In a recent paper, quantum vacuum was considered as a source of gravity, and the simplest, phenomenon, the gravitational polarization of the quantum vacuum by an immersed point-like body, was studied. In this paper, we have derived the effective gravitational charge density of the quantum vacuum, caused by two immersed point-like bodies. Among others, the obtained result proves that quantum vacuum can have regions with a negative effective gravitational charge density. Hence, quantum vacuum, the ‘ocean’ in which all matter of the Universe is immersed, acts as a complex fluid with a very variable gravitational charge density that might include both positive and negative densities; a crucial prediction that can be tested within the Solar system. In the general case of ${N \ge {\rm{3}}}$ point-like bodies, immersed in the quantum vacuum, the analytical solutions are not possible, and the use of numerical methods is inevitable. The key point is that an appropriate numerical method, for the calculation of the effective gravitational charge density of the quantum vacuum induced by N immersed bodies, might be crucial in description of galaxies, without the involvement of dark matter or a modification of gravity. The development of such a valuable numerical method, is not possible, without a previous (and in this study achieved) understanding of the impact of a two-body system.

中文翻译：

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两个点状体引起的量子真空引力极化

在最近的一篇论文中，量子真空被认为是引力的来源，并且研究了最简单的现象，即浸没的点状体对量子真空的引力极化。在本文中，我们推导出了由两个浸没的点状体引起的量子真空的有效引力电荷密度。其中，所获得的结果证明了量子真空可以具有具有负有效引力电荷密度的区域。因此，量子真空，即宇宙中所有物质都浸入其中的“海洋”，作为一种复杂的流体，具有非常可变的引力电荷密度，可能包括正密度和负密度。一个可以在太阳系内测试的重要预测。在 ${N \ge {\rm{3}}}$ 点状物体的一般情况下，沉浸在量子真空中，解析解是不可能的，使用数值方法是不可避免的。关键是，在不涉及暗物质或引力修改的情况下，一种适当的数值方法，用于计算由 N 个浸没物体引起的量子真空的有效引力电荷密度，可能对描述星系至关重要。如果没有先前（并且在本研究中实现）对二体系统影响的理解，就不可能开发出这种有价值的数值方法。没有暗物质的参与或引力的改变。如果没有先前（并且在本研究中实现）对二体系统影响的理解，就不可能开发出这种有价值的数值方法。没有暗物质的参与或引力的改变。如果没有先前（并且在本研究中实现）对二体系统影响的理解，就不可能开发出这种有价值的数值方法。