by Carlos Herdeiro and Eugen Radu, Guest Editors of Focus Issue: Hairy Black Holes.

One of the most recognizable statements about black holes is that they have “no-hair”. Close inspection, however, shows that this is a belief rather than a mathematically proven theorem. Moreover, decades of research on this topic have shown that, depending on what one precisely means, this statement may be simply wrong. That is, as solutions of Einstein’s equations, in a generic context, black holes are not necessarily “bald”. Then, less ambitious, but perhaps more relevant questions are: “Can astrophysical black holes have hair?” and “Can we test the existence of black hole hair with present and future astrophysical observations?”.

This CQG focus issue brings together a set of papers describing models in which black holes do have “hair”, as well as observational efforts that have the potential to assess if this is (or not) the case for astrophysical black hole candidates. This collection of research papers is by no means a faithful and complete description of all possible alternatives to the Kerr paradigm in the literature. Rather, the selected papers focus on contexts of recent discussion that have presented new angles on the topic. We hope, nevertheless, that it can help emphasizing the necessity of scrutinizing non-Kerr-like compact objects, in order to have more robust interpretations of observational data in both the gravitational waves and electromagnetic channels—in particular addressing possible model degeneracy—that are expected to become increasingly more precise in the next few years.

**What is black hole hair?**

Black hole spacetimes, like any other physical system, have a number of macroscopic degrees of freedom that need to be specified to fully describe their macroscopic physics. The typical ones are mass, angular momentum and electric charge. These have the property that can be measured just by looking at the behaviour of some physical quantities far away from the black hole. In other words they are associated to a Gauss-law. But are there degrees of freedom of a different nature?

The idea that black holes have “no hair” was introduced by John Wheeler in the early 1970s, and was borne out the black hole uniqueness theorems. These mathematical results show the most general physically acceptable (single) black hole solution in electro-vacuum is the Kerr-Newman solution, which is completely described by the three physical quantities above (a magnetic charge can also be added, but that does not change the spirit of the discussion). Thus, black holes are simple, certainly simpler than typical macroscopic objects. The reader can easily imagine a football and a rugby ball, both of which are electrically neutral, which could have precisely the same mass and both could be at rest (thus zero angular momentum); obviously they would be quite different. But black holes are distinct. Two (neutral) black holes with the same mass and angular momentum must be exactly equal.

The football and rugby ball can be distinguished, as sources of a gravitational field, by their multipole structure. For instance, they will have a different mass quadrupole. Kerr black holes, on the other hand, also possess higher gravitational multipoles, say a mass quadrupole, but these are not independent degrees of freedom; rather, they are completely determined by their mass and angular momentum, which as mentioned above can be measured very far away from the black hole. Thus, studying the black hole far away from it, an observer may infer everything that can be possibly known (macroscopically) about the black hole spacetime – as long as he/she studied the theory of the Kerr-Newman solution, of course!

Broadly speaking, black hole “hair” is a physical parameter, or degree of freedom, that needs to be specified to fully describe the black hole, but that, unlike the above examples, is not associated to a Gauss-law. Knowing this parameter, therefore requires probing the spacetime deeper inside. For instance one may imagine measuring the mass quadrupole of an astrophysical black hole candidate (which requires probing the spacetime “closer” to the compact object than a measurement of its mass) and check if it is related to the mass and angular momentum by the relation expected for a Kerr black hole. If it is not, the black hole has “hair”.

In a nutshell, in Einstein’s theory: black hole “hair” makes the black hole different from the Kerr solution, by quantities not measurable at spatial infinity.

**Around the no-“hair” paradigm: exotic matter or exotic gravity?**

The physical concept behind the no-hair conjecture, for black holes, is that, classically, black holes are “all eating machines”. Black holes end up consuming any type of substance that exists in their vicinity. Consequently, any physical “substance” not protected by a powerful conservation law, that may prevent it from disappearing without trace once it falls behind the horizon, cannot be present in the end state of gravitational collapse, and hence in physically relevant black hole spacetimes. The role of the Gauss-law is precisely that of creating a memory, for outside observers, of the corresponding physical quantity, even when its source falls behind the horizon.

The basic example of this behaviour is the following. In the gravitational collapse of a star, all mass and rotation gravitational multipoles end up disappearing as degrees of freedom. The only surviving ones are exactly the mass and angular momentum, precisely the ones associated to Gauss laws, anchored on spacetime symmetries.

There are, however, ways to try and go around this apparently natural picture. One way is to modify the theory of gravity away from Einstein’s general relativity, which we call exotic gravity. Another is to consider some sort of exotic matter that does not naturally fall into a black hole. Both types of situations have been shown to endow black holes with “hair”. Perhaps more surprisingly, recently it was found that even quite reasonable physical matter can create resonances around Kerr black holes which can endow the latter with hair and change its geometry away from that of Kerr, when a substantial amount of this matter is present.

All these situations are typically studied by using scalar fields, which (unless the scalar field is gauged) have no associated Gauss law. The exotic gravity could be a scalar-tensor theory and examples of hairy black holes in these models are exemplified in this volume. Similarly, the exotic matter could also be a scalar field, as also illustrated in this issue. But an example where the exotic matter is a massive vector field (a.k.a. Proca field) is also provided.

**Observational windows for black hole hair**

The recent discovery of gravitational waves, together with the interpretation of the signal as originating from a black hole binary merger is a major breakthrough for the field of strong gravity. In particular, this event can be taken as evidence for the existence of black holes in the universe and it has even been interpreted as evidence for the existence of Kerr black holes. This interpretation is only robust, however, if no other theoretically sound alternative compact object can produce a similar phenomenology, within current error bars. Thus, the awakening of the gravitational astronomy era, together with the ever increasing precision of electromagnetic observations, in particular some targeting black holes and their properties, motivates scrutinizing models of alternative compact objects and their phenomenology.

In this volume we have included discussions of electromagnetic observables, such as black hole shadows and features of the X-ray spectrum of a black hole surrounded by an accretion disk, such as the thermal spectrum or the iron line in the reflexion spectrum, that could be used to test the no-hair paradigm.

Assembling these various theoretical models, together with observational ways in which the existence of hair can be assessed, will hopefully help to demystify this concept and pave the way for a rigorous understanding if and under which circumstances, black holes in the Universe are really “bald”.

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