Influence of gravity on the quantum vacuum of a massless minimally coupled scalar field under Robin boundary conditions on parallel plates is investigated. We introduce the detailed calculation of the volume energy for the case the gravitational background is weak in its most general form for a static spacetime. It founds that the quantum vacuum usually reacts to the gravitational field by decreasing the Casimir energy. In addition, we find sufficient conditions under which the Casimir force increases. Interestingly, the first order perturbation corrections, are present in the obtained formula for the volume energy. We show that for some specific choices of parameters, the energy is independent of Robin coefficients. Consistency with the literature is shown in some limiting cases and well-known examples are presented for both an increase or decrease in the volume energy.

中文翻译：

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罗宾边界条件下的卡西米尔板块：弯曲时空的情况

研究了重力对平行板Robin边界条件下无质量最小耦合标量场量子真空的影响。我们介绍了在静态时空下最基本形式的引力背景较弱的情况下体积能量的详细计算。发现量子真空通常通过降低卡西米尔能量来对引力场作出反应。另外，我们找到了卡西米尔力增加的充分条件。有趣的是，在获得的体积能量公式中，存在一阶扰动校正。我们表明，对于某些特定的参数选择，能量与Robin系数无关。