We derive the quadratic action for the physical degrees of freedom of massless spin-0, spin-1, and spin-2 perturbations on a Schwarzschild-(A)dS background in arbitrary dimensions. We then use these results to compute the static response of asymptotically flat Schwarzschild black holes to external fields. Our analysis reproduces known facts about black hole Love numbers — in particular that they vanish for all types of perturbation in four spacetime dimensions — but also leads to new results. For instance, we find that neutral Schwarzschild black holes polarize in the presence of an electromagnetic background in any number of spacetime dimensions except four. Moreover, we calculate for the first time black hole Love numbers for vector-type gravitational perturbations in higher dimensions and find that they generically do not vanish. Along the way, we shed some light on an apparent discrepancy between previous results in the literature, and clarify some aspects of the matching between perturbative calculations of static response on a Schwarzschild background and the point-particle effective theory.

我们在任意维度的 Schwarzschild-(A)dS 背景上推导出无质量自旋 0、自旋 1 和自旋 2 扰动的物理自由度的二次作用。然后我们使用这些结果来计算渐近平坦的 Schwarzschild 黑洞对外部场的静态响应。我们的分析再现了关于黑洞洛夫数的已知事实——特别是它们在四个时空维度的所有类型的扰动中都消失了——但也导致了新的结果。例如，我们发现中性 Schwarzschild 黑洞在存在电磁背景的情况下会在除四个时空维度之外的任意数量的时空维度中极化。此外，我们首次计算了更高维度矢量型引力扰动的黑洞洛夫数，发现它们一般不会消失。一路上，