The observational evidence for horizons:

from echoes to precision gravitational-wave

physics

Vitor Cardoso1,2 and Paolo Pani3,1

1CENTRA, Departamento de Fısica, Instituto Superior Tecnico,Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa,

Portugal2Perimeter Institute for Theoretical Physics, 31 Caroline Street

North Waterloo, Ontario N2L 2Y5, Canada3Dipartimento di Fisica, “Sapienza” Universita di Roma & Sezione

INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy

April 15, 2019

Abstract

The existence of black holes and of spacetime singularities is a fun-damental issue in science. Despite this, observations supporting theirexistence are scarce, and their interpretation unclear. We overview howstrong a case for black holes has been made in the last few decades, andhow well observations adjust to this paradigm. Unsurprisingly, we con-clude that observational proof for black holes is impossible to come by.However, just like Popper’s black swan, alternatives can be ruled out orconfirmed to exist with a single observation. These observations are withinreach. In the next few years and decades, we will enter the era of precisiongravitational-wave physics with more sensitive detectors. Just as accelera-tors require larger and larger energies to probe smaller and smaller scales,more sensitive gravitational-wave detectors will be probing regions closerand closer to the horizon, potentially reaching Planck scales and beyond.What may be there, lurking?

1

arX

iv:1

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1v5

[gr

-qc]

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Apr

201

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Contents

1 Introduction 3

2 Setting the stage: escape cone, photospheres, quasinormal modes,and tidal effects 62.1 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Escape trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Photospheres, ECOs and ClePhOs . . . . . . . . . . . . . . . . . 82.4 Quasinormal modes . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Quasinormal modes, photospheres, and echoes . . . . . . . . . . . 132.6 The role of the spin . . . . . . . . . . . . . . . . . . . . . . . . . 142.7 BHs in binaries: tidal heating and tidal deformability . . . . . . 16

3 Beyond vacuum black holes 183.1 Are there alternatives? . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Formation and evolution . . . . . . . . . . . . . . . . . . . . . . . 203.3 On the stability problem . . . . . . . . . . . . . . . . . . . . . . . 22

4 Have exotic compact objects been already ruled out by electro-magnetic observations? 24

5 Testing the nature of BHs with GWs 275.1 Smoking guns: echoes and resonances . . . . . . . . . . . . . . . 275.2 Precision physics: QNMs, tidal deformability, heating, and mul-

tipolar structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Summary 31

2

“The crushing of matter to infinite density by infinite tidal gravita-tion forces is a phenomenon with which one cannot live comfortably.From a purely philosophical standpoint it is difficult to believe thatphysical singularities are a fundamental and unavoidable feature ofour universe [...] one is inclined to discard or modify that theoryrather than accept the suggestion that the singularity actually occursin nature.”

Kip Thorne, Relativistic Stellar Structure and Dynamics (1966)

“No testimony is sufficient to establish a miracle, unless the testi-mony be of such a kind, that its falsehood would be more miraculousthan the fact which it endeavors to establish.”

David Hume, An Enquiry concerning Human Understanding (1748)

1 Introduction

The discovery of the electron and the known neutrality of matter led in 1904to J. J. Thomson’s “plum-pudding” atomic model. Data from new scatteringexperiments was soon found to be in tension with this model, which was even-tually superseeded by Rutherford’s, featuring an atomic nucleus. The point-likecharacter of elementary particles opened up new questions. How to explain theapparent stability of the atom? How to handle the singular behavior of theelectric field close to the source? What is the structure of elementary particles?Some of these questions were elucidated with quantum mechanics and quantumfield theory. Invariably, the path to the answer led to the understanding ofhitherto unknown phenomena.

The history of elementary particles is a timeline of the understanding ofthe electromagnetic (EM) interaction, and is pegged to its characteristic 1/r2

behavior (which necessarily implies that other structure has to exist on smallscales within any sound theory).

Arguably, the elementary particle of the gravitational interaction are blackholes (BHs). Within General Relativity (GR), BHs are indivisible and are infact the simplest macroscopic objects that one can conceive. The uniquenesstheorems – establishing that the two-parameter Kerr family of BHs describesany vacuum, stationary and asymptotically flat, regular BH solution – haveturned BHs into somewhat of a miracle elementary particle [1].

Can BHs, as envisioned in vacuum GR, hold the same surprises that theelectron and the hydrogen atom did when they started to be experimentallyprobed? Are there any parallels that can be useful guides? The BH interioris causally disconnected from the exterior by an event horizon. Unlike theclassical description of atoms, the GR description of the BH exterior is self-consistent and free of pathologies. The “inverse-square law problem” – the GR

3

counterpart of which is the appearance of pathological curvature singularities –is swept to inside the horizon and therefore harmless for the external world.There are good indications that classical BHs are perturbatively stable againstsmall fluctuations, and attempts to produce naked singularities, starting fromBH spacetimes, have failed. BHs are not only curious mathematical solutionsto Einstein’s equations, but also their formation process is well understood. Infact, there is nothing spectacular with the presence or formation of an eventhorizon. The equivalence principle dictates that an infalling observer crossingthis region (which, by definition, is a global concept) feels nothing extraordinary:in the case of macroscopic BHs all of the local physics at the horizon is ratherunremarkable.

Why, then, should one question the existence of BHs? 1 There are a numberof important reasons to do so.

• Even if the BH exterior is pathology-free, the interior is not. The Kerrfamily of BHs harbors, singularities and closed timelike curves in its in-terior, and more generically it features a Cauchy horizon signaling thebreakdown of predictability of the theory. In fact, the geometry describ-ing the interior of an astrophysical spinning BH is currently unknown. Apossible resolution of this problem may require accounting for quantumeffects. It is conceivable that these quantum effects are of no consequencewhatsoever on physics outside the horizon. Nevertheless, it is conceiv-able as well that the resolution of such inconsistency leads to new physicsthat resolves singularities and does away with horizons, at least in theway we understand them currently. Such possibility is not too dissimilarfrom what happened with the atomic model after the advent of quantumelectrodynamics.

• In a related vein, quantum effects around BHs are far from being undercontrol, with the evaporation process itself leading to information loss.The resolution of such problems could include changing the endstate ofcollapse [4–8] (perhaps as-yet-unknown physics can prevent the formationof horizons), or altering drastically the near-horizon region [9]. The factof the matter is that there is no tested nor fully satisfactory theory ofquantum gravity, in much the same way that one did not have a quantumtheory of point particles at the beginning of the 20th century.

1It is ironic that one asks this question more than a century after the first BH solutionwas derived. Even though Schwarzschild and Droste wrote down the first nontrivial regular,asymptotic flat, vacuum solution to the field equations already in 1916 [2, 3], several decadeswould elapse until such solutions became accepted. The dissension between Eddington andChandrasekhar over gravitational collapse to BHs is famous – Eddington firmly believed thatnature would find its way to prevent full collapse – and it took decades for the communityto overcome individual prejudices. Progress in our understanding of gravitational collapse,along with breakthroughs in the understanding of singularities and horizons, contributed tochange the status of BHs. Together with observations of phenomena so powerful that couldonly be explained via massive compact objects, the theoretical understanding of BHs turnedthem into undisputed kings of the cosmos. The irony lies in the fact that they quickly becamethe only acceptable solution. So much so, that currently an informal definition of a BH mightwell be “any dark, compact object with mass above three solar masses.”

4

• It is tacitly assumed that quantum gravity effects become important onlyat the Planck scale. At such lengthscales

√c~/G, the Schwarzschild ra-

dius is of the order of the Compton wavelength of the BH. In the ordersof magnitude standing between the Planck scale and those accessible bycurrent experiments many surprises can hide (to give but one example,extra dimensions would change the numerical value of the Planck scale).

• Horizons are not only a rather generic prediction of GR, but their exis-tence is in fact necessary for the consistency of the theory at the classicallevel. This is the root of Penrose’s (weak) Cosmic Censorship Conjec-ture, which remains one of the most urgent open problems in fundamentalphysics. In this sense, the statement that there is a horizon in any space-time harboring a singularity in its interior is such a remarkable claim, that(in an informal description of Hume’s statement above) it requires similarremarkable evidence. This is the question we will entertain here, is thereany observational evidence for the existence of BHs?

• It is in the nature of science that paradigms have to be constantly ques-tioned and subjected to experimental and observational scrutiny. Mostspecially because if the answer turns out to be that BHs do not exist, theconsequences are so extreme and profound, that it is worth all the possibleburden of actually testing it. Within the coming years we will finally be inthe position of performing unprecedented tests on the nature of compactdark objects, the potential pay-off of which is enormous. As we will argue,the question is not just whether the strong-field gravity region near com-pact objects is consistent with the Kerr geometry, but rather to quantifythe limits of observations in testing the event horizon.

• If we don’t entertain alternatives, we won’t find them. We need to knowhow to model alternatives to understand how consistent the data is withaccepted paradigms.

• Finally, there are sociological issues, sometimes invoked to claim that oneshould invest taxpayers money in sure-fire science. The reality is thattaxpayers, all of us, want to know. More frequently than not, the thrilllies in the intellectual possibilities rather than on merely checking knownfacts. This is, after all, why pure science is relevant. Thus, it is a waste ofresources not to use (built and already paid for) detectors to investigateissues that lie at the heart of fundamental questions.

Known physics all but exclude BH alternatives. Nonetheless, the StandardModel of fundamental interactions is not sufficient to describe the cosmos – atleast on the largest scales – and also leaves all of the fundamental questionsregarding BHs open. It may be wise to keep an open mind. As we will argue,from the point of view of observers merely testing the spacetime structure with-out a priori theoretical bias, the evidence for horizons is nonexistent. In fact,the question to be asked is not if there is an event horizon in the spacetime, buthow close to it do experiments or observations go.

5

2 Setting the stage: escape cone, photospheres,quasinormal modes, and tidal effects

“Alas, I abhor informality.”

That Mitchell and Webb Look, Episode 2

Our framework will be that of GR. Let us start by focusing on spherical sym-metry for simplicity. Birkhoff’s theorem guarantees that any vacuum spacetimeis described by the Schwarzschild geometry, which in standard coordinates reads

ds2 = −fdt2 + f−1dr2 + r2dΩ2 , (1)

where M is the total mass of the spacetime (we use geometrical G = c = 1units, except if otherwise stated) and

f = 1− 2M

r. (2)

The existence of a null hypersurface at r = 2M is the defining feature of aBH. This hypersurface is called the event horizon of a Schwarzschild BH. Inthe following, we will compare the properties of the latter with those of anultracompact object with a effective surface at

r0 = 2M(1 + ε) , (3)

having often in mind the case where ε 1 (for instance, for Planckian cor-rections of an astrophysical BH, ε ∼ 10−40). Although the above definition iscoordinate-dependent, the proper distance between the surface and rg scales like∫ r02M

drf1/2 ∼ ε1/2. Most of the results discussed below show a dependence onlog ε, making the distinction irrelevant. There are objects for which the effectivesurface is ill-defined, because the matter fields are smooth everywhere. In suchcases, r0 can conservatively be taken to be the geometric center of the object.

2.1 Geodesics

The geodesic motion of timelike or null particles in the geometry (1) can be de-scribed with the help of two conserved quantities, the specific energy E = f t andangular momentum L = r2ϕ, where a dot stands for a derivative with respectto proper time [10,11]. The radial motion can be computed via a normalizationcondition,

r2 = E2 − f(L2

r2+ δ1

)≡ E2 − Vgeo , (4)

where δ1 = 1, 0 for timelike or null geodesics, respectively. The null limit canbe approached letting E,L → ∞ and re-scaling all quantities appropriately.At large distances, r M , the motion of matter resembles that in Newton’s

6

theory. For instance, circular orbits, defined by r = r = 0, exist and are stable atlarge distance: small fluctuations in the motion restore the body to its originalposition. However, circular trajectories are stable only when r ≥ 6M , andunstable for smaller radii. The r = 6M surface defines the innermost stablecircular orbit (ISCO), and has an important role in controlling the inner partof the accretion flow onto compact objects.

2.2 Escape trajectories

robs , ψesc

6M, 270°

4M, 227°

3M, 180°

2.5M, 137°

2.0001 M, 2°

-6 -4 -2 0 2 4 6-4

-2

0

2

4

x/M

y/M

numerical8 Log[Cot [ψ/2]]

1 2 5 10 20 5010-4

10-3

10-2

0.1

1

10

ψ [degree ]

t rou

ndtrip

/M

Figure 1: Left: Critical escape trajectories of radiation in the Schwarzschildgeometry. A locally static observer (located at r = robs) emits photons isotropi-cally, but those emitted within the colored conical sectors will not reach infinity.Right: Coordinate roundtrip time of photons as a function of the emission angleψ > ψesc and for ε 1.

There are other GR effects on the motion of particles around compact ob-jects, beyond the appearance of ISCOs in spacetimes. One of them concerns thebehavior of light rays and how they are bent by the spacetime curvature. Thereexists a critical value of the angular momentum L ≡ KME for a light ray tobe able to escape to infinity. By requiring that a light ray emitted at a givenpoint will not find turning points in its motion, Eq. (4) yields Kesc = 3

√3 [10].

Suppose that the light ray is emitted by a locally static observer at r = r0. Inthe local rest frame, the velocity components of the photon are [12]

vlocalϕ =MK

r0

√f0 , vlocalr =

√1−K2M2

f0r20, (5)

where f0 ≡ f(r0) = 1 − 2M/r0. With this, one can easily compute the escapeangle, sinψesc = 3M

√3f0/r0. In other words, the solid angle for escape is

∆Ωesc = 2π

(1−

√1− 27M2(r0 − 2M)

r30

)∼ 27π

(r0 − 2M

8M

), (6)

7

where the last step is valid for ε 1. For angles larger than these, the lightray falls back and either hits the surface of the object, if there is one, or willbe absorbed by the horizon. The escape angle is depicted in Fig. 1 for differentemission points r0. The rays that are not able to escape reach a maximumcoordinate distance,

rmax ∼ 2M

(1 +

4f0M2

r20 sin2 ψ

). (7)

This result is accurate away from ψesc, whereas for ψ → ψesc the photon ap-proaches the photosphere (r = 3M) discussed in the next section. The coordi-nate time that it takes for photons that travel initially outward, but eventuallyturn back and hit the surface of the object, is shown in Fig. 1 as a function ofthe locally measured angle ψ, and is of order ∼M for most of the angles ψ, forε 1. We find a closed form expression away from ψcrit, which describes wellthe full range (see Fig. 1),

troundtrip ∼ 8M log(cot (ψ/2)) . (8)

When averaging over ψ, the coordinate roundtrip time is then 32M Cat/π ≈9.33M , for any ε 1, where “Cat” is Catalan’s constant.

2.3 Photospheres, ECOs and ClePhOs

Figure 2: Schematic classification of compact objects according to their com-pactness (in a logarithmic scale). Objects in the same category have similardynamical properties on a timescale τ .

Another truly relativistic feature is the existence of circular null geodesics,i.e., of circular motion for high-frequency electromagnetic or gravitational waves(GWs). In the Schwarzschild geometry, this can happen only at r = 3M . Thislocation defines a surface called the photosphere, or, in the context of equatorialslices, a light ring. The photosphere has a number of interesting properties, and

8

is useful to understand certain features of spacetimes. For example, it controlshow BHs look like when illuminated by accretion disks or stars, thus definingtheir so-called “shadow”. Imaging these shadows for the supermassive darksource SgrA∗ is the main goal of the Event Horizon Telescope [13, 14]. Thephotosphere also has a bearing on the spacetime response to any type of high-frequency waves, and therefore describes how high-frequency gravitational waveslinger close to the horizon.

At the photosphere, V ′′geo = −2E2/(3M2) < 0. Thus, circular null geodesicsare unstable: a displacement δ of null particles grows exponentially [11,15]

δ(t) ∼ δ0eλt , λ =

√−f2V ′′r

2E2=

1

3√

3M. (9)

A geodesic description anticipates that light or GWs may persist at or close tothe photosphere on timescales 3

√3M ∼ 5M . Because the geodesic calculation

is local, these conclusions hold irrespectively of the spacetime being vacuumall the way to the horizon or not. An ultracompact object with surface atr0 = 2M(1 + ε), with ε 1, would feature exactly the same geodesics andproperties close to its photosphere, provided that on timescales ∼ 15M the nullgeodesics did not have time to bounce off the surface. We are requiring threee-fold times for the instability to dissipate more than 99.7% of the energy of theinitial pulse. This amounts to requiring that

ε . εcrit ∼ 0.0165 . (10)

Thus, the horizon plays no special role in the response of high frequency waves,nor could it: it takes an infinite (coordinate) time for a light ray to reach thehorizon. The above threshold on ε is a natural sifter between two classes ofcompact, dark objects. For objects characterized by ε & 0.0165, light or GWscan make the roundtrip from the photosphere to the object’s surface and back,before dissipation of the photosphere modes occurs. For objects satisfying (10),the waves trapped at the photosphere relax away by the time that the wavesfrom the surface hit it back.

We can thus use the properties of the ISCO and photosphere to distinguishbetween different classes of models: an object is defined as compact if it featuresan ISCO, or in other words if its surface satisfies r0 < 6M . Accretion disksaround compact objects of the same mass should have similar characteristics.It is customary to define Exotic Compact Objects (ECOs) as those horizonlesscompact objects more massive than a neutron star. Among the compact objects,some feature photospheres. These could be called ultracompact objects (UCOs).Finally, those objects which satisfy condition (10) have a “clean” photosphere,and will be designated by ClePhOs. The early-time dynamics of ClePhOs isexpected to be the same as that of BHs. At late times, ClePhOs should displayunique signatures of their surface. The zoo of compact objects is summarizedin Fig. 2 2.

2This diagram can be misleading in special cases. Some objects with a surface at finite ε can

9

2.4 Quasinormal modes

Low-frequency gravitational or EM waves are not well described by null par-ticles. In the linearized regime, massless waves are all described by a masterpartial differential equation of the form [16]

f2∂2Ψ(t, r)

∂r2+ ff ′

∂Ψ(t, r)

∂r− ∂2Ψ(t, r)

∂t2− V (r)Ψ(t, r) = S(t, r) . (11)

The source term S(t, r) contains information about the cause of the disturbanceΨ(t, r). The information about the angular dependence of the wave is encodedin the way the separation was achieved, and involves an expansion in scalar,vector or tensor harmonics for different spins s of the perturbation. Theseangular functions are labeled by an integer l > |s|, and the effective potential is

V = f

(l(l + 1)

r2+ (1− s2)

2M

r3

), (12)

with s = 0,±1,±2 for scalar, vector or axial tensor modes. The s = ±2 equationdoes not describe completely gravitational perturbations, the potential for polargravitational perturbations is more complicated [16–18].

It is hard to extract general information from the PDE (11), in particularbecause its solutions depend on the source term and initial conditions. Wecan gain some insight by studying the source-free equation in Fourier space. Bydefining the Fourier transform through Ψ(t, r) = 1√

2π

∫e−iωtψ(ω, r)dω, one gets

the following ODEd2ψ

dz2+(ω2 − V

)ψ = 0 , (13)

where r is implicitly written in terms of the new “tortoise” coordinate

z = r + 2M log( r

2M− 1), (14)

such that z(r) diverges logarithmically near the horizon. In terms of z, Eq. (13)is equivalent to the time-independent Schrodinger equation in one dimensionand it reduces to the wave equation governing a string when M = l = 0. Tounderstand a string of length L with fixed ends, one imposes Dirichlet boundaryconditions and gets an eigenvalue problem for ω. The boundary conditions canonly be satisfied for a discrete set of normal frequencies, ω = nπ

L (n = 1, 2, ...).The corresponding wavefunctions are called normal modes and form a basis ontowhich one can expand any configuration of the system. The frequency is purelyreal because the associated problem is conservative.

nevertheless behave as ClePhOs. For example, constant density stars near the Buchdahl limitbehave as ClePhOs for axial modes: these do not couple to the fluid and travel unimpeded tothe center of the star. Their travel time can be considerably long, hence these stars act, forall purposes, as ClePhOs for these fluctuations. A specific example is discussed in footnote3. Nevertheless, it should be clear that these are contrived examples which are not meant todescribe quantum corrections.

10

If one is dealing with a BH spacetime, the appropriate conditions (requiredby causality) correspond to having waves traveling outward to spatial infinity(Ψ ∼ eiωz as z →∞) and inwards to the horizon (Ψ ∼ e−iωz as z → −∞) [seeFig. 3]. Due to backscattering off the effective potential (12), the eigenvaluesω are not known in closed form, but they can be computed numerically [16–18]. The fundamental l = 2 mode (the lowest dynamical multipole in GR) ofgravitational perturbations reads [19]

Mω ≡M(ωR + iωI) ≈ 0.373672− i0.0889623 . (15)

Remarkably, the entire spectrum is the same for both the axial or the polargravitational sector [17]. The frequencies are complex and are therefore calledquasinormal frequencies. Their imaginary component describes the decay intime of fluctuations on a timescale τ ≡ 1/|ωI |, and hints at the stability of thegeometry. Unlike the case of a string with fixed end, we are now dealing with anopen system: waves can travel to infinity or down the horizon and therefore it isphysically sensible that any fluctuation damps down. The corresponding modesare the quasinormal modes (QNMs), which in general do not form a completeset.

Boundary conditions play a crucial role in the structure of the QNM spec-trum. If a reflective surface is placed at r0 = 2M(1 + ε) & 2M , where (say)Dirichlet boundary conditions have to be imposed, the spectrum changes con-siderably. For ε = 10−6 the fundamental mode of the eigenspectrum reads

Mωpolar ≈ 0.13377− i2.8385× 10−7 (16)

Mωaxial ≈ 0.13109− i2.3758× 10−7 . (17)

Not only are the two gravitational sectors no longer isospectral but, more im-portantly, the perturbations have smaller frequency and are much longer lived,since a decay channel (the horizon) has disappeared.

More generically, for Dirichlet or Neumann boundary conditions at r = r0,the QNMs in the ε→ 0 limit read [20]

MωR ' M

2|z0|(pπ − δ) ∼ | log ε|−1 , (18)

MωI ' −βlsM

|z0|(2MωR)2l+2 ∼ −| log ε|−(2l+3) , (19)

where z0 ≡ z(r0) ∼ 2M log ε, p is an odd (even) integer for Dirichlet (Neumann)boundary conditions, δ is the phase of the wave reflected at r = r0, and βls =[(l−s)!(l+s)!(2l)!(2l+1)!!

]2[21, 22]. The above scaling can be understood in terms of modes

trapped between the peak of the potential (12) at r ∼ 3M and the “hardsurface” at r = r0 [23–26] [see Fig. 3]. Low-frequency waves are almost trappedby the potential, so their frequency scales as the size of the cavity (in tortoisecoordinates), ωR ∼ 1/z0, just like the normal modes of a string. The (small)imaginary part is given by waves which tunnel through the potential and reach

11

-10 0 10 20 300.000.050.100.15

z/M

V(z)M

2

outgoingtrapped

0.00

0.05

0.10

0.15V(z)M

2 outgoingingoing

Figure 3: Typical effective potential for perturbations of a Schwarzschild BH(top panel) and of an horizonless compact object (bottom panel). In the BHcase, QNMs are waves which are outgoing at infinity (z → +∞) and ingoing atthe horizon (z → −∞), whereas the presence of a potential well (provided eitherby a reflective surface, a centrifugal barrier at the center, or by the geometry)supports quasi-trapped, long-lived modes.

infinity. The tunneling probability can be computed analytically in the small-frequency regime and scales as |A|2 ∼ (MωR)2l+2 1 [21]. After a timet, a wave trapped inside a box of size z0 is reflected N = t/z0 times, and

its amplitude reduces to A(t) = A0

(1− |A|2

)N ∼ A0

(1− t|A|2/z0

). Since,

A(t) ∼ A0e−|ωI |t ∼ A0(1− |ωI |t) in this limit, we immediately obtain

ωR ∼ 1/z0 , ωI ∼ |A|2/z0 ∼ ω2l+3R . (20)

This scaling agrees with exact numerical results and is valid for any l and anytype of perturbation.

Clearly, a perfectly reflecting surface is an idealization. In certain models,only low-frequency waves are reflected, whereas higher-frequency waves probethe internal structure of the specific object [27, 28]. In general, the location ofthe effective surface and its properties (e.g., its reflectivity) can depend on theenergy scale of the process under consideration.

12

2.5 Quasinormal modes, photospheres, and echoes

The effective potential Vgeo for geodesic motion reduces to that of wave propa-gation V in the high-frequency, high-angular momentum (i.e., eikonal) regime.Thus, some properties of geodesic motion have a wave counterpart [11]. Theinstability of light rays along the null circular geodesic translates into someproperties of waves around objects compact enough to feature a photosphere.A wave description needs to satisfy “quantization conditions.” Since GWs arequadrupolar in nature, the lowest mode of vibration should satisfy

MωgeoR = 2

ϕ

t=

2

3√

3∼ 0.3849 . (21)

In addition the mode is damped, as we showed, on timescales 3√

3M . Overallthen, the geodesic analysis predicts

Mωgeo ∼ 0.3849− i0.19245 . (22)

This crude estimate, valid in principle only for high-frequency waves, matchesrather well even the fundamental mode of a Schwarzschild BH, Eq. (15).

Nevertheless, QNMs frequencies can be defined for any dissipative system,not only for compact objects or BHs. Thus, the association with photosphereshas limits. Such an analogy is nonetheless enlightening in the context of objectsso compact that they have photospheres and resemble Schwarzschild deep intothe geometry, in a way that condition (10) is satisfied [23,24,29].

For a BH, it becomes clear that the excitation of the spacetime modes hap-pens at the photosphere [15]. The vibrations excited there travel outwards topossible observers or down the event horizon. The structure of GW signals atlate times is therefore expected to be relatively simple. This is shown in Fig. 4,which refers to the scattering of a Gaussian pulse of axial quadrupolar modes offa BH. The pulse crosses the photosphere, and excites its modes. The ringdownsignal, a fraction of which travels to outside observers, is to a very good leveldescribed by the lowest QNMs. The other fraction of the signal generated atthe photosphere travels downwards and into the horizon. It dies off and has noeffect on observables at large distances.

Contrast the previous description with the dynamical response of ultracom-pact objects for which condition (10) is satisfied (i.e., a ClePhO) [cf. Fig. 4].The initial description of the photosphere modes still holds, by causality. Thus,up to timescales of the order |z0| ∼ −M log ε (the roundtrip time of radiationbetween the photosphere and r0) the signal is identical to that of BHs [23, 24].At later times, however, the pulse traveling inwards is bound to see the objectand be reflected either at its surface or at its center. In fact, this pulse is semi-trapped between the object and the light ring. Upon each interaction with thelight ring, a fraction exits to outside observers, giving rise to a series of echoesof ever-decreasing amplitude. From Eqs. (18)–(19), repeated reflections occurin a characteristic echo delay time [23,24]

τecho ∼ 4M | log ε| . (23)

13

However, the main burst is typically generated at the photosphere and hastherefore a frequency content of the same order as the BH QNMs (15). Theinitial signal is of high frequency and a substantial component is able to crossthe potential barrier. Thus, asymptotic observers see a series of echoes whoseamplitude is getting smaller and whose frequency content is also going down 3.At very late times, the QNM analysis is valid, and the signal is described byEqs. (18)–(19).

A formal description of the the frequency content of the sequence of echoes,and how they can be obtained with the help of the BH response, can be foundin Ref. [26]. A method to reconstruct the potential of the echo source, after themodes have been extracted was proposed in Ref. [33].

2.6 The role of the spin

When spherical symmetry is broken by angular momentum J , Birkhoff’s the-orem does not apply, so the exterior metric of a spinning object is not uniqueand technically much more difficult to compute.4 Furthermore, BH uniquenesstheorems do not apply in the presence of matter. In particular, a stationaryobject is not necessarily axisymmetric, unlike the Kerr solution which uniquelydescribes a BH in stationary configurations. Even restricting to axisymmet-ric configurations, the linear response of a spinning object depends also on theazimuthal number m (such that |m| ≤ l), which characterizes the angular de-pendence of the perturbations and yields a Zeeman-like splitting of the QNMfrequencies. The relation between null geodesics and BH QNMs in the eikonallimit is more involved but conceptually similar to the static case [36].

Assuming that the exterior spacetime approaches the Kerr geometry whenε→ 0, the linear response of a ClePhO can be studied with a toy model [37–40]in which the Kerr horizon is replaced by a reflective surface at r0 = r+(1 + ε),

where r± = M(1 ±√

1− χ2) are the locations of the horizons and χ = J/M2

is the dimensionless spin. In this case, Eq. (18) is modified by the replacementωR → ωR −mΩ on the left-hand side [20] (where Ω is the angular velocity ofthe object when ε→ 0), whereas Eq. (19) is replaced by [21]

MωI ' −βls|z0|

(2M2r+r+ − r−

)[ωR(r+ − r−)]

2l+1(ωR −mΩ) , (24)

3 An interesting, nontrivial example was worked out years ago in a different context [30].The example concerns axial GWs emitted during the scatter of point particles around ultra-compact, constant-density stars. The GW signal shows a series of visible, well-space echoesafter the main burst of radiation, which are associated to the quasi-trapped s-modes of ul-tracompact stars [31]. It turns out that all the results of that work fit very well our descrip-tion: axial modes do not couple to the fluid and travel free to the geometrical center of thestar, which is therefore the effective surface in this particular case. The time delay of theechoes in Fig. 1 of Ref. [30] is very well described by the GW’s roundtrip time to the center,τecho ∼ 27π

8ε−1/2M , where r0 = 9

4M(1 + ε) is the radius of the star and r0 = 9

4M is the

Buchdahl’s limit [32].4Nonetheless, there are indications that all multipole moments of the external spacetime

approach those of a Kerr BH as ε → 0 [34, 35]. In this limit, it is natural to expect that theexterior geometry is extremely close to Kerr, unless some discontinuity occurs in the BH limit.It would be extremely important to confirm or confute this conjecture.

14

ClePhOBH

0 100 200 300 400 500 600-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

t/M

-ψ(r=50

M,t)

τecho ∼4M |log ϵ|

200 400 600 800 1000-0.2-0.10.00.10.2

Figure 4: Ringdown waveform for a BH (dashed black curve) compared to aClePhO (solid red curve) with a reflective surface at r0 = 2M(1 + ε) withε = 10−11. We considered l = 2 axial gravitational perturbations and a Gaussianwavepacket ψ(r, 0) = 0, ψ(r, 0) = e−(z−zm)2/σ2

(with zm = 9M and σ = 6M)as initial condition. Note that each subsequent echo has a smaller frequencycontent and that the damping of subsequent echoes is much larger than thelate-time QNM prediction (e−ωIt with ωIM ∼ 4× 10−10 for these parameters).Data available online [19].

where now z0 ∼M [1+(1−χ2)−1/2] log ε. The result (24) shows how angular mo-mentum can bring about substantial qualitative changes. The spacetime is un-stable for ωR(ωR−mΩ) < 0 (i.e., in the superradiant regime [22]), on a timescaleτinst ≡ 1/ωI . This phenomenon is called ergoregion instability [22,41,42]. In theε→ 0 limit and for sufficiently large spin, ωR ∼ mΩ and ωI ∼ | log ε|−1. Noticethat such description is only valid at small frequencies, and therefore becomesincreasingly less accurate at large spins and away from the instability threshold(where ωR, ωI ∼ 0). For large spin the result is more complex and can be foundin Ref. [43]. Furthermore, in the superradiant regime the “damping” factor,ωI/ωR > 0, so that, at very late times (when the pulse frequency content isindeed described by these formulas), the amplitude of the QNMs increases dueto the instability. Such increase is anyway small, for example ωI

ωR≈ 4 × 10−6

when ε = 0.001, l = 2 and χ = 0.7.In the spinning case, the echo delay time (23) reads

τecho ∼ 2M [1 + (1− χ2)−1/2] log ε , (25)

which corresponds to the period of the corotating mode, τecho ∼ (ωR −mΩ)−1.Note that, as we explained earlier, the signal can only be considered as a se-

ries of well-defined pulses at early stages. In this stage, the pulse still contains a

15

substantial amount of high-frequency components which are not superradiantlyamplified. Thus, amplification will occur at late times; the early-time evolutionof the pulse generated at the photosphere is more complex.

2.7 BHs in binaries: tidal heating and tidal deformability

The properties of an event horizon have also important consequences for thedynamics of binary systems containing a BH. As previously discussed, a spinningBH absorbs radiation of frequency ω > mΩ, but amplifies radiation of smallerfrequency. In this respect, BHs are dissipative systems which behave just likea Newtonian viscous fluid [44, 45]. Dissipation gives rise to various interestingeffects in a binary system – such as tidal heating [46,47], tidal acceleration, andtidal locking – the Earth-Moon system being no exception due to the friction ofthe oceans with the crust.

For low-frequency circular binaries, the energy flux associated to tidal heat-ing at the horizon corresponds to the rate of change of the BH mass [48,49],

M = EH ∝Ω5K

M2(ΩK − Ω) , (26)

where ΩK 1/M is the orbital angular velocity and the (positive) prefactordepends on the masses and spins of the two bodies. Thus, tidal heating isstronger for highly spinning bodies relative to the nonspinning by a factor ∼Ω/ΩK 1.

The energy flux (26) leads to a potentially observable phase shift of GWsemitted during the inspiral. Thus, it might be argued that an ECO binarycan be distinguished from a BH binary, because EH = 0 for the former [50].However, the trapping of radiation in ClePhOs can efficiently mimic the effectof a horizon [50]. In order for absorption to affect the orbital motion, it isnecessary that the time radiation takes to reach the companion, Trad, be muchlonger than the radiation-reaction time scale due to heating, TRR ' E/EH ,where E ' − 1

2M(MΩK)2/3 is the binding energy of the binary (assuming equalmasses). For BHs, Trad → ∞ because of time dilation, so that the conditionTrad TRR is always satisfied. For ClePhOs, Trad is of the order of the GWecho delay time, Eq. (23), and therefore increases logarithmically as ε → 0.Thus, an effective tidal heating might occur even in the absence of an eventhorizon if the object is sufficient compact. The critical value of ε increasesstrongly as a function of the spin. For orbital radii larger than the ISCO, thecondition Trad TRR requires ε 10−88 for χ . 0.8, and therefore even Planckcorrections at the horizon scale are not sufficient to mimic tidal heating. Thisis not necessarily true for highly spinning objects, for example Trad TRR atthe ISCO requires ε 10−16 for χ ≈ 0.9.

Finally, the nature of the inspiralling objects is also encoded in the way theyrespond when acted upon by the external gravitational field of their companion –through their tidal Love numbers [51]. An intriguing result in classical GR is

16

the fact that the tidal Love numbers of a BH are precisely zero [52–54]. Onthe other hand, those of ECOs are small but finite [55]. In particular, thetidal Love numbers of ClePhOs vanish logarithmically in the BH limit. Thus,any measurement of the tidal Love number k translates into an estimate of thedistance of the ECO surface from its Schwarzschild radius,

ε ∼ e−1/k . (27)

Owing to the above exponential dependence, a measurement of the tidal Lovenumber of an ECO at the level of k ≈ O(10−3) already probes Planck distancesaway from the gravitational radius rg [55]. This level of accuracy is in principlewithin reach future GW detectors [50].

17

3 Beyond vacuum black holes

“Mumbo Jumbo is a noun and is the name of a grotesque idol said tohave been worshipped by some tribes. In its figurative sense, MumboJumbo is an object of senseless veneration or a meaningless ritual.”

Concise Oxford English Dictionary

To entertain the possibility that dark, massive, compact objects are not BHs,requires one to discuss some outstanding issues. One can take two differentstands on this topic:

i. a pragmatic approach of testing the spacetime close to compact, darkobjects, irrespective of their nature, by devising model-independent ob-servations that yield unambiguous answers.

ii. a less ambitious and more theoretically-driven approach, which starts byconstructing objects that are very compact, yet horizonless, within someframework. It proceeds to study their formation mechanisms and stabilityproperties; then discard solutions which either do not form or are unsta-ble on relatively short timescales; finally, understand the observationalimprints of the remaining objects, and how they differ from BHs’.

In practice, when dealing with outstanding problems where our ignoranceis extreme, both approaches should be used simultaneously. Indeed, using con-crete models can sometimes be a useful guide to learn about broad, model-independent signatures. As it will hopefully become clear, one could designexotic horizonless models which mimic all observational properties of a BH witharbitrary accuracy. While the statement “BHs exist in our Universe” is funda-mentally unfalsifiable5, alternatives can be ruled out or confirmed to exist witha single observation, just like Popper’s black swans.

3.1 Are there alternatives?

A nonexhaustive summary of possible objects which could mimic BHs is shownin Table 1. Stars made of constant density fluids are perhaps the first knownexample of compact configurations, the literature on the subject is too long tolist. In GR, isotropic static spheres made of ordinary fluid satisfy the Buchdahllimit on their compactness, 2M/r0 < 8/9 [32], and can thus never be a ClePhO.Interestingly, polytrope UCO stars always have superluminal sound speed [57].

Compact solutions can be built with fundamental, massive bosonic fields.The quest for self-gravitating bosonic configurations started in an attempt to

5Arguably, this is true for any statement about the existence of a physical entity in ourUniverse, but this does not prevent us to accumulate more and more evidence to support agiven model or theory, nor to rule out competitors.

18

understand if gravity could produce “solitons” through the nonlinearities ofthe field equations. Such configurations are broadly referred to as boson stars.Boson stars have attracted considerable interest as light scalars are predicted tooccur in different scenarios, and ultralight scalars can explain the dark matterpuzzle [136]. When fastly spinning, such stars can be extremely compact. Thereseem to be no studies on the classification of such configurations (there aresolutions known to display photospheres, but it is unknown whether they canbe as compact as ClePhOs).

The other objects in the list require either unknown matter or large quantumeffects. For example, a specific model of a wormhole was discussed carefully inthe context of “BH foils” [102]. In the nonspinning case, the geometry of thismodel is described by

ds2 = − (f + λ) dt2 +dr2

f+ r2dΩ2 . (28)

The constant λ is assumed to be extremely small, λ ∼ e−M2/`2P where `P is

the Planck length. In the context of GR, this solution requires matter violatingthe null energy condition. Other objects, such as fuzzballs, were introducedas a microstate description of BHs in string theory. In this setup the indi-vidual microstates are horizonless and the horizon arises as a coarse-graineddescription of the microstate geometries. Phenomena such as Hawking radia-tion can be recovered, in some instances, from classical instabilities [111, 137].“Gravitational-vacuum stars” or gravastars [4, 89] are ultracompact configura-tions supported by a negative pressure, which might arise as an hydrodynamicaldescription of one-loop QFT effects in curved spacetime, so they do not neces-sarily require exotic new physics [138]. In these models, the Buchdahl limit isevaded because the internal effective fluid is anisotropic and also violates theenergy conditions [139]. Gravastars have been recently generalized to includeanti-de Sitter cores, in what was termed AdS bubbles, and which may allow forholographic descriptions [98].

For most objects listed in Table 1 which are inspired by quantum-gravity cor-rections, the changes in the geometry occur deep down in the strong field regime.Some of these models – including also black stars [124], superspinars [115] orcollapsed polymers [119,120] – predict that large quantum backreaction shouldaffect the horizon geometry even for macroscopic objects. In these models, ourparameter ε is naturally of the order ∼ O(`P /M) ∈ (10−39, 10−46) for massesin the range M ∈ (10, 108)M. The 2− 2-hole model predicts even more com-pact objects, with ε ∼ (`P /M)2 ∈ (10−78, 10−92) [118]. In all these cases, bothquantum-gravity or microscopic corrections at the horizon scale select ClePhOsas well-motivated alternatives to BHs.

Despite a number of supporting arguments – some of which urgent and wellfounded – it is important to highlight that there is no horizonless ClePhO forwhich we know sufficiently well the physics at the moment.

19

3.2 Formation and evolution

Although supported by sound arguments, the vast majority of the alternativesto BHs are, at best, incompletely described. Precise calculations (and ofteneven a rigorous framework) incorporating the necessary physics are missing.

One notable exception to our ignorance are boson stars. These configurationsare known to arise, generically, out of the gravitational collapse of massivescalars. Their interaction and mergers can be studied by evolving the Einstein-Klein-Gordon system, and there is evidence that accretion of less massive bosonstars makes them grow and cluster around the configuration of maximum mass.In fact, boson stars have efficient gravitational cooling mechanisms that allowthem to avoid collapse to BHs and remain very compact after interactions [68,73,75].

The remaining objects listed in Table 1 were built in a phenomenologicalway or they arise as solutions of Einstein equations coupled to exotic matterfields. For example, models of quantum-corrected objects do not include allthe (supposedly large) local or nonlocal quantum effects that might preventcollapse from occurring. In the absence of a complete knowledge of the missingphysics, it is unlikely that a ClePhO forms out of the merger of two ClePhOs.These objects are so compact that at merger they will be probably engulfed bya common apparent horizon. The end product is, most likely, a BH. On theother hand, if large quantum effects do occur, they would probably act on shorttimescales to prevent apparent horizon formation. In some models, Planck-scale dynamics naturally leads to abrupt changes close to the would-be horizon,without fine tuning [118]. Likewise, in the presence of (exotic) matter or if GRis classically modified at the horizon scale, Birkhoff’s theorem no longer holds,and a star-like object might be a more natural outcome than a BH. In summary,with the exception of boson stars, we do not know how or if UCOs and ClePhOsform in realistic collapse or merger scenarios.

20

Model Taxonomy Formation Stability EM signatures GWs

Fluid stars UCOs 7 X X X[18, 25,33,56–58] [18, 25,30,56]

Anisotropic stars ClePhOs 7 X X X[59–61] [62,63] [35,61,64] [35,64]

Boson stars & UCOs, (ClePhOs?) X X X Xoscillatons [65–72] [68, 71,73–75] [70, 76–80] [81–83] [24,50,55,84–88]

Gravastars COs – ClePhOs 7 X X ∼[4, 89] [79] [90–92] [23–25,33,50,55,92–97]

AdS bubbles UCOs – ClePhOs 7 X ∼ 7

[98] [98] [98]

Wormholes ClePhOs 7 X X ∼[99–103] [104,105] [106–109] [23,50,55]

Fuzzballs ClePhOs 7 7 7 ∼[5, 6] (but see [110–113]) (but see [23,24,114])

Superspinars COs – ClePhOs 7 X 7 ∼[115] [37,116] (but see [117]) [23,24]

2 − 2 holes ClePhOs 7 7 7 ∼[118] (but see [118]) (but see [118]) [23, 24]

Collapsed ClePhOs 7 7 7 ∼polymers [119,120] (but see [119,121]) [121]

Quantum bounces / ECO – ClePhOs 7 7 7 ∼black stars [7, 8, 122–125] (but see [123,126]) [125]

Quantum stars∗ UCOs – ClePhOs 7 7 7 7

[127,128]Fire-walls∗ ClePhOs 7 7 7 ∼

[129–131] [24, 132]

Table 1: Catalogue of some proposed horizonless compact objects. A X tickmeans that the topic was addressed. With the exception of boson stars, how-ever, most of the properties are not fully understood yet. An asterisk ∗ standsfor the fact that these objects are BHs, but could have phenomenology similarto the other compact objects in the list. This table does not include models ofquantum-corrected BHs (e.g., [133–135]), even when the latter predict large cor-rections near the horizon, because also in this case the object is a BH (althoughdifferent from Kerr).

21

3.3 On the stability problem

“There is nothing stable in the world; uproar’s your only music.”

John Keats, Letter to George and Thomas Keats, Jan 13 (1818)

Appealing solutions are only realistic if they form and remain as long-termstable solutions of the theory. In other words, solutions have to be stable whenslightly perturbed or they would not be observed. There are strong indicationsthat the exterior Kerr spacetime is stable, although a rigorous proof is still miss-ing [140]. There are good reasons to believe that some – if not all – horizonlesscompact solutions are linearly or nonlinearly unstable.

Some studies of linearized fluctuations of ultracompact objects are given inTable 1. We will not discuss specific models, but we would like to highlighttwo general results. Linearized gravitational fluctuations of any nonspinningUCO are extremely long-lived and decay no faster than logarithmically [56,112, 113, 141]. Indeed, such perturbations can be again understood in terms ofmodes quasi-trapped within the potential barrier shown in Fig. 3: they requirea photosphere but are absent in the BH case. For a ClePhO, these modes arevery well approximated by Eqs. (18)–(19). The long damping time of thesemodes has led to the conjecture that any UCO is nonlinearly unstable and mayevolve through a Dyson-Chandrasekhar-Fermi type of mechanism [56,141]. Theendstate is unknown, and most likely depends on the equation of the state of theparticular UCO: some objects may fragment and evolve past the UCO regioninto less compact configurations, via mass ejection, whereas other UCOs maybe forced into gravitational collapse to BHs.

The above mechanism is supposed to be active for any spherically symmetricUCO, and also on spinning solutions. However, it is nonlinear in nature. On theother hand, UCOs (and especially ClePhOs) can develop negative-energy regionsonce spinning. In such a case, there is a well-known linear instability, dubbedas ergoregion instability. The latter affects any horizonless geometry with anergoregion [39,41,42,79,142] and is deeply connected to superradiance [22].

In summary, there is good evidence that UCOs are linearly or nonlinearlyunstable. Unfortunately, the effect of viscosity is practically unknown [56], andso are the timescales involved in putative dissipation mechanisms that mightquench this instability. On the other hand, even unstable solutions are relevantif the timescale is larger than any other dynamical scale in the problem 6. Thenonlinear mechanism at work for spherically symmetric spacetimes presumablyacts on very long timescales only; a model problem predicts an exponentialdependence on the size of the initial perturbation [143]. At least in some portionof the parameter space the ergoregion-instability timescale is very large [39,41,79]. From Eq. (24), we can estimate the timescale of the instability of a spinning

6After all, “we are all unstable”.

22

ClePhO,

τ ≡ 1

ωI∼ −| log ε|1 + (1− χ2)−1/2

2βls

(r+ − r−r+

)[ωR(r+ − r−)]

−(2l+1)

ωR −mΩ. (29)

As previously discussed, a spinning ClePhO is (superradiantly) unstable onlywhen Ω > ωR/m. For example, for l = m = s = 2 and χ = 0.7, the aboveformula yields

τ ∈ (5, 1)

(M

106M

)yr when ε ∈ (10−45, 10−22) . (30)

Generically, the ergoregion instability acts on timescales which are parametri-cally longer than the dynamical timescale, ∼M , of the object. Given such longtimescales, it is likely that the instability can be efficiently quenched by somedissipation mechanism of nongravitational nature, although this effect would bemodel-dependent [39]. Furthermore, it is possible that the endstate of the in-stability is simply an ECO spinning at the superradiant threshold, Ω = ωR/m.In such case, the instability would only rule out highly-spinning ECOs in acertain compactness range. Likewise, EM or GW observations indicating sta-tistical prevalence of slowly-spinning compact objects, across the entire massrange, could be an indication for the absence of an horizon.

Finally, there are indications that instabilities of UCOs are merely the equiv-alent of Hawking radiation for these geometries, and that therefore there mightbe a smooth transition in the emission properties when approaching the BHlimit [102,111].

23

4 Have exotic compact objects been already ruledout by electromagnetic observations?

“And here - ah, now, this really is something a little recherche.”

Sherlock Holmes, The Musgrave Ritual, Sir Arthur Conan Doyle

Before the GW revolution, questions about the observational evidence forBHs were initially posed in the context of EM observations. A concise overviewof the arguments is presented in Refs. [144, 145], concluding that “it is funda-mentally impossible to give an observational proof for the existence of a BHhorizon.” Surprisingly however, there have been recent claims [146–148] thatobservations of some objects (most notably the compact radio source Sgr A∗) allbut rule out other candidates, including those whose surface is a Planck-distanceaway from the horizon (i.e., ε ∼ 10−45 for Sgr A∗, thus qualifying as ClePhOsaccording to our classification). The argument can, roughly, be described asfollows:

i. There are systems with a very low accretion rate and very low luminosity,for which the central object is very dark.

ii. The coordinate time that a particle in the accretion disk takes to reachthe surface scales as Ttravel ∼ M | log ε|. Even for Planck-sized ε this timeis very small.

iii. Assumption: because the above timescale is so small, a thermodynamicand dynamic equilibrium must be established between the accretion diskand the central object, on relatively short timescales.

iv. Assumption: Because there is equilibrium, the central object must bereturning in EM radiation most of the energy that it is taking in from thedisk.

v. Because we see no such radiation, the premise that there is a surface mustbe wrong.

The arguments above fail to take into account important physics:

Timescale to reach equilibrium. Strong lensing can prevent equilibriumfrom being achieved around ultracompact objects in sufficiently short time.Consider matter in the accretion disk falling in, releasing scattered radiationon the surface of the object which is then observed by our detectors. Sup-pose, for the sake of the argument, that once hitting the surface, it is scatteredisotropically. Then, as discussed in Section 2.2, only a fraction ∼ ε is able toescape during the first interaction with the star, cf. Eq. (6). The majority ofthe radiation, will fall back onto the surface after a time troundtrip ∼ 9.3M given

24

by the average of Eq. (8) 7. Suppose one injects, instantaneously, an energy δMonto the object. Then, after a time T , the energy emitted to infinity duringN = T/troundtrip interactions reads

∆E ∼[1− (1− ε)N

]δM . (31)

Clearly, ∆E → δM as N →∞ since all energy will eventually escape to infinity.However, ∆E ∼ εNδM if εN 1, i.e. when

ε 10−16(

M

106M

)(tHubble

T

), (32)

where we have been extremely conservative in normalizing T by the age of theUniverse, tHubble ≈ 1010 yr. The luminosity of ClePhOs satisfying the aboveproperty is roughly

E ∼ 10−17( ε

10−16

)(δMM

). (33)

When ε 10−16, this quantity is minute and it is impossible to achieve equilib-rium on any meaningful timescale. Indeed, any injected energy δM is eventuallyradiated in a timescale longer than tHubble whenever Eq. (32) is satisfied. Thus,gravastars, fuzzballs, and other objects inspired by quantum effects at the hori-zon scale (ε . 10−45) are not ruled out as exotic supermassive objects.Cascade into other channels. Even if the object were returning all ofthe incoming radiation on a sufficiently short timescale, a sizeable fraction ofthis energy could be in channels other than EM. Let us estimate the center-of-mass energy in collisions between matter being accreted and the compactobject. The infalling matter is assumed to be on radial free fall with four-velocity vµ(1) = (E/f,−

√E2 − f, 0, 0). For particles at the surface of the object

vµ(2) = (√f, 0, 0, 0). When the particles collide, their CM energy reads [149],

ECM = m0

√2√

1− gµνvµ(1)vµ(2) ∼

m0

√2E

ε1/4, (34)

with m0 the mass of the colliding particles (assumed to be equal). Therefore,the Lorentz factor of the colliding particles, in the CM frame, is γ 3×105 forε 10−22, and γ ∼ 1011 for Planck-scale modifications of supermassive objects.Therefore, ECM is orders of magnitude above the maximum energy achievableat the LHC. At these CM energies, it is possible that new degrees of freedomare excited (some of which might even be the unknown physics that forms thesecompact objects in the first place). It is well possible that if new fields existthey carry a substantial fraction of the luminosity (while going undetected in our

7One might wonder if the trapped radiation bouncing back and forth the surface of theobject might not interact with the accretion disk. As we showed in Section 2.2, this does nothappen, as the motion of trapped photons is confined to within the photosphere.

25

telescopes)8. Even without advocating new physics beyond the 10 TeV scale,extrapolation of known hadronic interactions to large energies suggests thatabout 20% of the collision energy goes into neutrinos, whose total energy is asizeable fraction of that of the photons emitted in the process [151].

In summary, we believe that some observations of supermassive objects mostlikely are compatible with ClePhOs (in particular they do not rule out gravastarsnor quantum corrections at the horizon scale); these same observations areprobably not compatible with the remaining of the UCO parameter space.

No argument whatsoever can be made to rule out exotic compact objects ofarbitrary compactness, simply because the transition to the BH limit should becontinuous. Therefore, any argument suggesting that exotic compact objectsare excluded for any ε has to be taken with great care.

8Incidentally, gravitational radiation makes only negligible contribution to the total flux,even for such large CM energies. The ratio of EM to gravitational radiation generated duringthese processes was estimated to be [150]

EEM

Egrav∼

log γ

γ

q

m0, (35)

with q/m0 the charge-to-mass ratio of the colliding particles. Thus, even at such large CMenergies, EM radiation is vastly larger than gravitational radiation, because q/m0 ∼ 1019 forprotons.

26

5 Testing the nature of BHs with GWs

“It is well known that the Kerr solution provides the unique solu-tion for stationary BHs in the universe. But a confirmation of themetric of the Kerr spacetime (or some aspect of it) cannot even becontemplated in the foreseeable future.”

S. Chandrasekhar, The Karl Schwarzschild Lecture,Astronomische Gesellschaft, Hamburg (September 18, 1986)

Testing the nature of dark, compact objects with EM observations is plaguedwith several difficulties. Some of these are tied to the incoherent nature of theEM radiation in astrophysics, and the amount of modeling and uncertaintiesassociated to such emission. Other problems are connected to the absorptionby the interstellar medium. As discussed in the previous section, testing quan-tum or microscopic corrections at the horizon scale with EM probes is verychallenging.

The historical detection of GWs by aLIGO [152] opens up the exciting pos-sibility of testing gravity in extreme regimes with unprecedented accuracy [50,132, 153–156]. GWs are generated by coherent motion of massive sources, andare therefore subjected to less modeling uncertainties (they depend on far fewerparameters) relative to EM probes. The most luminous GWs come from verydense sources, but they also interact very feebly with matter, thus providingthe cleanest picture of the cosmos, complementary to that given by telescopesand particle detectors.

Compact binaries are the preferred sources for GW detectors (and in factwere the source of the first detected events [152,157]). The GW signal from com-pact binaries is naturally divided in three stages, corresponding to the differentcycles in the evolution driven by GW emission [158–160]: the inspiral stage,corresponding to large separations and well approximated by post-Newtoniantheory; the merger phase when the two objects coalesce and which can only bedescribed accurately through numerical simulations; and finally, the ringdownphase when the merger end-product relaxes to a stationary, equilibrium solutionof the field equations [16,160,161].

All three stages provide independent, unique tests of gravity and of compactGW sources. The GW signal in the two events reported so far is consistentwith them being generated by BH binaries, and with the endstate being a BH.To which level, and how, are alternatives consistent with current and futureobservations? This question was recently addressed by several authors and maybe divided into two different schemes (see Table 2).

5.1 Smoking guns: echoes and resonances

For binaries composed of ClePhOs, the GWs generated during inspiral andmerger is expected to be very similar to those by a corresponding BH binary with

27

BH ECO ClePhO

rin

gdown

GW echoes 7 X(only UCOs) X(τecho ∼M | log ε|)Modified prompt ringdown 7 X 7

Extra modes 7 X X

insp

iral Multipolar structure (2PN) δMl = δSl = 0 δMl 6= 0, δSl 6= 0 δMl ' 0, δSl ' 0

Tidal heating (2.5− 4PN) X 7 7Tidal Love number (5PN) k = 0 k . O(kNS) k ∼ [log ε]−1

Resonances 7 [80, 96] ωM ∼ [log ε]−1

Table 2: Summary of the properties of exotic compact objects that affect theGW signal relative to BHs in the ringdown phase and in the inspiral phase ata given post-Newtonian (PN) order. In the multipolar structure, δMl and δSlare the corrections to the mass and current multipole moments relative to theno-hair relation for a Kerr BH in isolation [162], Ml + iSl = M l+1 (iχ)

l. Tidal

absorption at the horizon enters at 2.5PN (4PN) order for spinning (nonspin-ning) BHs. The typical tidal Love numbers of a neutron star are denoted bykNS.

the same mass and spin. The arguments were detailed in Section 2. However,clear distinctive features appear due to the absence of a horizon. These featuresare:

• The appearance of late-time echoes in the waveforms [23,24,29,40]. Afterthe merger, ClePhOs give rise to two different ringdown signals: the firststage is dominated by the photosphere modes, and is indistinguishablefrom a BH ringdown. However, after a time τecho ∼ M | log ε| (see Sec-tion 2.4), the waves trapped in the “photosphere+ClePhO” cavity startleaking out as echoes of the main burst. This is a smoking-gun for newphysics, potentially reaching microscopic or even Planckian corrections atthe horizon scale [23,24].

Strategies to dig GW signals containing “echoes” out of noise are notfully under control, first efforts are underway [163, 164]. The ability todetect such signals depends on how much energy is converted from themain burst into echoes (i.e., on the relative amplitude between the firstecho and the prompt ringdown signal in Fig. 4). Define the ratio of en-ergies to be γecho. Then, the signal-to-noise ratio ρ necessary for echoesto be detectable separately from the main burst, for a detection thresh-old of ρ = 8, is ρprompt ringdown & 80√

γecho(%). In such case the first echo

is detectable with a simple ringdown template (an exponentially dampedsinusoid [165]). Space-based detectors will see prompt ringdown eventswith very large ρ [166]. For γecho = 20%, we estimate that the plannedspace mission LISA [167] will see at least one event per year, even forthe most pessimistic population synthesis models used to estimate therates [166]. The proposed Einstein telescope [168] or Voyager-like [169]third-generation Earth-based detectors will also be able to distinguish

28

ClePhOs from BHs with such simple-minded searches. The event ratesfor LIGO are smaller, and more sophisticated searches need to be im-plemented. Preliminary analysis of GW data using the entire echoingsequence was reported recently [38, 170, 171]. More detailed characteriza-tion of the echo waveform (e.g., using Green’s function techniques [26] andaccounting for spin effects [40]) is necessary to reduce the systematics indata analysis. In several models of ClePhOs, γecho is large enough to pro-duce effects that – also owing to the logarithmic dependence of τecho forall models known so far – are within reach of near-future GW detectors,even if the ClePhO corrections occurs at the Planck scale [23,24]. This isa truly remarkably prospect. As the sensitivity of GW detectors increases,the absence of echoes might be used to rule out many quantum-gravitymodels completely, and to push tests of gravity closer and closer to thehorizon scale, as now routinely done for other cornerstones of GR, e.g. intests of the equivalence principle [155,172].

• Model-dependent fluid modes are also excited. Due to redshift effects,these will presumably play a subdominant role in the GW signal.

• Certain models of ECOs arise naturally in effective theories with extragravitational degrees of freedom [173]. In such case, the detection of extrapolarizations (as achievable in the future with a global network of inter-ferometers) might provide evidence for new physics at the horizon scale.

• Resonant mode excitation during inspiral. The echoes at late times arejust vibrations of the ClePhO. These vibrations have relatively low fre-quency and can, in principle, also be excited during the inspiral processitself, leading to resonances in the motion as a further clear-cut signal ofnew physics [80,87,96].

• Spin-mass distribution of compact objects skewed towards low spin, acrossthe mass range. The development of the ergoregion instability depletesangular momentum from spinning ClePhOs, independently of their mass.Although the effectiveness of such process is not fully understood, it wouldlead to slowly-spinning objects as a final state.

• It is also possible that, at variance with the BH case, the merger of twoClePhOs can be followed by a burst of EM radiation associated with thepresence of high-density matter during the collision. Although such emis-sion might be strongly redshifted, searches for EM counterparts of candi-date BH mergers can provide another distinctive signature of new physicsat the horizon scale.

5.2 Precision physics: QNMs, tidal deformability, heat-ing, and multipolar structure

The GW astronomy era will also gradually open the door to precision physics,for which smoking signs may not be necessary to test new physics.

29

• If the product of the merger is an ECO but not a ClePhO, it will simplyvibrate differently from a BH. Thus, precise measurements of the ring-down frequencies and damping times allow one to test whether or notthe object is a BH [174, 175]. Such tests are in principle feasible for wideclasses of objects, including boson stars [80,87,175], gravastars [95,97,139],wormholes [176,177], or other quantum-corrected objects [121,125].

• The merger phase can also provide information about the nature of thecoalescing objects. ECOs which are not sufficiently compact will likelydisplay a merger phase resembling that of a neutron-star merger ratherthan a BH merger [24,178]. The situation for ClePhOs is unclear since nosimulations of a full coalescence are available.

• The absence of a horizon affects the way in which the inspiral stage pro-ceeds. In particular, three different features may play a role, all of whichcan be used to test the BH-nature of the objects.

– No tidal heating for ECOs. Horizons absorb incoming high frequencyradiation, and serve as sinks or amplifiers for low-frequency radiationable to tunnel in (see Section 2.7). UCOs and ClePhOs, on theother hand, are not expected to absorb any significant amount ofGWs. Thus, a “null-hypothesis” test consists on using the phase ofGWs to measure absorption or amplification at the surface of theobjects [50]. LISA-type GW detectors [167] will place stringent testson this property, potentially reaching Planck scales near the horizonand beyond [50].

– Nonzero tidal Love numbers for ECOs. In a binary, the gravitationalpull of one object deforms its companion, inducing a quadrupolemoment proportional to the tidal field. The tidal deformability isencoded in the Love numbers, and the consequent modification ofthe dynamics can be directly translated into GW phase evolution athigher-order in the post-Newtonian expansion [179]. It turns out thatthe tidal Love numbers of a BH are zero [52, 53, 180–183], allowingagain for null tests. By devising these tests, existing and upcomingdetectors can rule out or strongly constraint boson stars [50, 55, 88,184] or even generic ClePhOs [50,55].

– Different multipolar structure. Spinning objects in a binary are alsoexpected to possess multipole moments that differ from those of theKerr geometry. This property is true at least for UCOs and for lesscompact objects. The impact of the multipolar structure on the GWphase will allow one to estimate and constrain possible deviationsfrom the Kerr geometry [185].

30

6 Summary

“The important thing is not to stop questioning. Curiosity has itsown reason for existing.”

Albert Einstein, From the memoirs of William Miller, an editor,quoted in Life magazine, May 2, 1955; Expanded, p. 281

Thomson’s atomic model was carefully constructed, and tested theoreticallyfor inconsistencies9. Rutherford’s incursion into scattering of α particles werenot meant to disprove the model, they were aimed at testing its accuracy. Ac-cording to Marsden, “...it was one of those ’hunches’ that perhaps some effectmight be observed, and that in any case that neighbouring territory of thisTom Tiddler’s ground might be explored by reconnaissance. Rutherford wasever ready to meet the unexpected and exploit it, where favourable, but he alsoknew when to stop on such excursions” [187].

All the current observational evidence gathered around massive, compactand dark objects is compatible with the BH hypothesis. There is no equally-well-motivated alternative, satisfying known laws of physics and composed ofordinary matter, which is compatible with those same observations. Despitethis, there are long-standing problems associated with horizons and singulari-ties, which hint at some inconsistency between classical gravity and quantummechanics at the scale of the horizon. Furthermore, since the majority of thematter in our Universe is unknown, we cannot exclude that dark compact ob-jects made of exotic matter are lurking in the cosmos. Even, and if, these issuesare resolved within classical physics, some outstanding questions remain. Whatevidence do we have that compact objects are BHs? How deep into the poten-tial can observations probe, and up to where are they still compatible with thecurrent paradigm? At one hand’s reach, we have the possibility to dig deeperand deeper into the strong-field region of compact objects, for free. As the sen-sitivity of EM and GW detectors increases, so will our ability to probe regionsof ever increasing redshift. Perhaps the strong-field region of gravity holds thesame surprises that the strong-field EM region did?

9It is amusing that some alternative models – which we now know to be closer to Ruther-ford’s model – were studied, shown to be unstable and therefore ruled out [186].

31

Acknowledgments. We are indebted to Carlos Barcelo, Silke Britzen, RamBrustein, Raul Carballo-Rubio, Bob Holdom, Luis Garay, Marios Karouzos,Gaurav Khanna, Joe Keir, Kostas Kokkotas, Claus Laemmerzahl, Jose Lemos,Caio Macedo, Samir Mathur, Emil Mottola, Ken-ichi Nakao, Richard Price,Ana Sousa, Bert Vercnocke, Frederic Vincent, Sebastian Voelkel and AaronZimmerman for providing detailed feedback, useful references and for suggest-ing corrections to an earlier version of the manuscript. We thank also themany other colleagues who provided constructive criticism and comments, in-cluding all the participants of the Models of Gravity: Black Holes, NeutronStars and the structure of space-time meeting in Oldenburg, of the Foundationsof the theory of Gravitational Waves meeting in Stockholm, of the Gravita-tional Waves and Cosmology workshop at DESY, of the Workshop on Modernaspects of Gravity and Cosmology in Orsay, of the WE-Heraeus-Seminar onDo Black Holes Exist? - The Physics and Philosophy of Black Holes in BadHonnef, of the GW161212: The Universe through gravitational waves workshopat the Simons Center, and of the Quantum Vacuum and Gravitation: Test-ing General Relativity in Cosmology workshop at the MITP. V.C. acknowledgesfinancial support provided under the European Union’s H2020 ERC Consolida-tor Grant “Matter and strong-field gravity: New frontiers in Einstein’s theory”grant agreement no. MaGRaTh–646597. Research at Perimeter Institute issupported by the Government of Canada through Industry Canada and by theProvince of Ontario through the Ministry of Economic Development & Innova-tion. This article is based upon work from COST Action CA16104 “GWverse”,and MP1304 “NewCompstar” supported by COST (European Cooperation inScience and Technology). This work was partially supported by FCT-Portugalthrough the project IF/00293/2013, by the H2020-MSCA-RISE-2015 Grant No.StronGrHEP-690904.

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