Testing the black hole 'no-hair' hypothesis

We celebrate this year the first direct detection of gravitational waves (GWs) and the first detection of a BH binary, in its last stages of coalescence [1]. In this context, it is appropriate to also honor the centenary of the Schwarzschild solution, which describes any regular asymptotically flat, static and spherically symmetric vacuum spacetime in GR. In standard Schwarzschild coordinates, the solution reads

$$\begin{eqnarray}&&{{\rm{d}}{s}}^{2}=-{c}^{2}\left(1-\displaystyle \frac{2{GM}}{{c}^{2}r}\right){{\rm{d}}{t}}^{2}+{\left(1-\displaystyle \frac{2{GM}}{{c}^{2}r}\right)}^{-1}{{\rm{d}}{r}}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2},\end{eqnarray} \tag{ 1.1 }$$

where G is Newton's constant and c the speed of light. The most distinctive feature of the solution above is a coordinate singularity at $r=2{GM}/{c}^{2}$, describing a null surface (the event horizon) which causally separates the inside and outside regions. For physical setups where (static, spherically symmetric) matter is present, the solution should be truncated at the radius of the object and merged smoothly with some interior solution. For pure vacuum spacetimes, it describes a BH. In other words, any static BH in the Universe which is non-spinning and that lives in approximately empty surroundings is described by the geometry (1.1).

The Schwarzschild solution is fully characterized by a single parameter, the total gravitational mass M. In this respect, it is not dissimilar from its Newtonian counterpart: a spherically symmetric, vacuum, static solution of Newton's gravity is also described by only a mass parameter. It turns out to be difficult to construct BH solutions described by more parameters. For example, let's try to 'anchor' a weak, static massless spin-s field onto the Schwarzschild solution. For the sake of illustration we focus on minimally coupled scalars, vector fields described by Maxwell's theory and gravitational fluctuations within vacuum GR [2]. At a linearized level (i.e., keeping the background geometry fixed), these fields can be expanded in scalar, vector or tensor harmonic functions, parametrized by an integer number $l=0,1,2,\ldots$, which carry information on the angular dependence of the field. These fields are all described by the equation [2]

$$\begin{eqnarray}&&{\left(\left(1-\displaystyle \frac{2{GM}}{{c}^{2}r}\right){\rm{\Psi }}^{\prime} \right)}^{\prime }-\left(\displaystyle \frac{l(l+1)}{{r}^{2}}+\displaystyle \frac{2{GM}(1-{s}^{2})}{{c}^{2}{r}^{3}}\right){\rm{\Psi }}=0,\end{eqnarray} \tag{ 1.2 }$$

where Ψ denotes the field amplitude, $s=0,1,2$ for scalars, vectors and tensors, respectively, and primes stand for radial derivatives. One can now multiply the above equation by the complex conjugate ${{\rm{\Psi }}}^{* }$, integrate from the horizon to infinity, and look for regular solutions of the above equation. We get, upon performing an integration by parts and dropping a boundary term (regular solutions evaluate it to zero)

$$\begin{eqnarray}&&-{\int }_{\tfrac{2{GM}}{{c}^{2}}}^{+\infty }{\rm{d}}{r}\left(1-\displaystyle \frac{2{GM}}{{c}^{2}r}\right)| {\rm{\Psi }}^{\prime} {| }^{2}+\displaystyle \frac{l(l+1)}{{r}^{3}}\left(r+\displaystyle \frac{2{GM}(1-{s}^{2})}{{c}^{2}\,l(l+1)}\right)| {\rm{\Psi }}{| }^{2}=0.\end{eqnarray} \tag{ 1.3 }$$

For any $l\geqslant s$ the integrand is negative-definite outside the horizon, and the only solution to the above equation is the trivial one, ${\rm{\Psi }}=0$. For vector fields, there is a nontrivial solution ${\rm{\Psi }}=\mathrm{const}$ for l = 0, describing a weakly charged BH. For tensors, the only non-trivial solutions (for which the integrand is not positive-definite) have $l=0,1$ and correspond to a slight change of mass and addition of small amount of angular momentum [3]. This simple exercise then leads one to the following conclusions:

• (i)  
  It is impossible to 'anchor' non-interacting scalars (and it turns out, also fermions) onto a Schwarzschild BH.
• (ii)  
  The Schwarzschild solution allows, in principle, for a generalization that includes (only) electric charge and rotation.

Note that the conclusions above are drawn in the context of GR; other theories would lead to different equations of motion that could (and do, sometimes) lead to other types of solutions. Thus, non-rotating BHs do not exhibit any 'protuberance'. John Wheeler and others summarized these results with the expression 'BHs have no hair', where 'hair' is a measure of the complexity of the gravitational field of the geometry. We will soon see that, within GR, BHs can rotate and be electromagnetically charged. Thus in fact BHs have a finite number of hairs, although the term 'no-hair' is still loosely applied to these solutions as well. In general, as we show below, if one characterizes the geometry through its multipole moments, BHs have only a finite and small number of independent quantities that suffice to completely describe its multipolar structure.

In fact, a charged and rotating BH solution was discovered many decades after Schwarzschild's work, and is now known as the Kerr–Newman BH, carrying mass, electric charge and angular momentum [4, 5]. In addition, in a series of uniqueness theorems (see [6–10] for reviews), it was established,

Theorem 1. An isolated, stationary and regular BH in Einstein–Maxwell theory is described by the Kerr–Newman family.

In other words, the structure of asymptotically flat, stationary BHs of Einstein–Maxwell theory is completely determined by its global charges defined at infinity, in particular its mass, angular momentum and electric charge. Astrophysical BHs are thought to be neutral to a very good approximation, because of quantum discharge effects, electron–positron pair production and charge neutralization by astrophysical plasmas [11]⁶ . Because of this, we will focus almost entirely on electrically neutral geometries. The neutral version of the Kerr–Newman solution is simply described by the two-parameter Kerr metric [4], which in standard Boyer–Lindquist coordinates reads

$$\begin{eqnarray}{{\rm{d}}{s}}^{2} & = & -{c}^{2}\left(1-\displaystyle \frac{2{GMr}}{{c}^{2}{\rm{\Sigma }}}\right){{\rm{d}}{t}}^{2}+\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}{{\rm{d}}{r}}^{2}+{\rm{\Sigma }}{\rm{d}}{\theta }^{2}-\displaystyle \frac{4{GJr}{{\rm{\sin }}}^{2}\theta }{{c}^{2}{\rm{\Sigma }}}{\rm{d}}{t}{\rm{d}}\phi \\ & & +\left[{r}^{2}+\displaystyle \frac{{J}^{2}}{{M}^{2}{c}^{2}}+\displaystyle \frac{2{{GJ}}^{2}r{{\rm{\sin }}}^{2}\theta }{{{Mc}}^{4}{\rm{\Sigma }}}\right]{{\rm{\sin }}}^{2}\theta {\rm{d}}{\phi }^{2},\end{eqnarray} \tag{ 1.4 }$$

where ${\rm{\Sigma }}\equiv {r}^{2}+\tfrac{{J}^{2}}{{M}^{2}{c}^{2}}{\cos }^{2}\theta ,{\rm{\Delta }}={r}^{2}-\tfrac{2{GMr}}{{c}^{2}}+\tfrac{{J}^{2}}{{M}^{2}{c}^{2}}$.

Note that theorem 1 does not require axi-symmetry, it being a consequence of stationarity. Note also that theorem 1 as well as all results discussed in the text, makes further analiticity assumptions, which we do not discuss here (but see [13, 14]). Due to these assumptions (which are required to infer axi-symmetry from stationarity), there are also those who advocate that the no-hair theorems are less general than commonly stated (see [15] and references therein).

From now on, we will use geometric units with G = c = 1, and express the Kerr solution as above with ${\rm{\Sigma }}\equiv {r}^{2}+{a}^{2}{\cos }^{2}\theta ,{\rm{\Delta }}={r}^{2}-2{Mr}+{a}^{2}$. This metric describes the gravitational field of a spinning BH of mass M and angular momentum $J={aM}$. The event horizon is located at ${r}_{+}=M+\sqrt{{M}^{2}-{a}^{2}}$, and the BH spin is bounded from above by $| a| \leqslant M$.

This uniqueness result is also an example of a no-hair theorem, and the horizon plays a crucial role in exclusion of 'hair', as it prevents information traveling at finite speeds (other then that conveyed by conserved charges) to cross the horizon. This expectation, originally conjectured by Wheeler, was put on firmer ground in a series of works. For example, Hartle [16] and Teitelboim [17–19] showed that it is impossible to measure baryon or lepton numbers of BHs. In flat space there is a long-range $1/{r}^{5}$ potential between two collections of matter arising from the exchange of neutrino pairs between them. But once one of the neutrinos crosses the horizon, a weak-interaction force of this kind ceases to exist. A Kerr BH has no exterior neutrino field with classical effects.

In fact, two strong results emerged regarding BHs in the presence of fundamental fields. The first concerns the existence of stationary BHs surrounded by stationary matter, in the form of fundamental fields without self-interactions⁷ .

Theorem 2. An isolated, stationary and regular BH in the Einstein–Klein–Gordon or Einstein–Proca⁸ theory with a time-independent boson is described by the Kerr family [21–24].

The fields can be also massless and complex, but the theorem requires them to inherit the spacetime symmetries, i.e., no time-dependence. Physically, time-dependent fields will scatter to infinity and/or enter the horizon and therefore break the assumed stationarity. A non-trivial time-dependence can be excluded for real scalars⁹ .

Theorem 3. An isolated, stationary and regular BH in the Einstein–Klein–Gordon theory with one real scalar is described by the Kerr family [21–24].

Real scalars with a nontrivial time dependence will give rise to a nontrivial quadrupole moment, therefore emitting GWs, again breaking the stationarity assumption. Notice that complex scalars are able to avoid this result, by producing a time-independent stress–energy tensor. At the same time, they are also able to avoid being absorbed at the horizon through a superradiant mechanism. We will see in section 3.2 that in fact complex fields are able to produce hairy BHs.

We should mention that there are also some 'no-hair' results for dynamical BH systems, and that these may even include evolving BH binaries. We refer the reader to [26] for a discussion of the general situation, but we will not discuss these examples any further here.

In summary, the 'no-hair' theorem(s) are a set of proofs that—under some conditions—Kerr–Newman is the only possible asymptotically flat and regular solution of the field equations in the presence of fundamental fields (see [8, 27–30] for further details).

However, there is no proof that Kerr–Newman is the most generic solution of the field equations (see below in section 3 for the reasons why such proof cannot exist). The 'no-hair' or 'Kerr' hypothesis states simply that the Kerr geometry—which depends on only two parameters—describes any BH in the Universe (with the exception of those involved in highly-dynamical phenomena). This is no modest proposal!

One thus is led to two questions:

• (a)  
  If one sets up some arbitrary initial conditions how is the final, Kerr state approached?
• (b)  
  Is Kerr inevitably the final state?
• (c)  
  How can one test for (a) or (b)?

We will try to answer these questions below. The no-hair hypothesis is pivotal to interpret observations of massive astrophysical bodies. The only other compact object that is agreed to populate the Universe are neutron stars. But neutron stars in GR cannot be more massive than $\sim 3{M}_{\odot }$ [31], as not even degeneracy pressure can be sustained for stars more massive than this limit. Thus, within the framework of GR with a standard matter sector, and of the no-hair hypothesis, any compact object¹⁰ with mass larger than $\sim 3{M}_{\odot }$ is a Kerr BH. Conversely, any observation of a compact object with mass larger than $\sim 3{M}_{\odot }$ and with metric different from the Kerr geometry would inevitably signal a departure from standard physics (either in the gravitational or in the matter sector). Therefore tests of strong-field gravity targeting BH systems aim at verifying the 'Kerr hypothesis' in various ways.

One intuitively expects that stationary—or at least very long-lived—hairy BHs are possible if one encloses a small spherical BH at the center of a large spherical 'wall', with pressure just enough to keep the wall static. In other words, we do not expect anything drastic to occur if a, say, $1\,\mathrm{Kg}$ or ${10}^{-24}\,\mathrm{mm}$ BH is placed inside a big room.

This expectation stands up to scrutiny. Analytical solutions describing an infinitely-thin, spherically symmetric shell surrounding a static BH were constructed (and shown to be stable in some regions of parameter space [44, 45]). They are described by the geometry

$$\begin{eqnarray}{{\rm{d}}{s}}^{2} & = & -\alpha \left(1-\displaystyle \frac{2{M}_{-}}{r}\right){{\rm{d}}{t}}^{2}+{\left(1-\displaystyle \frac{2{M}_{-}}{r}\right)}^{-1}{{\rm{d}}{r}}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2},\quad r\lt R,\\ {{\rm{d}}{s}}^{2} & = & -\left(1-\displaystyle \frac{2{M}_{+}}{r}\right){{\rm{d}}{t}}^{2}+{\left(1-\displaystyle \frac{2{M}_{+}}{r}\right)}^{-1}{{\rm{d}}{r}}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2},\quad r\gt R,\end{eqnarray} \tag{ 3.1 }$$

with $\alpha =\tfrac{1-2{M}_{+}/R}{1-2{M}_{-}/R}$ and where R stands for the radius of the shell, which is described by an energy surface density σ and pressure P, related through

$$\begin{eqnarray}&&{k}_{-}-{k}_{+}=4\pi R\sigma ,\qquad \displaystyle \frac{{M}_{+}}{{k}_{+}}-\displaystyle \frac{{M}_{-}}{{k}_{-}}=4\pi {R}^{2}(\sigma +2P),\end{eqnarray} \tag{ 3.2 }$$

with ${k}_{\pm }={(1-2{M}_{\pm }/R)}^{1/2}$. A finite-thickness shell version was studied recently [46, 47].

The previous solution is somewhat artificial, in that the shell of matter is infinitely thin, and supported outside the horizon. However, even reasonable 'short-hair' ('short' because it can be localized arbitrarily close to the horizon) solutions are possible, and can be found in closed analytical form. An example of a BH solution surrounded by an anisotropic fluid is described by [48]

$$\begin{eqnarray*}\begin{array}{rcl}{{\rm{d}}{s}}^{2} & = & -{{f}{\rm{d}}{t}}^{2}+\displaystyle \frac{{{\rm{d}}{r}}^{2}}{f}+{r}^{2}{\rm{d}}{\theta }^{2}+{r}^{2}{{\rm{\sin }}}^{2}\theta \,{\rm{d}}{\phi }^{2}\,,\\ f & = & 1-\displaystyle \frac{2M}{r}+\displaystyle \frac{{Q}_{m}^{2k}}{{r}^{2k}},\\ \rho & = & \displaystyle \frac{{Q}_{m}^{2k}(2k-1)}{8\pi {r}^{2k+2}},\quad P=k\rho ,\end{array}\end{eqnarray*}$$

where ρ and P are the density and (angular) pressure, respectively, of the anisotropic fluid, and Q_m is a constant, describing the 'matter-hair'. The stress–energy tensor of the fluid is given by

$$\begin{eqnarray}&&{T}_{\mu \nu }=\rho {U}_{\mu }{U}_{\nu }+P{\sigma }_{{ab}}-\rho {u}_{\mu }{u}_{\nu },\end{eqnarray} \tag{ 3.3 }$$

where U_μ is the fluid four-velocity, u_μ is a unit radial vector such that ${u}^{\mu }{U}_{\mu }=0$ and ${\sigma }_{\mu \nu }={g}_{\mu \nu }+{U}_{\mu }{U}_{\nu }-{u}_{\mu }{u}_{\nu }$. The solution above corresponds to an anisotropic stress–tensor, specified by (3.2) in [48], and it reduces to the Reissner–Nordstrom BH when k = 1.

It seems unlikely that these hairy solutions form in practice, in any kind of realistic environment. However, hairy solutions that have been observed in numerical simulations are compact torii around the final BH formed as the endstate of the coalescence and merger of two neutron stars [49]. These torii can have masses as large as 10% of the total mass, and seem to be stable on dynamical timescales.

The above solutions refer to fluids, which are themselves effective descriptions of fermions, and have not—with the exception of torii—been observed to form in realistic scenarios. We will now discuss a more natural framework for hairy BHs, in the context of fundamental fields. We will not try to make a systematic classification of solutions in this setup, and instead refer the reader to some of the outstanding reviews out there [8, 27–30, 50]. We would like to stress however, that all solutions need massive or self-interacting fields, in line with the fluid counterpart discussed previously.

Conceptually, the discovery of BHs with 'color' (static BH solutions in Einstein–Yang–Mills (EYM) theory [51, 52], that require for their complete specification an additional parameter—besides the mass—not associated with any conserved charge), was the first example showing that the no-hair hypothesis needs revision. The theory is EYM with SU(2) gauge group, described by the action

$$\begin{eqnarray}&&S=\int {{\rm{d}}}^{4}x\sqrt{-g}(R/16\pi G-{F}_{\mu \nu }^{a}{F}^{a\mu \nu }/{g}^{2}),\end{eqnarray} \tag{ 3.4 }$$

where ${F}_{\mu \nu }^{a}$ is the Yang–Mills field strength and $1/{g}^{2}$ the coupling constant. To see how new solutions are possible, let's go back to the perturbative framework of section 1.1 and 'freeze' gravity by sending $1/{g}^{2}\to 0$. Then, gravity effectively decouples from the YM field, and any vacuum solution solves the Einstein equations. Let us take again a Schwarzschild background. The YM field can be expressed in terms of a field ${\rm{\Psi }}(t,r)=\psi (r){{\rm{e}}}^{-{\rm{i}}\omega t}$. The radial wavefunction $\psi (r)$ satisfies the equation [53, 54]

$$\begin{eqnarray}&&f(f\psi ^{\prime} )^{\prime} +\left({\omega }^{2}+f\displaystyle \frac{1-{\psi }^{2}}{{r}^{2}}\right)\psi =0,\end{eqnarray} \tag{ 3.5 }$$

with $f=1-2M/r$. Note that trivial static solutions to this equation are $\psi =0,1$. Equation (3.5) is stable on the $\psi =1$ branch: its linearized (around $\psi =1$) version is identical to that of l = 1 electromagnetic modes, and a fundamental frequency $M\omega =0.248\,263-i0.092\,4877$ satisfies the necessary boundary conditions [54]. On the other hand, linearization around ${\rm{\Psi }}=0$ yields unstable solutions, as is easy to prove. We find the unstable mode $M\omega =i0.123\,2877$. This indicates that there is a nontrivial static ($\omega =0$) solution of equation (3.5). Such solution can be found imposing the appropriate asymptotic behavior at the horizon and demanding that $\psi (\infty )=1$.

The violation of the no-hair hypothesis was confirmed with the discovery of static BH solutions in theories like Einstein–Skyrme [55–57], and Einstein-non abelian-Proca [58, 59].

As Bekenstein pointed out, these hairy solutions are possible in a way that parallels the existence of charged BHs: the gauge invariance of electrodynamics causes the Coulomb potential to propagate instantaneously in an appropriate gauge. Thus, the argument that information about hair cannot leave the horizon because it would need to travel faster than light, does not apply to the Coulomb potential, and charged BHs exist. Likewise, it seems intuitive that gauge invariance of non-abelian gauge theories should allow one or more of the gauge field components generated by sources in a BH to 'escape' from it [6].

It is possible that a different mechanism might act to allow hairy BHs to exist: instead of allowing the hair to leave the horizon, one could in principle prevent it from falling in, in the first place! Superradiance in BH physics allows for precisely this, amplifying low-frequency bosonic waves, at the expense of the 'horizon's rotational energy' [60]. For massive fields, a finite-height barrier exists at large distances, confining the field and triggering an instability. Thus Kerr BHs are unstable against massive bosonic perturbations [60–64]. To be specific, the theory of a minimally coupled scalar with a mass term ${\mu }_{S}$

$$\begin{eqnarray}&&S=\int {{\rm{d}}}^{4}x\sqrt{-g}\left(\displaystyle \frac{R}{\kappa }-\displaystyle \frac{1}{2}{g}^{\mu \nu }{\bar{{\rm{\Psi }}}}_{,\mu }^{}{{\rm{\Psi }}}_{,\nu }^{}-\displaystyle \frac{{\mu }_{S}^{2}\bar{{\rm{\Psi }}}{\rm{\Psi }}}{2}\,\right).\end{eqnarray} \tag{ 3.6 }$$

admits Kerr BHs solutions which are linearly unstable against fluctuations of the scalar Ψ. The nonlinear development of the instability is not yet known [20, 65, 66], but linearized evolutions show that sufficiently small dimensionless coupling $M{\mu }_{S}$ leads to configurations that can live longer than a Hubble time around a Kerr BH. An example of the evolution of a Gaussian wavepacket around a Kerr BH is shown in figure 4. Notice how differently massless and massive fields behave (compare with figure 3). As such, even minimally coupled massive scalars produce what for all purposes are hairy solutions.

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It seems intuitive that for real fields, the instability acts in such a way as to reduce the angular momentum off the BH, producing a slowly-spinning Kerr BH [67]. This is also a consequence of theorem 3 in section 1.2. Complex fields on the other hand, may give rise to a time-independent stress–energy tensor and therefore avoid energy loss, and both theorems 1 and 2. It is thus not surprising that new solutions might branch off Kerr. These solutions were described in [29, 68] (and previously, at the linear level, in [69]) and are the perfect (because of their simplicity) example of how no-hair theorems can be circumvented. These 'hairy' solutions can be generalized to complex vectors [70] (for which the Kerr geometry is also linearly unstable [20, 60, 71, 72]) and also to self-interacting scalars [73, 74].

Models of minicharged dark matter predict the existence of new fermions which possess a fractional electric charge or are charged under a hidden U(1) symmetry [75–80]. Their corresponding charge is naturally much smaller than the electron charge and their coupling to the Maxwell sector is suppressed. The following classical Lagrangian captures these theories [77]

$$\begin{eqnarray}&&{ \mathcal L }=\sqrt{-g}\left(\displaystyle \frac{R}{16\pi }-\displaystyle \frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\displaystyle \frac{1}{4}{B}_{\mu \nu }{B}^{\mu \nu }+4\pi {{ej}}_{\mathrm{em}}^{\mu }{A}_{\mu }+4\pi {e}_{h}{j}_{h}^{\mu }{B}_{\mu }+4\pi \epsilon {{ej}}_{h}^{\mu }{A}_{\mu }\right),\end{eqnarray} \tag{ 3.7 }$$

where ${F}_{\mu \nu }:= {\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ and ${B}_{\mu \nu }:= {\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }$ are the field strengths of the ordinary photon and of the dark photon, respectively, ${j}_{\mathrm{em}}^{\mu }$ and ${j}_{h}^{\mu }$ are the electromagnetic and the hidden number currents, e is the electron charge, and e_h is the gauge coupling of the hidden sector. The model (3.7) describes a theory in which a charged fermion is coupled to ordinary photons with coupling ${\epsilon }^{2}{e}^{2}$ and to dark photons with coupling ${e}_{h}^{2}:= {\epsilon }_{h}^{2}{e}^{2}$. The parameters and ${\epsilon }_{h}$ are free. In this theory, BHs are described by the Kerr–Newman family of geometries, but now the classical and quantum discharge mechanisms can be suppressed. Thus, BHs can acquire electric 'hair' [12].

Finally, another possible mechanism for hair growth, consists on dropping the stationarity assumption, and use time-dependent boundary conditions. Under certain conditions, this can lead to non-trivial BH geometries [26, 81, 82].

In addition to simple extensions of GR, there are a number of proposed modifications of GR that actually change the gravity sector. The 'zoo' is too big to describe here, we refer the reader to recent reviews on the topic [28, 83]. Some modifications are string-theory motivated and are also expected to arise in a completion of GR, and in this sense it is not too surprising to find theories which include higher-order-in-curvature terms, like Chern–Simons gravity [84] or Einstein-dilaton Gauss–Bonnet (EDGB) gravity [85, 86]. These theories give rise to hairy BH solutions [85–89]. We should also point out that even scalar–tensor theories can give rise to BHs surrounded by scalar fields, although the scalar fields needs nontrivial matter content to be anchored on [90, 91].

Other modifications of GR are motivated by a search for massive gravitons, a quest related to solutions of the cosmological constant problem, but also to nonlinear completions of GR. BHs in some of these theories also circumvent uniqueness and no-hair results and may be surrounded by massive-graviton hair [30, 92, 93]. Hairy BH solutions also appear in Lorentz-violating gravity theories. BH solutions naturally have hair in these theories, because the preferred foliation can be described in terms of a scalar field (see e.g. [94] and references therein).

Recent studies of quantum effects in BH geometries argue that BHs should be surrounded by 'soft' hair at quantum level [95], while some advocate that hair is a rule rather than the exception [96].

To conclude, BHs acquire hair in a variety of setups.

The no-hair hypothesis in astrophysical settings is tangled with the assumption that the compact object is too massive and compact to be anything else than a BH. Theories as simple as gravitating massive scalars or vectors give rise to objects—boson stars and oscillatons—which could mimic BHs: they can be very massive, they are dark because the interaction cross-section with normal matter is small, and they can be very compact [97–102]. It turns out that even the optical appearance of boson stars can be similar to that of BHs [103]. Other proposals for compact massive objects include gravastars [104], superspinars [105], wormholes [106] and even mixed wormhole-star systems [107]. Superspinars invoke the existence of unknown quantum effects that allow the existence of Kerr geometries without classical horizons.

Very compact objects with a hard and thermally emitting surface, such as gravastars, can be strongly constrained by observations [108, 109]. However, these constraints assume that the radiation channel is all in some standard model particles and is unlikely to strongly constrain boson stars, for example.

One of the most natural ways to test the spacetime metric of a compact object is to study—through astrophysical or GW observations—the motion of stellar objects in its surroundings. We here discuss the multipole expansions framework, which is probably the most appropriate to describe and perform these (non-dynamical) tests of the 'no-hair' hypothesis. Multipole expansions have been first introduced in Newtonian mechanics, to describe the gravitational (or electrostatic) potential generated by a distribution of masses (or charges) in terms of a set of scalar quantities, the multipoles (see e.g., [124] and the first chapter of [125], and references therein). They have then been extended to GR [126–129].

Multipole expansions are a powerful tool to extract physical content from a (gravitational or electrostatic) potential, or from a spacetime metric. Indeed, the multipoles capture all the properties of the potential or of the metric, and in many cases the observable quantities can be expressed in terms of a multipolar decomposition. The measurement of the different multipole moments, through electromagnetic and GW observations, would allow the mapping of a BH spacetime.

In the following, we shall briefly introduce multipolar expansions in Newtonian gravity and in GR; further details are discussed in the appendix.

5.1.1. Multipolar expansions in Newtonian gravity and in general relativity

In Newtonian gravity, the gravitational potential ${\rm{\Phi }}(t,\vec{x})$ in the exterior of a body with mass density $\rho (t,\vec{x})$ is the solution—in vacuum—of Poisson's equation ${{\rm{\nabla }}}^{2}{\rm{\Phi }}=4\pi G\rho$. This solution can be written as a series expansion in $1/r$ (where $r=| \vec{x}|$), called multipolar expansion of the potential, which, in the case of a stationary, axisymmetric body, is

$$\begin{eqnarray}&&{\rm{\Phi }}(\vec{x})=-G\displaystyle \sum _{l=0}^{\infty }\displaystyle \frac{{M}_{l}}{{r}^{l+1}}{P}_{l}({\rm{\cos }}\theta ),\end{eqnarray} \tag{ 5.1 }$$

where ${P}_{l}({\rm{\cos }}\theta )$ are the usual Legendre polynomials, and

$$\begin{eqnarray}&&{M}_{l}=\int \rho (\vec{x}){r}^{l}{P}_{l}({\rm{\cos }}\theta ){{\rm{d}}}^{3}x\end{eqnarray} \tag{ 5.2 }$$

are the multipole moments of the body. The first moments are well known: ${M}_{0}=M$ is the body's mass; M₁ is its dipole moment, vanishing in the center-of-mass frame; ${M}_{2}=Q$ is the quadrupole moment, M₃ is the octupole moment, and so on. Introducing dimensionless multipole moments ${J}_{l}=-{M}_{l}/({{MR}}^{l})$, where R is a characteristic length of the body (in the case of the Earth, R is the equatorial radius)

$$\begin{eqnarray}&&{\rm{\Phi }}=-G\left[\displaystyle \frac{M}{r}-{J}_{2}\displaystyle \frac{{{MR}}^{2}}{{r}^{3}}{P}_{2}(\cos \theta )+...\right].\end{eqnarray} \tag{ 5.3 }$$

The multipole moments of an astronomical body are a measure of its departure from spherical symmetry. Generally, planets and stars rotate so slowly that their deviation from spherical symmetry is tiny (other sources of asymmetry are even smaller). For instance, for the Earth ${J}_{2}=1.083\times {10}^{-3}$, and the $l\gt 2$ dimensionless multipoles are at least a thousand times smaller. These quantities have been determined by studying the motion of satellites orbiting around the Earth [130–132]. A depiction of the gravitational field of the Earth as reconstructed with data from these mission is shown in figure 5, where differences in the gravitational field across the globe are represented by elevation and color.

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The generalization of multipolar expansions to GR is not straightforward, due to the nonlinearity of Einstein's equations. The relativistic multipole moments of a stationary, asymptotically flat spacetime have been defined by Geroch and Hansen [126, 127], with a complex mathematical construction that allows to describe the deviation of the asymptotic geometry from flatness in terms of two sets of tensorial quantities evaluated at the point at infinity: the mass multipole moments and the current multipole moments (see the appendix). In the Newtonian limit, the mass multipole moments reduce to the moments in Newtonian theory¹⁴ . It has been shown that the spacetime is uniquely determined by its multipole moments [138–140]: in other words, they completely characterize the spacetime geometry outside any stationary body. It has also been shown (in the axisymmetric case) that it is possible to reconstruct the full spacetime from any 'well-behaved' set of relativistic multipole moments [141, 142].

As in the Newtonian case, when the body (and the spacetime) is axisymmetric, the relativistic (mass and current) multipoles reduce to a set of scalar quantities $({M}_{l},{J}_{l})$ [127, 129]; if the spacetime is reflection-symmetric (i.e., symmetric with respect to a reflection on the equatorial plane), the odd mass moments and the even current moments identically vanish: ${M}_{2l+1}={S}_{2l}=0$. As in the Newtonian case, ${M}_{0}=M$ is the mass, ${M}_{2}=Q$ is the quadrupole moment; moreover, ${S}_{1}=J$ is the angular momentum. For the Kerr spacetime

$$\begin{eqnarray}{M}_{2l} & = & {(-1)}^{l}{{Ma}}^{2l},\\ {S}_{2l+1} & = & {(-1)}^{l}{{Ma}}^{2l+1}.\end{eqnarray} \tag{ 5.4 }$$

Notice that this expression is a manifestation of ('no-hair') theorem 1 (section 1.2), as it fixes all multipole moments as function of two parameters only.

An alternative definition of relativistic multipole moments of stationary, asymptotically flat spacetimes has been introduced by Thorne [128]. This is an extension of the standard procedure of extracting the mass and the angular momentum from the far-field limit of the spacetime metric [143]. In Thorne's construction, all multipole moments can be read out from the asymptotic spacetime metric. In the axisymmetric case

$$\begin{eqnarray}{g}_{00} & = & -1+\displaystyle \frac{2M}{r}+\displaystyle \sum _{l\geqslant 2}\displaystyle \frac{2}{{r}^{l+1}}[{M}_{l}{P}_{l}(\cos \theta )+(l^{\prime} \lt l\,\mathrm{harmonics})]\\ & = & -1+\displaystyle \frac{2M}{r}+\displaystyle \frac{2Q}{{r}^{3}}{P}_{2}(\cos \theta )\,+\,...,\\ {g}_{0\phi } & = & -2{\sin }^{2}\theta \displaystyle \sum _{l\geqslant 1}\displaystyle \frac{1}{{r}^{l}}\displaystyle \frac{{S}_{l}}{l}[P{^{\prime} }_{l}(\cos \theta )+(l^{\prime} \lt l\,\mathrm{harmonics})]\\ & = & -{\sin }^{2}\theta \left(\displaystyle \frac{2J}{r}+\displaystyle \frac{2{S}_{3}}{3{r}^{3}}P{^{\prime} }_{3}(\cos \theta )+...\right).\end{eqnarray} \tag{ 5.5 }$$

This definition requires that the coordinate system belongs to a special class, the 'asymptotically Cartesian and mass centered' (ACMC) coordinates, which means that it becomes Minkowski with sufficient rapidity at large radii (see the appendix for details), and that the origin of the space coordinates lies at the center of mass of the source. As long as the coordinates are ACMC, the multipole moments $({M}_{l},{S}_{l})$ are coordinate-independent. Moreover, it has been shown that the definitions of relativistic multipole moments given by Geroch–Hansen and by Thorne are actually coincident [144]¹⁵ .

In the case of a weak-field source, the multipole moments of a stationary spacetime can also be expressed as integrals over the source of the energy and momentum densities [145], i.e. (in the axisymmetric case) by equation (5.2) for mass multipoles, and a similar expression for current multipoles [146, 147]. This definition can be extended, using properly defined 'effective' (energy and momentum) densities, to the case of strong-field sources, as long as the source can be covered by a so-called 'de Donder' coordinate frame [128]; this is the case, for instance, of compact stars. Multipole moments of BHs, instead, can only be defined in terms of their asymptotic geometry.

Given a spacetime metric expressed in a specific coordinate frame, there is a simple method to compute the Geroch–Hansen multipole moments of the spacetime. This procedure is based on Ryan's formula [148], which was introduced as a phenomenological tool to express observable quantites in terms of multipole moments (and will be discussed in more detail in section 5.3), but is also a powerful computational tool. One computes the energy of (geodesic) circular orbits as a function of the orbital frequency $E(\nu )$. A particle moving in that geodesic would emit GWs at frequency $f=2\nu$, and the so-called 'gravitational spectrum', i.e. the amount of GW energy emitted per logarithmic interval of frequency, is ${\rm{\Delta }}E(f)={{f}{\rm{d}}{E}}_{{gw}}/{\rm{d}}{f}=-\nu {\rm{d}}{E}/{\rm{d}}\nu$. Ryan has found a general, explicit expression of ${\rm{\Delta }}E(f)$ in terms of the multipole moments of the spacetime (equation (5.19) in section 5.3); comparing the function ${\rm{\Delta }}E(f)$ with this expression, one can extract the entire set of (Geroch–Hansen) multipole moments. This approach can be useful to express the asymptotic expansion of the metric—which in general in not expressed in an ACMC coordinate frame—in terms of the gauge-invariant multipole moments, for instance when the metric is the result of a numerical integration [149–152], or of a perturbative computation [153, 154]. When the metric is known in ACMC coordinates, instead, the multipoles can be extracted by comparison with Thorne's expansion (5.5), as e.g. in [155].

5.1.2. The multipole moments of some astrophysical objects

To conclude this discussion, we show in figure 6 typical values of the normalized quadrupole moment ${J}_{2}=-Q/({{MR}}^{2})$, as a function of the compactness M/R, for different astrophysical objects: the Earth and the Sun [160]; a neutron star; the final BH observed in the first LIGO detection [1], having $a=0.67M$ and $R=1.74M;$ a near-extremal BH with $a=0.98M$ (and thus $R=1.20M$). This figure shows that the normalized quadrupole moment of astrophysical objects comes in a variety of ranges, and that therefore the BH multipole moments are not accidentally shared by many other objects.

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However, we should also add that (exotic) ultra-compact objects can have multipole moments arbitrarily close to that of BHs. For example, in the ultra-compact limit, the mass quadrupole moment of gravastars is [161]

$$\begin{eqnarray}&&Q=\displaystyle \frac{{J}^{2}}{M}\left(1+\displaystyle \frac{8}{45\mathrm{log}(1-2M/R)}\right),\end{eqnarray} \tag{ 5.6 }$$

showing that it can be arbitrarily close to that of a BH with the same mass, i.e., $Q={J}^{2}/M$. This somewhat contrived example shows that the law (5.4) is not, by itself, evidence for the existence of a horizon, but the alternatives usually invoke somewhat more exotic physics.

5.2.1. Using two or more modes to test the Kerr hypothesis

Tests of the no-hair hypothesis based on the motion and observation of stars at large distances probe the background geometry in the weak-field regime. A multipolar decomposition is meaningful here, because the contribution of higher multipoles to the motion of stars is suppressed. Nevertheless, such tests are unable to probe different theories with the same background solution. To probe the dynamical content of the field equations, dynamical tests are necessary, and GWs are the ideal tool for this.

As we discussed in section 2, the vibration modes of BHs in GR are completely determined by the two parameters specifying the Kerr solutions. In any other theory with the same Kerr background, the functional relation among modes would be different. Thus, conceptually one can proceed as follows: measure the dominant mode of oscillation, which is characterized by a ringing frequency ω and a damping time τ. In GR, the dominant mode is invariably (when the oscillations are the result of a BH binary coalescence, which is likely to be the most efficient way to excite the QNMs) that with $l=m=2$, n = 0 mode, of frequency ${\omega }_{220}$ and damping time ${\tau }_{220}$. In the Schwarzschild limit these are the numbers shown in equation (2.3). These numbers allow one to determine the (redshifted) mass and angular momentum of the BH. Now measure the most important subdominant mode (typically $l=m=3,n=0$) and check that it is located at the GR prediction. One thus has at hand a null test of GR based on the observation of ringdown modes of BHs.

In practice, there are measurement errors associated with the different sources of noise in the detector, and each of the stages in the test carries associated uncertainties. These uncertainties are best described in units of the 'signal-to-noise' ratio (SNR) ρ [162, 163]. As a rule-of-thumb, a total $\rho \gtrsim 8$ is required for a detection. In the following we focus on the SNR for the ringdown part of the signal only. Fortunately, accurate measurements of the mass and angular momentum of BHs are feasible [39, 164]: for detection of the fundamental $l=m=2\,\mathrm{bar}$ mode, for example, figure 7 shows the estimated error (multiplied by ρ) in measuring the mass M, angular momentum parameter $j\equiv a/M$, QNM amplitude A_lmn, and phase ${\phi }_{{lmn}}$ (see equation (2.1) for definitions of these quantities and [39] for further details). At large ρ, the uncertainties are well approximated by [39, 165]

$$\begin{eqnarray}&&{\sigma }_{j}=\displaystyle \frac{2}{\rho }\left|\displaystyle \frac{{Q}_{{lmn}}}{{Q}_{{lmn}}^{\prime }}\right|,\qquad {\sigma }_{M}=\displaystyle \frac{2}{\rho }\left|\displaystyle \frac{{{MQ}}_{{lmn}}{f}_{{lmn}}^{\prime }}{{f}_{{lmn}}{Q}_{{lmn}}^{\prime }}\right|,\end{eqnarray} \tag{ 5.7 }$$

where primes stand for derivatives with respect to the dimensionless angular momentum j, and can be performed with the help of the fits in equations (2.4) and (2.5) , or taken from the numerical data [2, 38, 39]. Even for marginal detection events, the accuracy of mass and spin measurements is at the level of 10% or better.

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For large SNRs, one enters the regime where a second mode can be disentangled in the signal. To understand what criteria needs to be met, consider first disentangling frequencies. The error associated with the direct measurement of the frequency and damping time of one mode ('1') in the signal is [39, 165]

$$\begin{eqnarray}&&\rho {\sigma }_{{f}_{1}}=\displaystyle \frac{1}{2\sqrt{2}}{\left\{\displaystyle \frac{{f}_{1}^{3}(3+16{Q}_{1}^{4})}{{{ \mathcal A }}_{1}^{2}{Q}_{1}^{7}}\left[\displaystyle \frac{{{ \mathcal A }}_{1}^{2}{Q}_{1}^{3}}{{f}_{1}(1+4{Q}_{1}^{2})}+\displaystyle \frac{{{ \mathcal A }}_{2}^{2}{Q}_{2}^{3}}{{f}_{2}(1+4{Q}_{2}^{2})}\right]\right\}}^{1/2},\end{eqnarray} \tag{ 5.8 }$$

$$\begin{eqnarray}&&\rho {\sigma }_{{\tau }_{1}}=\displaystyle \frac{2}{\pi }{\left\{\displaystyle \frac{(3+4{Q}_{1}^{2})}{{{ \mathcal A }}_{1}^{2}{f}_{1}{Q}_{1}}\left[\displaystyle \frac{{{ \mathcal A }}_{1}^{2}{Q}_{1}^{3}}{{f}_{1}(1+4{Q}_{1}^{2})}+\displaystyle \frac{{{ \mathcal A }}_{2}^{2}{Q}_{2}^{3}}{{f}_{2}(1+4{Q}_{2}^{2})}\right]\right\}}^{1/2}.\end{eqnarray} \tag{ 5.9 }$$

These errors refer to mode '1' in a pair. By considering the 'symmetric' case ${\phi }_{1}={\phi }_{2}=0$, the errors on f₂ and ${\tau }_{2}$ are simply obtained by exchanging indices ($1\leftrightarrow 2$).

A natural criterion (á la Rayleigh) to resolve frequencies and damping times is

$$\begin{eqnarray}&&| {f}_{1}-{f}_{2}| \gt \max ({\sigma }_{{f}_{1}},{\sigma }_{{f}_{2}}),\qquad | {\tau }_{1}-{\tau }_{2}| \gt \max ({\sigma }_{{\tau }_{1}},{\sigma }_{{\tau }_{2}}).\end{eqnarray} \tag{ 5.10 }$$

In interferometry this would mean that two objects are (barely) resolvable if 'the maximum of the diffraction pattern of object 1 is located at the minimum of the diffraction pattern of object 2'. We can introduce two 'critical' SNRs required to resolve frequencies and damping times

$$\begin{eqnarray}&&{\rho }_{\mathrm{crit}}^{f}=\displaystyle \frac{\max (\rho {\sigma }_{{f}_{1}},\rho {\sigma }_{{f}_{2}})}{| {f}_{1}-{f}_{2}| },\qquad {\rho }_{\mathrm{crit}}^{\tau }=\displaystyle \frac{\max (\rho {\sigma }_{{\tau }_{1}},\rho {\sigma }_{{\tau }_{2}})}{| {\tau }_{1}-{\tau }_{2}| },\end{eqnarray} \tag{ 5.11 }$$

and recast our resolvability conditions as

$$\begin{eqnarray}&&\rho \gt {\rho }_{\mathrm{crit}}=\min ({\rho }_{\mathrm{crit}}^{f},{\rho }_{\mathrm{crit}}^{\tau }),\qquad \rho \gt {\rho }_{\mathrm{both}}=\max ({\rho }_{\mathrm{crit}}^{f},{\rho }_{\mathrm{crit}}^{\tau }).\end{eqnarray} \tag{ 5.12 }$$

The first condition implies resolvability of either the frequency or the damping time, the second implies resolvability of both.

A related question is the magnitude of the SNR in order to detect a multi-mode signal, and to resolve two signals of different amplitudes. This problem was studied in [165], and quantified by deriving a critical SNR for amplitude resolvability ${\rho }_{\mathrm{GLRT}}$ based on the generalized likelihood ratio test. This criteria is almost (but not quite) equivalent to requiring that the second, sub-dominant mode alone has enough energy that it could be detected on its own.

The different SNRs required to resolve the two modes of a final BH which is the end-product of an unequal-mass merger of two BHs, is shown in figure 8. The figure shows that tests of the no-hair hypothesis require large, but not-too-large-to-be-impossible SNRs.

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The implementation of this method in an actual search pipeline for two planned GW detectors ET and NGO was described in [166]. The authors consider a theory-independent parametrization of the QNM frequencies given by

$$\begin{eqnarray}&&{\omega }_{{lm}}={\omega }_{{lm}}^{(\mathrm{GR})}(1+{\rm{\Delta }}{\hat{\omega }}_{{lm}}),\end{eqnarray} \tag{ 5.13 }$$

$$\begin{eqnarray}&&{\tau }_{{lm}}={\tau }_{{lm}}^{(\mathrm{GR})}(1+{\rm{\Delta }}{\hat{\tau }}_{{lm}}),\end{eqnarray} \tag{ 5.14 }$$

where ${\omega }_{{lm}}^{(\mathrm{GR})}$ are the values of the fundamental QNM frequencies and damping times of Kerr BHs, and ${\rm{\Delta }}{\hat{\omega }}_{{lm}},{\rm{\Delta }}{\hat{\tau }}_{{lm}}$ parametrize the relative deviations to these numbers. The BH is assumed to be the endpoint of a mass ratio q = 2, 10 binary BH merger and its spin is assumed to be $j=0.6,0.26$.

When the signal is described by GR, estimates of mass and spin from, say, three modes will be consistent and yield (generically accurate) measurements of these quantities. This is depicted in figure 9: the three estimates intersect at a common point, which will be the best estimate for the mass and spin of the final BH (the injected signal corresponds to a BH of mass $(500,{10}^{8}){M}_{\odot }$ for NGO and ET respectively, and the best estimate is very close to the injected value).

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By contrast, if the signal is inconsistent with GR the intersections of the confidence regions will not agree, as in figure 10.

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The accuracy with which the deviation constants can be estimated and the mode parameters resolvable is shown in figure 11. For each BH system considered, the width of the 90% confidence intervals for the extracted values of ${\rm{\Delta }}{\hat{\omega }}_{{lm}}$ and ${\rm{\Delta }}{\hat{\tau }}_{{lm}}$ were plotted against luminosity distance, for injections with GR waveforms

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The previous results all assume comparatively large SNRs. This analysis was recently extended to multiple detections with larger noise, using a model selection scheme called TIGER (Test Infrastructure for GEneral Relativity), for the ET telescope [167].

As an example, assume that GR is correct ($\delta {\hat{\omega }}_{22}=\delta {\hat{\omega }}_{33}=\delta {\hat{\tau }}_{22}=0$). How would GW observations of ringdown actually constrain the deviation parameters? Results are summarized in figure 12. The top panels show the probability distribution function (PDF) both for an example single source at ${D}_{{\rm{L}}}=20.69$ Gpc (z = 2.47), and for a catalog of 20 sources. As expected, the single-source PDFs are quite wide and uninformative. For $\delta {\hat{\omega }}_{22}$ and $\delta {\hat{\omega }}_{33}$, with 20 sources the PDFs become strongly peaked, with very little bias. There is a clear advantage in using all available detections.

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If the BH progenitor is known with some accuracy, further tests are possible, even with a single ringdown mode: the inspiral phase allows, in principle, the mass and spin of the two components to be determined. GR then predicts a well-defined final BH with a certain mass and spin, a prediction which can be tested using the dominant ringdown mode [168]. In fact, in some cases it might even be possible to infer the properties of the progenitor from ringdown observations  [169, 170].

These results were recently used, in conjunction with population synthesis models of the formation and evolution of BH binaries, to estimate how many detections would yield tests of GR. The results are optimistic for space-based detectors or future generations of Earth-based ones [171].

5.2.2. Constraining alternative theories

Up to now, we studied how two modes can test GR. We can turn this around and use measurements of two (or more) ringdown modes to constraint specific modified theories, converting the errors (5.8) and (5.9) on the frequency and damping time to errors on physical quantities by using a simple propagation of errors [12]. Specifically, let $M,j=a/M$ be the mass and dimensionless angular momentum of the BH, and Q an extra parameter which presumably enters the description of the geometry. Since Q measures deviations from GR or from the Kerr geometry, we will always take it to be small. Then

$$\begin{eqnarray}&&{\sigma }_{X}=\displaystyle \frac{\partial X}{\partial M}{\sigma }_{M}+\displaystyle \frac{\partial X}{\partial \chi }{\sigma }_{\chi }+\displaystyle \frac{\partial X}{\partial Q}{\sigma }_{Q},\end{eqnarray} \tag{ 5.15 }$$

where $X=({f}_{1},{f}_{2},{\tau }_{1})$. It is straightforward to solve the system of three equations above for ${\sigma }_{M}$, ${\sigma }_{\chi }$ and ${\sigma }_{Q};$ this yields

$$\begin{eqnarray}&&\rho {\sigma }_{M}={F}_{1}({f}_{1},{f}_{2},{\tau }_{1},{{ \mathcal A }}_{2}/{{ \mathcal A }}_{1}),\end{eqnarray} \tag{ 5.16 }$$

$$\begin{eqnarray}&&\rho {\sigma }_{\chi }={F}_{2}({f}_{1},{f}_{2},{\tau }_{1},{{ \mathcal A }}_{2}/{{ \mathcal A }}_{1}),\end{eqnarray} \tag{ 5.17 }$$

$$\begin{eqnarray}&&\rho {\sigma }_{Q}={F}_{3}({f}_{1},{f}_{2},{\tau }_{1},{{ \mathcal A }}_{2}/{{ \mathcal A }}_{1}),\end{eqnarray} \tag{ 5.18 }$$

where F_i are, usually, cumbersome analytical functions. Finally, and because Q is small, we can view ${\sigma }_{Q}$ as an upper bound on the quantity Q itself and use it to estimate the constraint that can be imposed by a ringdown detection with a certain SNR ρ. This strategy was used to study constraints on the electric charge of BHs, resulting in the constraint $\tfrac{| Q| }{M}\lesssim 0.1\sqrt{\tfrac{100}{\rho }}$ [12]. A similar procedure can used to constraint the magnitude of the Gauss–Bonnet coupling constant, or of any other theories, as long as the QNMs are well understood in these theories [172].

5.2.3. Environmental effects

The QNMs of BHs are intimately connected to the null circular geodesic, or light ring [11, 37, 121, 173]. In addition, orbits that pass inside the light ring must plunge into the BH. Indeed, the innermost stable circular orbit (ISCO) for example, is always outside the light ring. As such, stars or matter debris composing accretion disks are not expected to populate the region close to the light ring. In turn, this means that 'environmental effects' (i.e., the impact of matter surrounding the BH) are expected to have a small impact on the QNMs of BHs and consequently on tests of the Kerr-hypothesis.

An exhaustive list of environmental effects, such as the impact of electric charge, accretion disks, dark matter, cosmological constant, etc are presented and quantified in [11, 173]. The results are summarized in table 1. These effects represent the ultimate precision with which one can test the no-hair hypothesis, in the absence of more detailed knowledge about the matter surrounding the BH. To summarize, no-hair tests are possible with ringdown waves from perturbed BHs, with threshold SNRs shown in figure 8. Third-generation detectors will be able to perform accurate tests of GR, or possibly infer details of the environment surrounding BHs.

Table 1.  Upper limits on the environmental corrections to the BH QNMs. We define ${\delta }_{R,I}=1-{\omega }_{R,I}/{\omega }_{R,I}^{(0)}$, where ω_{R, I} is the real (imaginary) part of the ringdown frequency in the presence of environmental effects, whereas ${\omega }_{R,I}^{(0)}$ is the same for an isolated BH with the same total mass. Conservative environmental reference values are $q={10}^{-3},B={10}^{8}\,\mathrm{Gauss},{\rho }_{3}^{\mathrm{DM}}={\rho }_{\mathrm{DM}}/({10}^{3}{M}_{\odot }/{\mathrm{pc}}^{3})$. We assume a Shakura–Sunyaev disk model with viscosity parameter α = 0.1 and Eddington ratio f_Edd = 10⁻⁴ (f_Edd = 1) for thick and thin disks, respectively. The spherical and ring-like matter distributions have mass δ M ∼ 10⁻³M. The scaling with the parameters is shown in [11]. Taken from [173].

Correction	$| {\delta }_{R}| ( \% )$	$| {\delta }_{I}| ( \% )$
Spherical near-horizon distribution	0.05	0.03
Ring at ISCO	0.01	0.01
Electric charge	10−5	10−6
Magnetic field	10−8	10−7
Gas accretion	10−11	10−11
DM halos	${10}^{-21}{\rho }_{3}^{\mathrm{DM}}$	${10}^{-21}{\rho }_{3}^{\mathrm{DM}}$
Cosmological effects	10−32	10−32

5.2.4. QNMs as a probe of the event horizon

As shown by Ryan [148], the multipole moments of a central object which is assumed to be stationary, axisymmetric and reflection-symmetric, can be measured by looking at the motion of a test body orbiting around the central object. Indeed, observable, gauge-invariant quantities can be expressed in terms of the multipole moments, as defined by Geroch and Hansen. These quantities are the GW spectrum, i.e. (as mentioned in section 5.1.1) the amount of GW energy emitted per logarithmic interval of frequency; the perihastron precession and the orbital plane precession ('epyciclic frequencies'); the number of cycles of the GW signal emitted per logarithmic interval. For instance, the GW spectrum is

$$\begin{eqnarray}{\rm{\Delta }}E(f) & = & f\displaystyle \frac{{{\rm{d}}{E}}_{{gw}}}{{\rm{d}}{f}}\\ & = & \mu \left(\displaystyle \frac{1}{3}{v}^{2}-\displaystyle \frac{1}{2}{v}^{4}+\displaystyle \frac{20}{9}\displaystyle \frac{{S}_{1}}{{M}^{2}}{v}^{5}+\left(-\displaystyle \frac{27}{8}+\displaystyle \frac{{M}_{2}}{{M}^{3}}\right){v}^{6}\,+\,...\right),\end{eqnarray} \tag{ 5.19 }$$

where f is the frequency of the emitted GWs (twice the orbital frequency ν), μ is the mass of the test body, and $v={(\pi {Mf})}^{1/3}$. Since the frequency is associated with the Killing vector of the stationary spacetime, all of these quantities are gauge-invariant. Note that the post-Newtonian expansion (5.19) is only accurate for $v\ll 1;$ near the ISCO, $v\sim {(6)}^{-1/2}$ (or larger for rotating BHs) and higher-order terms should be included in the expansion [177].

Ryan's construction [148], in principle, seems to be an extremely powerful phenomenological tool. GW and electromagnetic observations can be used to measure the multipole moments of the central object, thus providing a mapping of the spacetime. In particular, the moments can be extracted from the GW signal emitted by an extreme mass–ratio inspiral (EMRI), i.e., a stellar-mass BH inspiralling around a supermassive BH—a target source for space-based detectors such as eLISA [178, 179]. A comparison with the moments of Kerr's solution (5.4) would provide a test of the 'no-hair' hypothesis.

In practice, the ambitious program of 'mapping the BH spacetime' with its multipole moments turned out to be less practical than expected. Firstly, it has been observed that multipole expansions are poorly convergent in the strong-field region near the BH horizon, where a large number of terms is required to describe the spacetime [180]. Although the detection of an EMRI waveform by an eLISA-like detector could allow to measure the quadrupole $Q={M}_{2}$ with good accuracy [181, 182], a deviation from Kerr spacetime can affect the higher-order multipoles as well, which are difficult to measure independently. Secondly, the beautiful expression of ${\rm{\Delta }}E$ in terms of the multipole moments (5.19) is useless if we do not also know the geodesic motion and the emitted GW flux, which are required to compute the gravitational waveform [183]; and both geodesic motion and GW flux are very difficult to describe for a non-Kerr spacetime with arbitrary multipole moments. We also remark that the GW flux depends on the dynamical equations of gravity.

Several different approaches have been developed to deal with these concerns. Ryan's construction has been extended to specific modified gravity theories, such as scalar–tensor gravity [184, 185] and, to some extent, to EDGB gravity [186], in order to make the 'theory bias' towards general relativity less severe. Other authors have constructed parametrized deviations of Kerr spacetimes ('bumpy' or 'quasi-Kerr' BHs), studying geodesics and EMRI waveforms In some of these solutions, the deviation only affects the quadrupole moment [180, 183]; in others, higher-order multipoles are also affected [187–193]. However, in many of these 'bumpy' BH solutions the spacetime metric is not parametrized by multipole moments. These solutions have also been used to devise non-dynamical tests of the 'no-hair' hypotheses based on astrophysical, electromagnetic observations (see section 5.4).

Non-dynamical tests of the 'no-hair' hypothesis can also be done with astrophysical observations of the electromagnetic emission from the surroundings of BH candidates (see e.g. [194–196] and references therein).

5.4.1. Motion of stars

A promising measurement of the multipole moments of BH candidates is the (electromagnetic) observation of stars on tight orbits around supermassive BHs, most specially Sgr A*, the compact object at the center of our Galaxy [197–200]. Although such tests are complicated by several additional factors (related to the fact that such orbits have typical radii much larger than those involved in GW observations), progress in instrumentation makes them attractive possibilities in the near-future.

The idea is very simple, and consists on measuring the pericenter and orbital plane precession of stars orbiting a massive BH, on tight and eccentric enough orbits. The precession depends on the mass of the central object (the Schwarzschild—S—part of the geometry), on the spin J and on the quadrupole moment Q of the BH and can therefore be inverted to estimate each of these quantities, testing the no-hair hypothesis.

Define

$$\begin{eqnarray}&&{A}_{S}=\displaystyle \frac{6\pi }{{c}^{2}}\displaystyle \frac{{GM}}{a(1-{e}^{2})},\end{eqnarray} \tag{ 5.20 }$$

$$\begin{eqnarray}&&{A}_{J}=\displaystyle \frac{4\pi \,j}{{c}^{3}}{\left[\displaystyle \frac{{GM}}{a(1-{e}^{2})}\right]}^{3/2},\end{eqnarray} \tag{ 5.21 }$$

$$\begin{eqnarray}&&{A}_{Q}=\displaystyle \frac{3\pi \,{j}^{2}}{{c}^{4}}{\left[\displaystyle \frac{{GM}}{a(1-{e}^{2})}\right]}^{2},\end{eqnarray} \tag{ 5.22 }$$

with M the central, massive object, j its dimensionless spin, e the star's eccentricity and a its semi-major axis. In the orbit-averaged approximation, and assuming that the orbiting stars are much lighter than the central object, stars experience an advance of the orbital periapse given by (to lowest post-Newtonian order)

$$\begin{eqnarray}&&{\boldsymbol{\delta }}\varpi ={A}_{S}-2{A}_{J}-\displaystyle \frac{1}{2}{A}_{Q}(1-3{{\rm{\cos }}}^{2}i),\end{eqnarray} \tag{ 5.23 }$$

per orbit. In addition the orbital plane also precesses due to coupling between the orbital angular momentum and the central object spin J. The nodal precession is

$$\begin{eqnarray}&&\delta {\rm{\Omega }}={A}_{J}-{A}_{Q}{\rm{\cos }}\,{i},\end{eqnarray} \tag{ 5.24 }$$

with i the orbital inclination.

It can be seen that the Schwarzschild contribution exceeds the spin and quadrupole terms for in-plane precession, for most of the parameter space. Fortunately, the orbital plane precession depends only on the spin and quadrupole contributions. However, to test the no-hair hypothesis, one must determine five parameters (the BH mass, quadrupole moment and the three spin components), thus the orbits of two stars are required [197].

In addition, there are a number of effects that complicate the determination of the BH's multipole moments, such as [197–199]

• (i)  
  The contribution of other orbiting stars to the mass, spin and quadrupole moment of the central object. This contribution can be estimated once a stellar density distribution is known or prescribed. Assuming a number density of stars

  $$\begin{eqnarray}&&n(a,\theta )={N}_{0}/{a}_{0}^{3}{(a/{a}_{0})}^{-2}{n}_{\theta }(\theta ),\end{eqnarray} \tag{ 5.25 }$$

  one gets for the relative contributions to the mass, spin and quadrupole moment [198, 199]

  $$\begin{eqnarray}&&\displaystyle \frac{\delta M}{M}=4.8\times {10}^{-8}{M}_{6}{\left(\displaystyle \frac{{a}_{0}}{1\mathrm{pc}}\right)}^{-1}{a}_{1},\end{eqnarray} \tag{ 5.26 }$$

  $$\begin{eqnarray}&&j\displaystyle \frac{\delta J}{J}=3\times {10}^{-8}{M}_{6}{\left(\displaystyle \frac{{a}_{0}}{1\mathrm{pc}}\right)}^{-1}{a}_{1}^{3/2},\end{eqnarray} \tag{ 5.27 }$$

  $$\begin{eqnarray}&&Q/{M}^{3}\displaystyle \frac{\delta Q}{Q}=4\times {10}^{-10}{M}_{6}\displaystyle \frac{{\tilde{n}}_{\theta }}{0.1}{\left(\displaystyle \frac{{a}_{0}}{1\mathrm{pc}}\right)}^{-1}{a}_{1}^{3}.\end{eqnarray} \tag{ 5.28 }$$

  Here, ${a}_{1}\equiv \tfrac{a}{M}$, ${M}_{6}\equiv \tfrac{{M}_{* }}{{10}^{6}{M}_{\odot }}$, M_* being the total mass of stars inside a characteristic orbital separation a₀, and

  $$\begin{eqnarray}&&{\tilde{n}}_{\theta }={\int }_{-1}^{1}(3{{\rm{\cos }}}^{2}\theta -1){n}_{\theta }(\theta ){\rm{d}}\,{\rm{\cos }}\,\theta ,\end{eqnarray} \tag{ 5.29 }$$

  characterizes the angular distribution of stars. The strongest constraint here comes from the quadrupole corrections, which depend sensitively on the semi-major axis a. For the corrections to be under control, $a\lesssim 1000M$.
• (ii)  
  Perturbations to the orbit of the stars being measured. These effects include
• (iii)  
  Decoherence of the orbit due to Newtonian interactions with other stars, on a timescale

  $$\begin{eqnarray}&&{t}_{N}=12.6\times {10}^{8}{M}_{6}^{-1/2}{\left(\displaystyle \frac{{a}_{0}}{1\mathrm{pc}}\right)}^{-1/2}{a}_{1}{\left(\displaystyle \frac{{m}_{* }}{{M}_{\odot }}\right)}^{-1/2}{\left(\displaystyle \frac{M}{4.3\times {10}^{6}{M}_{\odot }}\right)}^{3/2}\,\sec ,\end{eqnarray} \tag{ 5.30 }$$

  where m_* is the average mass of a star within the stellar cluster.
• (iv)  
  Decoherence is the dominant perturbing effect, but in hydrodynamic interactions with the accretion flow (drag), stellar winds and tidal effects on the stars being measured introduce additional sources of error. These are quantified in [199].

In [199] the authors have studied how one can constrain the spin and quadrupole moment of Sgr A*, by measuring the motion of two stars on eccentric orbits. The results are shown in figure 13, assuming an astrometric precision of $10\mu$as. It turns out that, even at the small orbital separations used in their study, tracing the orbits of stars primarily measures the spin of the BH.

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We note, however, that even such large errors on the BH quadrupole moment already impose some constraints on some hairy BH solutions (see, e.g, [68], where some of the BH solutions have extremely large quadrupole moments).

5.4.2. Motion of pulsars

5.4.3. Accretion disks

5.4.4. BH 'shadows'

As discussed in section 5.3, when matter is moving very close to a BH (or any compact object) a multipolar decomposition of the gravitational field is not particularly useful since all, or a substantial number of, multipoles contribute to the gravitational potential and to the motion of matter. However, the spacetime around compact objects possesses unique features—such as ISCOs, light rings (unstable null geodesics), etc [222]—that might be used as smoking guns of the BH-nature of the object and even of GR. In particular, the null geodesics carry information about the effective size of BHs, since in essence any particle or light ray penetrating the light ring will never reach asymptotic observers. Thus, BHs create 'shadows' of matter around them [223–225]. The exact shape and appearance of BHs depends on the source illuminating them, but a crucial ingredient determining the optical appearance is the rotation rate, which determines how close to the horizon the co-rotating light ring is, and how far away from the horizon the counter-rotating light ring is. The rotation rate also determines how tight the accretion disk can bind to the BH, and how much the gravitational and Doppler shift emitted from the disk will be. An example of the optical appearance of BHs is shown in figure 14, for a Kerr BH rotating with $a/M=0.9$ and being illuminated by an accretion torus [103].

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Observations of BH shadows as a tool to probe the geometry close to the event horizon became possible with the advent of powerful instruments such as the Event Horizon Telescope [231]. The main obstacles to performing tests of GR with observations of BH shadows are (a) the large number of parameters that describe the shadow, including the inclination angle of the object, the mass and angular momentum of the BH, and all the details of the accretion mechanism (the illuminating source); (b) the lack of a robust parametrization of deviations from the Kerr geometry, in the strong-field limit.

Recently, a proof-of-principle describing tests of the Kerr hypothesis with observations of BH shadows was put forward [230]. As a background, the following metric (belonging to the class of 'quasi-Kerr' spacetimes introduced in [193]) was proposed

$$\begin{eqnarray}{{\rm{d}}{s}}^{2} & = & -\displaystyle \frac{{\rm{\Sigma }}({r}^{2}-2{Mr}+{a}^{2}{{\rm{\cos }}}^{2}\theta )}{{({A}_{1}({r}^{2}+{a}^{2})-{a}^{2}{{\rm{\sin }}}^{2}\theta )}^{2}}{{\rm{d}}{t}}^{2}-\displaystyle \frac{2a{{\rm{\sin }}}^{2}\theta {\rm{\Sigma }}({A}_{1}({r}^{2}+{a}^{2})-\widetilde{{\rm{\Delta }}})}{{({A}_{1}({r}^{2}+{a}^{2})-{a}^{2}{{\rm{\sin }}}^{2}\theta )}^{2}}{\rm{d}}{t}{\rm{d}}\phi \\ & & +\displaystyle \frac{{\rm{\Sigma }}}{\widetilde{{\rm{\Delta }}}}{{\rm{d}}{r}}^{2}+{\rm{\Sigma }}{\rm{d}}{\theta }^{2}+\left[\displaystyle \frac{{A}_{1}^{2}{({r}^{2}+{a}^{2})}^{2}-{a}^{2}\widetilde{{\rm{\Delta }}}{{\rm{\sin }}}^{2}\theta }{{({A}_{1}({r}^{2}+{a}^{2})-{a}^{2}{{\rm{\sin }}}^{2}\theta )}^{2}}\right]{\rm{\Sigma }}{{\rm{\sin }}}^{2}\theta {\rm{d}}{\phi }^{2},\end{eqnarray} \tag{ 5.31 }$$

where $\widetilde{{\rm{\Delta }}}={\rm{\Delta }}+\beta {M}^{2}$ and ${A}_{1}=1+{\alpha }_{13}{M}^{3}/{r}^{3}$. This geometry is parametrized by the BH mass M, angular momentum J = Ma and the constants ${\alpha }_{13},\beta$. We should point out that this particular form of the metric is rather ad hoc and does not describe any known theory, but that in any case the results are to be taken only as proof-of-principle.

The shadow resulting from the metric (5.31) was compared to simulations of observations by the Event Horizon Telescope of the BH at the center of our Galaxy, Sgr A*. The result is summarized in figure 15. The study uses a reconstructed image of Sgr A* based on a simulated image of a radiatively-inefficient accretion flow. Within the working hypothesis, the prospects for constraining the unknown parameters $\beta ,{\alpha }_{13}$ are good: with 100 observations, the parameters can be determined to be

$$\begin{eqnarray}&&{\alpha }_{13}={0.13}_{-0.21}^{+0.43}{}_{-0.34}^{+0.9},\end{eqnarray} \tag{ 5.32 }$$

$$\begin{eqnarray}&&\beta ={0.03}_{-0.10}^{+0.05}{}_{-0.24}^{+0.07}.\end{eqnarray} \tag{ 5.33 }$$

The uncertainties associated with these parameters are such that they are effectively of order ${ \mathcal O }(1)$, precisely the order at which they enter in the metric¹⁶ .

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Hairy BHs arising naturally from simple theories, such as the minimally coupled scalar theory (3.6) [29, 68], will also give rise to shadows which can, in principle, be discriminated from those of Kerr BHs [233].

It should be noted that shadow observations can also, in principle, be used to test the BH-nature of compact objects. One could expect that objects without a horizon, and presumably with a surface, will feature a bright surface, allowing immediately to discriminate between BHs and, say, gravastars. It was pointed out in [103] that the main factor determining the shadow is the light ring. As such, horizonless compact objects with a light ring can mimick Kerr BHs very efficintly, provided the illuminating source is not accreted towards the center of the object.

The Newtonian gravitational potential in the exterior of a massive body with density $\rho (t,\vec{x})$ is the solution of Poisson's equation ${{\rm{\nabla }}}^{2}{\rm{\Phi }}=4\pi G\rho$ in vacuum:

$$\begin{eqnarray}&&{\rm{\Phi }}(t,\vec{x})=-G\int \displaystyle \frac{\rho (t,\vec{x}^{\prime} )}{| \vec{x}-\vec{x}^{\prime} | }{{\rm{d}}}^{3}x^{\prime} .\end{eqnarray} \tag{ 6.1 }$$

With a Taylor expansion of $1/| \vec{x}-\vec{x}^{\prime} |$ around $\vec{x}^{\prime} =\vec{0}$ and a spherical harmonic decomposition, this solution can be written as a series in $1/r$ (where $r=| \vec{x}|$), which is called multipolar expansion of the potential:

$$\begin{eqnarray}&&{\rm{\Phi }}(t,\vec{x})=-G\displaystyle \sum _{{lm}}\displaystyle \frac{1}{{r}^{l+1}}\displaystyle \frac{4\pi }{2l+1}{I}_{{lm}}(t){Y}_{{lm}}(\theta ,\phi ),\end{eqnarray} \tag{ 6.2 }$$

where Y_lm are the usual spherical harmonics, and

$$\begin{eqnarray}&&{I}_{{lm}}(t)=\int \rho (t,\vec{x}){r}^{l}{Y}_{{lm}}(\theta ,\phi ){{\rm{d}}}^{3}x\end{eqnarray} \tag{ 6.3 }$$

are the multipole moments of the potential.

This expansion can also be expressed in terms of symmetric-trace-free (STF) tensors [124, 128]:

$$\begin{eqnarray}&&{\rm{\Phi }}(t,\vec{x})=-G\displaystyle \sum _{l=0}^{\infty }\displaystyle \frac{1}{{r}^{l+1}}\displaystyle \frac{(2l-1)!!}{l!}{I}^{\langle {i}_{1}\cdots {i}_{l}\rangle }{n}^{{i}_{1}}\cdots {n}^{{i}_{l}},\end{eqnarray} \tag{ 6.4 }$$

where $(2l-1)!!\,=\,(2l-1)(2l-3)(2l-5)\cdots 1$, ${n}^{i}={x}^{i}/r$ (in polar coordinates, ${n}^{i}=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$), the brackets $\langle \cdots \rangle$ denote the symmetric and trace-free part of a tensor, and ${I}^{\langle {i}_{1}\cdots {i}_{l}\rangle }(t)=\int \rho (t,\vec{x}^{\prime} )({x}^{{i}_{1}}\cdots {x}^{{i}_{l}}-{\rm{trace}}\,{\rm{parts}}){{\rm{d}}}^{3}x$ are the multipole moments. For a given value of l, the components of the STF tensor ${n}^{\langle {i}_{1}}\cdots {n}^{{i}_{l}\rangle }$ are combinations of the spherical harmonics Y_lm with $m=-l,\ldots ,l$, and ${I}^{\langle {i}_{1}\cdots {i}_{l}\rangle }$ are combinations of the multipoles (6.3) I_lm. The first terms of this expansion are

$$\begin{eqnarray}&&{\rm{\Phi }}=-\displaystyle \frac{{GM}}{r}-\displaystyle \frac{3G}{2}{I}^{\langle {ij}\rangle }{x}^{i}{x}^{j}\end{eqnarray} \tag{ 6.5 }$$

where I = M is the mass, ${I}^{i}=\int \rho {x}^{i}{{\rm{d}}}^{3}x=0$ in the center-of-mass frame, ${I}^{\langle {ij}\rangle }=\int \rho ({x}^{i}{x}^{i}-{\delta }^{{ij}}{r}^{2}/3){{\rm{d}}}^{3}x$ is the quadrupole moment¹⁷ .

In most cases, one considers the multipole expansion of a stationary, axisymmetric body with symmetry axis $\hat{k}=(0,0,1)$. Then, the only non-vanishing moments are I_l0 and, defining

$$\begin{eqnarray}&&{M}_{l}\equiv \sqrt{\displaystyle \frac{4\pi }{2l+1}}{I}_{l0}=\int \rho (\vec{x}){r}^{l}{P}_{l}({\rm{\cos }}\theta ){{\rm{d}}}^{3}x,\end{eqnarray} \tag{ 6.6 }$$

the expansion (6.2) reduces to equation (5.1):

$$\begin{eqnarray}&&{\rm{\Phi }}(\vec{x})=-G\displaystyle \sum _{l=0}^{\infty }\displaystyle \frac{{M}_{l}}{{r}^{l+1}}{P}_{l}(\cos \theta ),\end{eqnarray} \tag{ 6.7 }$$

where, as discussed in section 5.1, ${M}_{0}=M$, ${M}_{1}=0$ in the center-of-mass frame, ${M}_{2}=Q$ quadrupole moment. With the further assumption that the body (and then, the gravitational potential) is reflection-symmetric across the equatorial plane, i.e., symmetric for $\theta \to \pi -\theta$, then ${M}_{2l+1}=0$: the only non-vanishing multipoles are those with even values of l.

In STF notation, axi-symmetry with respect to $\hat{k}$ implies that ${I}^{\langle {i}_{1}\cdots {i}_{l}\rangle }\propto {k}^{\langle {i}_{1}}\cdots {k}^{{i}_{l}\rangle }$. Using normalization properties of STF tensors [125], it can be shown that

$$\begin{eqnarray}&&{I}^{\langle {i}_{1}\cdots {i}_{l}\rangle }={M}_{l}{k}^{\langle {i}_{1}}\cdots {k}^{{i}_{l}\rangle }.\end{eqnarray} \tag{ 6.8 }$$

For instance, the quadrupole STF tensor is ${I}^{\langle {ij}\rangle }={M}_{2}({k}^{i}{k}^{j}-{\delta }^{{ij}}/3)$ = Q diag(−1/3, −1/3, 2/3).

Geroch and Hansen developed a formalism to define and compute the multipole moments of stationary, asymptotically flat solutions of Einstein's equations in vacuum [126, 127]. Their approach has then been generalized to non-vacuum spacetimes, including the electromagnetic field [235] (see also [236]) and a scalar field [184]. The latter work can also be seen as a generalization to modified gravity theories: the multipole structure with a scalar field, derived in [184], also applies to vacuum spacetimes in Bergmann–Wagoner scalar–tensor theories [237, 238]. Other scalar–tensor theories with additional terms falling off rapidly enough, such as EDGB theory [86], enjoy the same multipole structure, as can be seen by comparing the derivation of the quadrupole moment in EDGB BHs [186] with the computation of multipole moments in Bergmann–Wagoner theories [184].

In the Geroch–Hansen construction, multipole moments are introduced as tensors at infinity, generated by a set of potentials. The asymptotic behavior is defined without introducing a specific coordinate frame, through an asymptotic completion of spacetime in which, after a conformal rescaling of the metric, the spacetime is extended to include the 'infinite point' Λ. To understand this procedure, let us consider for simplicity the Euclidean flat space in Newtonian theory, with metric ${{\rm{d}}{s}}^{2}={\delta }^{{ij}}{{\rm{d}}{x}}^{i}{{\rm{d}}{x}}^{j}$. The multipole expansion of a potential ${\rm{\Phi }}(\vec{x})$ is

$$\begin{eqnarray}&&{\rm{\Phi }}=\displaystyle \sum _{l\geqslant 0}\displaystyle \frac{1}{l!}\displaystyle \frac{1}{{r}^{l+1}}{Q}^{\langle {i}_{1}\cdots {i}_{l}\rangle }{x}^{{i}_{1}}\cdots {x}^{{i}_{l}}=\displaystyle \frac{Q}{r}+\displaystyle \frac{{Q}^{i}}{{r}^{3}}{x}^{i}+\displaystyle \frac{1}{2}\displaystyle \frac{{Q}^{\langle {ij}\rangle }}{{r}^{5}}{x}^{i}{x}^{j}+....\end{eqnarray} \tag{ 6.9 }$$

Changing coordinates to ${\bar{x}}^{i}={x}^{i}/{r}^{2}$, the metric reads ${{\rm{d}}{s}}^{2}=({\delta }^{{ij}}{\rm{d}}{\bar{x}}^{i}{\rm{d}}{\bar{x}}^{j})/{\bar{r}}^{4}$. With a conformal transformation the metric becomes ${\overline{{\rm{d}}{s}}}^{2}={{\rm{\Omega }}}^{2}{{\rm{d}}{s}}^{2}={\delta }^{{ij}}{\rm{d}}{\bar{x}}^{i}{\rm{d}}{\bar{x}}^{j}$ (where ${\rm{\Omega }}={\bar{r}}^{2}$), regular at the infinite point Λ ($\bar{r}=0$). In the new coordinate frame, the conformally rescaled field is

$$\begin{eqnarray}&&\widetilde{{\rm{\Phi }}}({\bar{x}}^{i})={{\rm{\Omega }}}^{-1/2}{\rm{\Phi }}=Q+{Q}^{i}{\bar{x}}^{i}+\displaystyle \frac{1}{2}{Q}^{\langle {ij}\rangle }{\bar{x}}^{i}{\bar{x}}^{j}+\displaystyle \frac{1}{6}{Q}^{\langle {ijk}\rangle }{\bar{x}}^{i}{\bar{x}}^{j}{\bar{x}}^{k}+....\end{eqnarray} \tag{ 6.10 }$$

The STF derivatives of $\widetilde{{\rm{\Phi }}}$ evaluated at the infinite point Λ yield the multipole moments: ${\widetilde{{\rm{\Phi }}}}_{| {\rm{\Lambda }}}=Q$, ${\widetilde{{\rm{\Phi }}}}_{,i| {\rm{\Lambda }}}={Q}^{i}$, ${\widetilde{{\rm{\Phi }}}}_{,\langle {ij}\rangle | {\rm{\Lambda }}}={Q}^{\langle {ij}\rangle }$, ${\widetilde{{\rm{\Phi }}}}_{,\langle {ijk}\rangle | {\rm{\Lambda }}}={Q}^{\langle {ijk}\rangle }$, and so on.

In GR the procedure is the same, with some differences:

• The three-space is the space of orbits of the timelike Killing vector (i.e., choosing the time variable corresponding to the Killing vector, any $t={\rm{const}}.$ submanifold). This is a curved space, with its own covariant derivative D_i (${\bar{D}}_{i}$ after the conformal transformation).
• The conformal transformation is not given explicitly; it is only required that the conformally-rescaled metric is smooth at the infinite point Λ, and that the conformal factor and its first derivative (but not the second) vanish in Λ, i.e. that Ω falls off 'like $1/{r}^{2}$'. Actually, this requirement is the definition of 'asymptotic flatness' of the spacetime.
• The derivatives on the fields are covariant derivatives ${\bar{D}}_{i}$. Moreover, the requirement that the moments do not depend on the choice of Ω determines extra terms involving the Ricci tensor (of the three-space) in the definition of the moments. For instance ${Q}^{\langle {ij}\rangle }={({\bar{D}}_{\langle i}{\bar{D}}_{j\rangle }{\rm{\Phi }}-1/2{{ \mathcal R }}_{\langle {ij}\rangle }{\rm{\Phi }})}_{| {\rm{\Lambda }}}$.

In this way, the asymptotic expansion of the potentials can be constructed in a coordinate-independent way; the multipoles are defined as tensors in Λ.

The next step is to define the potentials corresponding to the multipoles of the four-dimensional spacetime. Given the Killing vector field ${\xi }^{\mu }$, the mass potential and the angular potential are ${{\rm{\Phi }}}_{M}=({\lambda }^{2}+{\omega }^{2}-1)/(4\lambda )$ and ${{\rm{\Phi }}}_{J}=\omega /(2\lambda )$, where

$$\begin{eqnarray}\lambda & = & -{\xi }^{\mu }{\xi }_{\mu },\\ {\omega }_{\alpha } & = & {\epsilon }_{\alpha \beta \gamma \delta }{\xi }^{\beta }{\xi }^{\delta ;\gamma }={\omega }_{,\alpha }\end{eqnarray} \tag{ 6.11 }$$

(the fact that ${\omega }_{\alpha }$ is a gradient is a consequence of Einstein's field equations). The multipole expansion of these potentials, defined with the procedure outlined above, yields the mass multipoles ${M}^{\langle {i}_{1}\cdots {i}_{l}\rangle }$ and the current multipoles ${S}^{\langle {i}_{1}\cdots {i}_{l}\rangle }$. In the weak-field limit, the Geroch–Hansen mass multipole moments reduce to the Newtonian multipole moments ${I}^{\langle {i}_{1}\cdots {i}_{l}\rangle }$ defined in equation (6.4), with a different normalization: ${M}^{\langle {i}_{1}\cdots {i}_{l}\rangle }=(2l-1)!!{I}^{\langle {i}_{1}\cdots {i}_{l}\rangle }$ (STF mass monopoles in Thorne's paper [128], instead, have the same normalization as the Newtonian multipole moments).

When extra fields are present, they correspond to new potentials and then to new sets of multipole moments. For instance, a scalar field ϕ brings a potential ${{\rm{\Phi }}}_{S}=\phi$ [184] which yields a set of scalar multipoles ${S}^{\langle {i}_{1}\cdots {i}_{l}\rangle }$. It should be noted that the Ricci terms ${{ \mathcal R }}_{{ij}}$ in the definition of multipole moments induce a mixing of the moments associated to different fields.

When the spacetime is symmetric with respect to an axis $\hat{k}$

$$\begin{eqnarray}{M}^{\langle {i}_{1}\cdots {i}_{l}\rangle } & = & (2l-1)!!{M}_{l}{k}^{\langle {i}_{1}}\cdots {k}^{{i}_{l}\rangle },\\ {S}^{\langle {i}_{1}\cdots {i}_{l}\rangle } & = & (2l-1)!!{S}_{l}{k}^{\langle {i}_{1}}\cdots {k}^{{i}_{l}\rangle }.\end{eqnarray} \tag{ 6.12 }$$

The mass and current STF moments reduce to two sets of scalar moments M_l, S_l. The Geroch–Hansen procedure can be cast in a simpler form [129], exploiting the powerful Ernst potential formalism. Indeed, it can be shown that the complex combination $\zeta ={{\rm{\Phi }}}_{M}+{\rm{i}}{{\rm{\Phi }}}_{J}$ is the (secondary) Ernst potential, and Einstein's equations can be formulated in terms of the Ernst potential.

The multipole moments of a stationary, isolated object can also be defined in terms of the asymptotic behavior of the spacetime metric. Indeed, as shown by Thorne [128], the (asymptotic) spacetime metric can be expanded in inverse powers of a suitable radial coordinate; the coefficients of this expansion can be interpreted as the multipole moments:

$$\begin{eqnarray}{g}_{00} & = & -1+\displaystyle \frac{2M}{r}+\displaystyle \sum _{l\geqslant 2}\displaystyle \frac{1}{{r}^{l+1}}\left(\displaystyle \frac{2}{l!}{M}^{\langle {a}_{1}\cdots {a}_{l}\rangle }{n}^{{a}_{1}}\cdots {n}^{{a}_{l}}+(l^{\prime} \lt l\,\mathrm{harmonics})\right),\\ {g}_{0j} & = & -2\displaystyle \sum _{l\geqslant 1}\displaystyle \frac{1}{{r}^{l+1}}\left(\displaystyle \frac{1}{l!}{\epsilon }^{{{jka}}_{l}}{S}^{\langle {{ka}}_{1}...{a}_{l-1}\rangle }{n}^{\langle {a}_{1}}\cdots {n}^{{a}_{l}\rangle }+(l^{\prime} \lt l\,\mathrm{harmonics})\right.\\ & & +(l-\mathrm{harmonics}\,\mathrm{with}\,\mathrm{parity}{(-1)}^{l})).\end{eqnarray} \tag{ 6.13 }$$

The coordinate systems in which the spacetime metric has the form (6.13) are called ACMC coordinates: they are asymptotically cartesian and (when the source is weak-field and covered by the same coordinate system) the origin of the coordinate system lies at the center of mass of the source. It has been shown [144] that the definitions of multipole moments by Thorne and by Geroch–Hansen are equivalent. Actually, the moments appearing in [128] have different normalization; however, for simplicity of notation, in equations (6.13) (and throughout this review) we have used the normalizations of Geroch–Hansen, which are also those adopted in most of the recent literature on the subject.

As discussed above, when the spacetime is symmetric with respect to an axis $\hat{k}$, the multipole moments ${M}^{\langle {a}_{1}\cdots {a}_{l}\rangle }$, ${S}^{\langle {a}_{1}\cdots {a}_{l}\rangle }$ reduce to the scalar quantities M_l, S_l (see equations (6.12)). Using the normalization properties of STF tensors [125, 239], it can be shown that

$$\begin{eqnarray}{M}^{\langle {a}_{1}\cdots {a}_{l}\rangle }{n}^{\langle {a}_{1}}\cdots {n}^{{a}_{l}\rangle } & = & l!\,{M}_{l}{P}_{l}({\rm{\cos }}\,\theta ),\\ {\epsilon }^{{{jka}}_{l}}{S}^{\langle {{ka}}_{1}...{a}_{l-1}\rangle }{n}^{\langle {a}_{1}}\cdots {n}^{{a}_{l}\rangle } & = & (l-1)!\,{\epsilon }^{{ijk}}{n}^{i}{k}^{k}\,{S}_{l}P{^{\prime} }_{l}({\rm{\cos }}\,\theta ),\end{eqnarray} \tag{ 6.14 }$$

where $\cos \theta =\hat{n}\cdot \hat{k}$, and primes denote derivatives with respect to $\cos \theta$. The metric expansion can then be written as

$$\begin{eqnarray}{g}_{00} & = & -1+\displaystyle \frac{2M}{r}+\displaystyle \sum _{l\geqslant 2}\displaystyle \frac{1}{{r}^{l+1}}({M}_{l}{P}_{l}(\cos \theta )+(l^{\prime} \lt l\,\mathrm{harmonics})),\\ {g}_{0j} & = & -2\displaystyle \sum _{l\geqslant 1}\displaystyle \frac{1}{{r}^{l+1}}\left({\epsilon }^{{ijk}}{n}^{i}{k}^{k}\displaystyle \frac{{S}_{l}}{l}P{^{\prime} }_{l}(\cos \theta )+(l^{\prime} \lt l\,\mathrm{harmonics})\right)\end{eqnarray} \tag{ 6.15 }$$

(note that when the spacetime is symmetric with respect to the axis $\hat{k}$, the vector harmonics with parity ${(-1)}^{l}$ in the expansion of g_0j—see equation (6.13)—identically vanish). In polar coordinates $\hat{k}=(0,0,1)$, $\hat{n}=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$ and ${\epsilon }^{{ijk}}{n}^{i}{k}^{k}{{\rm{d}}{x}}^{j}=r{{\rm{\sin }}}^{2}\theta {\rm{d}}\phi$, therefore equations (6.15) yield equations (5.5).

If the source can be covered by de Donder coordinates (in which ${\bar{h}}^{\mu \nu }\equiv -{(-g)}^{1/2}{g}^{\mu \nu }+{\eta }^{\mu \nu }$ satisfies ${\bar{h}}_{\,\,,\nu }^{\mu \nu }=0$, and Einstein's equations can be written as flat-space wave equations for ${\bar{h}}^{\mu \nu }$), it is possible to define an 'effective' stress–energy tensor ${\tau }^{\mu \nu }$, which is the source of the wave equations. In these coordinates, Thorne's multipole moments can be expressed as integrals of ${\tau }^{\mu \nu }$ over the source [128]:

$$\begin{eqnarray}{M}^{\langle {a}_{1}\cdots {a}_{l}\rangle } & = & (2l-1)!!\displaystyle \int {\tau }^{00}{x}^{\langle {a}_{1}}\cdots {x}^{{a}_{l}\rangle }{{\rm{d}}}^{3}x,\\ {S}^{\langle {a}_{1}\cdots {a}_{l}\rangle } & = & \displaystyle \frac{2l(2l-1)!!}{l+1}\displaystyle \int {x}^{i}{\tau }^{0j}{\epsilon }^{{ij}\langle {a}_{l}}{x}^{{a}_{1}}\cdots {x}^{{a}_{l-1}\rangle }{{\rm{d}}}^{3}x.\end{eqnarray} \tag{ 6.16 }$$

When the spacetime is symmetric with respect to an axis $\hat{k}$, equations (6.16) give

$$\begin{eqnarray}{M}_{l} & = & \displaystyle \int {\tau }^{00}{r}^{l}{P}_{l}({\rm{\cos }}\theta ){{\rm{d}}}^{3}x,\\ {S}_{l} & = & \displaystyle \frac{2}{l+1}\displaystyle \int {\epsilon }^{{ijk}}{\tau }^{0j}{n}^{i}{k}^{k}{r}^{l}P{^{\prime} }_{l}({\rm{\cos }}\theta ){{\rm{d}}}^{3}x.\end{eqnarray} \tag{ 6.17 }$$

In the case of a weak-field source, ${\tau }^{00}=\rho$, ${\tau }^{0j}=\rho {v}^{j}$, and equations (6.17) reduce to [146, 147]

$$\begin{eqnarray}{M}_{l} & = & \displaystyle \int \rho {r}^{l}{P}_{l}({\rm{\cos }}\theta ){{\rm{d}}}^{3}x,\\ {S}_{l} & = & \displaystyle \frac{2}{l+1}\displaystyle \int \rho {v}^{\phi }{r}^{l}P{^{\prime} }_{l}({\rm{\cos }}\theta ){{\rm{\sin }}}^{2}\theta {{\rm{d}}}^{3}x.\end{eqnarray} \tag{ 6.18 }$$