We study the quantum vacuum zero point energy in the Schwarzschild black hole as well as in the Nariai limit of the de Sitter–Schwarzschild backgrounds. We show that the regularized vacuum energy density near the black hole and also in the Nariai setup match exactly with the corresponding value in the flat background, scaling with the fourth power of the mass of the quantum field. The horizon radius of the de Sitter (dS) space created from the vacuum zero point energy introduces a new length scale which should be compared with the black hole horizon radius. There is an upper limiting mass for the black hole immersed in the vacuum zero point energy which is determined by the mass of the Nariai metric associated to the dS background constructed from zero point energy. We calculate the variance in the distribution of the vacuum zero point energy and the density contrast and show that it develops strong inhomogeneities on subhorizon scales.

中文翻译：

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黑洞背景中的宇宙常数和真空零点能量

我们研究了史瓦西黑洞以及德西特-史瓦西背景的 Nariai 极限中的量子真空零点能量。我们表明，黑洞附近以及 Nariai 设置中的正则化真空能量密度与平坦背景中的相应值完全匹配，与量子场质量的四次方成比例。由真空零点能量产生的德西特 (dS) 空间的视界半径引入了一个新的长度尺度，应该与黑洞视界半径进行比较。沉浸在真空零点能量中的黑洞有一个上限质量，该质量由与由零点能量构成的 dS 背景相关的 Nariai 度量的质量决定。