The set of all metrics that can be placed on a given manifold defines an infinite-dimensional `superspace' that can itself be imbued with the structure of a Riemannian manifold. Geodesic distances between points on Met$(M)$ measure how close two different metrics over $M$ are to one another. Restricting our attention to only those metrics that describe physical black holes, these distances may therefore be thought of as measuring the level of geometric similarity between different black hole structures. This allows for a systematic quantification of the extent to which a black hole, possibly arising as an exact solution to a theory of gravity extending general relativity in some way, might be `non-Kerr'. In this paper, a detailed construction of a superspace for stationary black holes with an arbitrary number of hairs is carried out. As an example application, we are able to strengthen a recent claim made by Konoplya and Zhidenko about which deviation parameters describing a hypothetical, non-Schwarzschild black hole are likely to be most relevant for astrophysical observables.

中文翻译：

---

量化黑洞时空之间的接近度：一种超空间方法

可以放在给定流形上的所有度量的集合定义了一个无限维的“超空间”，它本身可以充满黎曼流形的结构。Met$(M)$ 上的点之间的测地距离衡量 $M$ 上的两个不同指标彼此之间的接近程度。将我们的注意力限制在那些描述物理黑洞的度量上，因此这些距离可以被认为是衡量不同黑洞结构之间几何相似性的水平。这允许系统量化黑洞可能是“非克尔”的程度，该黑洞可能是对以某种方式扩展广义相对论的引力理论的精确解。在本文中，详细构建了具有任意数量头发的静止黑洞的超空间。