation is shown to have multiple solutions, just as Einstein thought. But these multiple solutions can be distinguished by the different physical meaning that each metric solution attaches to the local coordinates used to write it. Thus the hole argument, while formally correct, does not prohibit the subsequent rejection of spurious solutions and the selection of a physically unique metric. This conclusion is illustrated using the Schwarzschild metric. It is suggested that the Einstein hole argument therefore cannot be used to argue against substantivalism.

中文翻译：

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爱因斯坦孔论的有效性

爱因斯坦从他的“空洞”思想实验中辩解说，他确信，当能量动量张量源在时空洞中消失时，他的广义相对论场方程的解不能由该源唯一地确定。在回顾了有源微分形的定义之后，本文使用它们来概述爱因斯坦结果的数学证明。正如爱因斯坦所认为的那样，相对论场方程具有多种解。但是，可以通过每个度量标准解决方案附加到用于写入它的局部坐标的不同物理含义来区分这些多个解决方案。因此，空洞论点虽然在形式上是正确的，但并不禁止随后拒绝虚假解和选择物理上唯一的度量。使用Schwarzschild度量说明了这一结论。因此建议，爱因斯坦孔论不能被用来反对实体主义。