The Weiss variation of the Einstein-Hilbert action with an appropriate boundary term has been studied for general boundary surfaces; the boundary surfaces can be spacelike, timelike, or null. To achieve this we introduce an auxiliary reference connection and find that the resulting Weiss variation yields the Einstein equations as expected, with additional boundary contributions. Among these boundary contributions, we obtain the dynamical variable and the associated conjugate momentum, irrespective of the spacelike, timelike or, null nature of the boundary surface. We also arrive at the generally non-vanishing covariant generalization of the Einstein energy-momentum pseudotensor. We study this tensor in the Schwarzschild geometry and find that the pseudotensorial ambiguities translate into ambiguities in the choice of coordinates on the reference geometry. Moreover, we show that from the Weiss variation, one can formally derive a gravitational Schrödinger equation, which may, despite ambiguities in the definition of the Hamiltonian, be useful as a tool for studying the problem of time in quantum general relativity. Implications have been discussed.

已经研究了具有适当边界项的 Einstein-Hilbert 作用的 Weiss 变化，用于一般边界表面；边界表面可以是类空间的、类时的或空的。为了实现这一点，我们引入了一个辅助参考连接，并发现由此产生的 Weiss 变化产生了预期的爱因斯坦方程，并具有额外的边界贡献。在这些边界贡献中，我们获得了动力学变量和相关的共轭动量，而与边界表面的类空间、类时间或零性质无关。我们还得出了爱因斯坦能量-动量赝张量的一般非零协变推广。我们在 Schwarzschild 几何中研究了这个张量，并发现在参考几何上的坐标选择中，伪张量歧义转化为歧义。此外，我们表明，从 Weiss 变式中，人们可以正式推导出引力薛定谔方程，尽管哈密顿量的定义有歧义，但它可以作为研究量子广义相对论中时间问题的工具。已经讨论了影响。