![FIG. 4. Left panels: quadrupolar GWenergy spectrum [cf. Eq. (4)] for a point particle crossing a traversable wormhole and compared to the case of a particle plunging into a Schwarzschild BH with the same energy E. Top and bottom panels refer to r0 ¼ 2.1M, E ¼ 1.1, and to r0 ¼ 2.001M, E ¼ 1.5, respectively (different parameters give qualitatively similar results). Vertical dashed lines denote the frequency of the first QNMs of the wormhole (cf. Fig. 2) which correspond to narrow resonances in the flux [37,38]. Right panels: the corresponding GWwaveforms compared to the BH case. The BH waveform was shifted in time by Δt [cf. Eq. (6)] to account for the dephasing due to the light travel time from the throat to the light ring.](/figures/fig-4-left-panels-quadrupolar-gwenergy-spectrum-cf-eq-4-for-hwcvugwc.webp)
Is the Gravitational-Wave Ringdown a Probe of the Event Horizon?
Vitor Cardoso,
1,2
Edgardo Franzin,
3,1
and Paolo Pani
4,1
1
CENTRA, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa,
Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
2
Perimeter Institute for Theoretical Physics, 31 Caroline Street North Waterloo, Ontario N2L 2Y5, Canada
3
Dipartimento di Fisica, Università di Cagliari & Sezione INFN Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
4
Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Piazzale Aldo Moro 5, 00185 Roma, Italy
(Received 25 February 2016; published 27 April 2016)
It is commonly believed that the ringdown signal from a binary coalescence provides a conclusive proof
for the formation of an event horizon after the merger. This expectation is based on the assumption that the
ringdown waveform at intermediate times is dominated by the quasinormal modes of the final object. We
point out that this assumption should be taken with great care, and that very compact objects with a light
ring will display a similar ringdown stage, even when their quasinormal-mode spectrum is completely
different from that of a black hole. In other words, universal ringd own waveforms indicate the presence of
light rings, rather than of horizons. Only precision observations of the late-time ringdown signal, where the
differences in the quasinormal-mode spectrum eventually show up, can be used to rule out exotic
alternatives to black holes and to test quantum effects at the horizon scale.
DOI: 10.1103/PhysRevLett.116.171101
Introduction.—The first direct gravitational-wave (GW)
detection of a compact-binary coalescence by aLIGO [1]
opens up the exciting possibility of testing gravity in
extreme regimes [2–4]. The detected GW signal is char-
acterized by three phases [5–7]: the inspiral stage, corre-
sponding to large separations and well approximated by
post-Newtonian theory; the merger phase when the two
objects coalesce and which can only be described accu-
rately through numerical simulations; and the ringdown
phase when the merger end-product relaxes to a stationary,
equilibrium solution of the field equations [7–9].
It is commonly believed that the ringdown waveform is
dominated by the quasinormal modes (QNMs) [8,10,11] of
the final object. If the latter is a Kerr black hole (BH), the
entire QNM spectrum is characterized only by the BH mass
and angular momentum. Thus, the detection of a few modes
from the ringdown signal can allow for precision measure-
ments of the BH mass and spin, and possibly of higher
multipole moments, which can be used to perform null-
hypothesis tests of the no-hair theorems of general relativity
[4,12–14]. This reasoning suggests that the GW ringdown
signal provides a way to prove the existence of an event
horizon in dark, compact objects. In light of the intrinsic
limitations that inevitably plague any electromagnetic test
of an event horizon, ringdown detections might arguably
provide the only conclusive proof of the existence of
BHs [15].
Light ring, ringdown, and QNMs.—The argument
above relies on the assumption that the ringdown modes
coincide with the QNM frequencies, defined as the poles of
the appropriate Green’s function in the complex plane [8].
We stress that this correspondence does not hold in general
[as far as we are aware, Refs. [16,17] discuss this issue
correctly for the first time (cf. also a related discussion in
Ref. [18])]. The QNMs of a BH are intimately related to the
peculiar boundary conditions required at the event horizon,
namely absence of outgoing waves. If the final object does
not possess a horizon, the boundary conditions change
completely, thus drastically affecting the QNM structure.
On the other hand, the ringdown waves of the distorted
compact object are closely related to the null, unstable,
geodesics in the spacetime [8,19–22], their frequency and
damping time being associated with the orbital frequency
and with the instability time scale of circular null geodesics,
respectively. Thus, in principle, the ringdown phase should
not depend on the presence of a horizon as long as the final
object has a light ring.
If the final object is a BH, the ingoing condition at the
horizon simply takes the ringdown waves and “carries”
them inside the BH. In this case, the BH QNMs incidentally
describe also the ringdown phase. However, if the horizon
is replaced by a surface of different nature (as, e.g., in the
gravastar [23] or in the firewall [24] proposals), the
relaxation of the corresponding horizonless compact object
should then consist on the usual light-ring ringdown modes
(which are no longer QNMs), followed by the proper
modes of vibration of the object itself. The former are
insensitive to the boundary conditions and similar to the
BH case, whereas the latter (which one usually refers to as
QNMs) can differ dramatically from their BH counterpart,
since they are defined by different boundary conditions.
Setup.—To the best of our knowledge, the above picture
has never been verified in the context of GW tests
of an event horizon. Here, we perform such analysis
by considering the ringdown signal and the QNMs
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associated with a horizonless compact object with a light
ring. For definiteness, we focus on the gravitational
radiation emitted by a point particle in radial motion
towards a traversable wormhole [25,26], cf. Fig. 1 for an
illustration (the main qualitative features of our analysis are
independent of the specific horizonless object and apply
also to spherical shells of matter, gravastars, compact boson
stars, and others [17,27–29]).
The specific solution is obtained by identifying two
Schwarzschild metrics with the same mass M at the throat
r ¼ r
0
> 2M (we use G ¼ c ¼ 1 units). In Schwarzschild
coordinates, the two metrics are identical and described by
ds
2
¼ −Fdt
2
þ F
−1
dr
2
þ r
2
dΩ
2
, where F ¼ 1 − 2M=r.
Because Schwarzschild’s coordinates do not extend to
r<2M, we use the tortoise coordinate dr=dr
¼F,
where henceforth the upper and lower signs refer to the two
different universes connected at the throat. Without loss of
generality, we assume r
ðr
0
Þ¼0, so that one domain is
r
> 0 whereas the other domain is r
< 0. The surgery at
the throat requires a thin shell of matter with surface density
and surface pressure [26]
σ ¼ −
1
2πr
0
ffiffiffiffiffiffiffiffiffiffiffi
Fðr
0
Þ
p
;p¼
1
4πr
0
ð1 − M=r
0
Þ
ffiffiffiffiffiffiffiffiffiffiffi
Fðr
0
Þ
p
; ð1Þ
respectively. As required for traversable wormholes in
general relativity, the weak energy condition is violated
(σ < 0) [25,26], whereas the strong and null energy
conditions are satisfied when the throat is within the light
ring, r
0
< 3M (the weak energy condition is not necessarily
violated in modified gravity; e.g., in Einstein-dilaton
Gauss-Bonnet gravity traversable wormholes satisfying
all energy conditions exist [30]).
The four velocity of a particle with mass μ
p
≪ M and
conserved energy E in this spacetime reads
u
μ
p
≔dx
μ
p
=dτ ¼ðE=F; ∓
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
2
− F
p
; 0; 0Þ, where τ is the
proper time, and the coordinate time t
p
is governed by
t
0
p
ðrÞ¼∓
E
F
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
2
− F
p
; ð2Þ
where a prime denotes a derivative with respect to r.An
infalling object reaches the throat in finite time [we set
t
p
ðr
0
Þ¼0] and emerges in the other universe. In the point-
particle limit, Einstein equations coupled to the stress-
energy tensor T
μν
¼μ
p
R
ðdτ=
ffiffiffiffiffiffi
−g
p
Þu
μ
p
u
ν
p
δ½x
μ
−x
μ
p
ðτ Þ reduce
to a pair of Zerilli equations, d
2
ψ
l
ðω;rÞ=dr
2
þ
½ω
2
− V
l
ðrÞψ
l
ðω;rÞ¼S
l
, with [31]
V
l
¼2F
9M
3
þ9M
2
rΛ þ3Mr
2
Λ
2
þr
3
Λ
2
ð1 þΛÞ
r
3
ð3M þrΛÞ
2
;
S
l
¼
2
ffiffiffi
2
p
μ
p
Eð9 þ8ΛÞ
1=4
e
iωt
p
F ð3M þrΛÞ
2
ωt
0
p
ðrÞ
× fF
2
t
0
p
½2iΛ þð3M þrΛÞωt
0
p
− ð3M þrΛÞωg; ð3Þ
where Λ ¼ðl − 1Þðl þ 2Þ=2 and l ≥ 2 is the index
of the spherical-harmonic expansion. The source term is
different in the two universes due to the presence of
t
p
ðrÞ. The time-domain wave function can be recovered
via Ψ
l
ðt; rÞ¼1=
ffiffiffiffiffiffi
2π
p
R
dωe
−iωt
ψ
l
ðω;rÞ.
With the master equation in both universes at hand, we
only miss the junction conditions for ψ
l
at the throat.
The latter depend on the properties of the matter confined in
the thin shell [32]. For simplicity, here we assume that the
microscopic properties of the shell are such that ψ
l
and
dψ
l
=dr
are continuous at r
¼ 0. This assumption is not
crucial and can be modified without changing our quali-
tative results.
Finally, the energy flux emitted in GWs reads [31]
dE
dω
¼
1
32π
X
l≥2
ðl þ 2Þ!
ðl − 2Þ!
ω
2
jψ
l
ðω;r→ ∞Þj
2
; ð4Þ
and the solution ψ
l
can be obtained through the standard
Green’s function as
ψ
l
ðrÞ¼
ψ
þ
W
Z
r
−∞
dr
S
l
ψ
−
þ
ψ
−
W
Z
∞
r
dr
S
l
ψ
þ
; ð5Þ
where ψ
are the solutions of the corresponding homo-
geneous problem with correct boundary conditions at
r
→ ∞, and the Wronskian W ¼ ψ
−
dψ
þ
=dr
−
ψ
þ
dψ
−
=dr
is constant by virtue of the field equations.
We validated the results presented below by comparing this
procedure with a direct integration of the master equation
through a shooting method, obtaining the same results up to
numerical accuracy.
QNM spectrum.—The QNMs of the wormhole are
defined by the eigenvalue problem associated with the
master equation above with S
l
¼ 0 and supplemented by
regularity boundary conditions [8,10,11]. The latter are
ψ
l
∼ e
iωr
at the asymptotic boundaries of both universes.
Note that, because r
→ r at infinity, in Schwarzschild
FIG. 1. Illustration of a dynamical process involving a compact
horizonless object. A point particle plunges radially (red dashed
curve) in a wormhole spacetime and emerges in another “uni-
verse”. The black curve denotes the wormhole’s throat, the two
gray curves are the light rings. When the particle crosses each of
these curves, it excites characteristic modes which are trapped
between the light-ring potential wells, see Figs. 3 and 4.
PRL 116, 171101 (2016)
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coordinates, both homogeneous equations and boundary
conditions are the same. At the throat, we impose con-
tinuity of dψ
l
=dr
which—given the symmetry of the
problem and the homogeneity of the master equation—can
be achieved only in two ways: by imposing either
dψ
l
ð0Þ=dr
¼ 0 or ψ
l
ð0Þ¼0. Correspondingly, we find
two families of QNMs, ω ¼ ω
R
þ iω
I
, that can be obtained
by a straightforward direct integration supplied by a high-
order asymptotic expansion of the solution [33] in either of
the two domains.
A representative example of the polar QNM spectrum is
shown in Fig. 2. Remarkably, in the BH limit (r
0
→ 2M),
the spectrum is dramatically different from that of a
Schwarzschild BH. While the fundamental mode of a
Schwarzschild BH is ω
BH
M ∼ 0.3737 − 0.0890i,as
r
0
→ 2M, the QNMs of the wormhole approach the real
axis and becomes long lived; e.g., the fundamental mode is
ω
WH
M ≈ 0.0788–6.93 × 10
−9
i when r
0
¼ 2.00001M.In
fact, as r
0
→ 2M the deviations from the BH QNMs are
arbitrarily large.
This behavior can be understood by investigating the
effective potential shown in Fig. 3. Due to the presence of
the throat at r
¼ 0, the effective potential is Z
2
symmetric
and develops another barrier at r
< 0. Therefore, for any
r
0
≲ 3M, wormholes can support long-lived modes trapped
between the two potential wells near the light rings. These
modes are analog to the “slowly damped” modes of
ultracompact stars [34–36] (cf. Ref. [29] for a detailed
discussion).
0 0.1 0.2 0.3 0.4 0.5 0.6
ω
R
M
10
-8
10
-6
10
-4
10
-2
10
0
ω
I
M
n=0, family I
n=1, family I
n=2, family I
Schwarzschild BH
r
0
=2.001 M
r
0
=2.0001 M
r
0
=2.00001 M
n=0, family II
n=1, family II
n=2, family II
l=2
FIG. 2. The first three tones (n ¼ 0, 1, 2) for the two families
of polar l ¼ 2 QNMs of a wormhole parametrically shown in
the complex plane for different values of the throat location r
0
and compared to the first QNMs of a Schwarzschild BH. In the
BH limit (r
0
→ 2M), all QNMs of the wormhole approach the
real axis.
0.00
0.05
0.10
0.15
V(r
*
) M
2
-50 -40 -30 -20 -10 0 10 20 30 40 50
r
*
/M
0.00
0.05
0.10
0.15
V(r
*
) M
2
wormhole
black hole
outgoing at infinity
trapped
outgoing at infinity
outgoing at infinity
ingoing at horizon
FIG. 3. Effective (l ¼ 2) potential in tortoise coordinates for a
static traversable wormhole (top panel) with r
0
¼ 2.001M and for
a Schwarzschild BH (bottom panel).
0.05
0.10
0.15
0.20
0.25
dE/dωμ
p
-2
wormhole
black hole
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
ω M
0.00
0.05
0.10
0.15
dE/dωμ
p
-2
l=2, r
0
=2.1M, E=1.1
l=2, r
0
=2.001M, E=1.5
ω
n=0
I
ω
n=0
II
ω
n=1
I
ω
n=0
I
-0.20
0.00
0.20
0.40
0.60
ψ
2
-20 0 20 40 60 80 100 120
t/M
-0.20
0.00
0.20
0.40
ψ
2
wormhole
black hole
r
0
=2.1M, E=1.1
r
0
=2.001M, E=1.5
FIG. 4. Left panels: quadrupolar GW energy spectrum [cf. Eq. (4)] for a point particle crossing a traversable wormhole and compared
to the case of a particle plunging into a Schwarzschild BH with the same energy E. Top and bottom panels refer to r
0
¼ 2.1M, E ¼ 1.1,
and to r
0
¼ 2.001M, E ¼ 1.5, respectively (different parameters give qualitatively similar results). Vertical dashed lines denote
the frequency of the first QNMs of the wormhole (cf. Fig. 2) which correspo nd to narrow resonances in the flux [37,38]. Right panels:
the corresponding GW waveforms compared to the BH case. The BH waveform was shifted in time by Δt [cf. Eq. (6)] to acc ount for the
dephasing due to the light travel time from the throat to the light ring.
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Excitation of light-ring modes vs QNMs.—Given the
drastically different QNM spectrum of a wormhole relative
to the BH case, one might be tempted to expect a
completely different ringdown signal in actual dynamical
processes. This expectation seems to be confirmed by the
energy spectrum shown in the left panel of Fig. 4 and
compared to the case of a particle plunging into a
Schwarzschild BH. The spectra coincide only at low
frequencies but are generically very different.
Furthermore, in the BH limit, the long-lived QNMs of
the wormhole can be excited and correspond to narrow,
Breit-Wigner resonances in the spectrum [37,38].
However, as previously discussed, the BH QNMs are
light-ring modes and should play a role for any object with
a light ring. In fact, the striking difference in the energy
spectra does not leave a trace in the initial ringdown
waveform. This is shown in the right panel of Fig. 4 for
the time-domain wave function Ψ
2
ðt; rÞextracted at infinity
as a function of time. As the wormhole approaches the BH
limit, r
0
→ 2M, the initial ringdown is precisely the same
as in the Schwarzschild case: the waveform oscillates with
the same fundamental QNM of a Schwarzschild BH,
although the QNM spectrum of the wormhole is completely
different from that of the BH. We stress that the funda-
mental BH QNM does not appear as a pole of the
corresponding Green’s function of the wormhole, but
nevertheless dominates the ringdown.
The QNMs of the wormhole contain low energy and get
excited only at late times, namely after the particle crosses
the throat in the characteristic time scale
Δt ¼
Z
3M
r
0
dr
F
∼−2M log
l
M
; ð6Þ
where in the last step we considered r
0
¼ 2M þ l with
l ≪ M. Finally, in the BH limit (l → 0), all QNMs are
long lived and have similar frequencies (cf. Fig. 2), which
gives rise to a peculiar beating pattern at late times.
Discussion.—Our results give strong evidence for a
highly counterintuitive phenomenon: in the postmerger
phase of a compact-binary coalescence, the initial ring-
down signal chiefly depends on the properties of the light
ring—and not on the QNMs—of the final object. If the
latter is arbitrarily close to a BH, the ringdown modes will
correspond to the BH QNMs, even if the object does not
possess a horizon. In particular, this also means that mass
(and probably spin) estimates from current ringdown
templates perform well even if the compact object is
horizonless. The actual QNMs of the object are excited
only at late times and typically do not contain a significant
amount of energy. Therefore, they play a subdominant role
in the merger waveforms, but will likely dominate over
Price’s power-law tails [39].
Clearly, our model is heuristic and could be extended in
several ways, e.g., by including rotation, finite-size and
self-force effects, and more generic orbits. None of these
effects are expected to change the qualitative picture
discussed above [environmental effects (such as accretion
disks, magnetic fields, dark-matter distributions, or a
cosmological constant) are typically negligible [17] and
should not affect the waveform significantly]. In particular,
the motion of the particle before crossing the innermost-
stable circular orbit is irrelevant for the ringdown signal,
which depends almost entirely on the subsequent plunge
and on the particle’s motion after crossing the light ring. It
would be interesting to extend our analysis by performing a
numerical simulation of a compact-binary merger produc-
ing a horizonless compact object.
Our results are relevant to test possible consequences of
quantum effects at the horizon scale [40], e.g., the firewall
[24] and the gravastar [23] proposals. In these models, the
QNM spectrum might considerably differ from the Kerr
case [17,29], but this will not prevent GW observatories
from detecting their ringdown signal using standard
BH-based templates. For various BH mimickers, the
horizon is removed by a quantum phase transition, which
would naturally occur on Planckian length scales
[16,17,23,24,40]. In this case, the changes to the QNM
spectrum are more dramatic and, if detected, they will
provide a smoking gun for quantum corrections at the
horizon scale. In the l ≪ M limit, we expect that our
results will be qualitatively valid for any model.
Interestingly, Eq. (6) shows that the delay Δt for the
QNMs to appear after the main burst of radiation produced
at the light ring depends only logarithmically on l. For a
final object with M ≈ 60M
⊙
, Δt ∼ 16τ
BH
(τ
BH
≈ 3 ms
being the fundamental damping time of a Schwarzschild
BH with the same mass) even if the length scale is
Planckian, l ∼ L
p
¼ 2 × 10
−33
cm. For l ∼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2L
p
M
p
∼
10
−13
cm as in the original gravastar model [23], such
delay is only halved.
Our results suggest that future GW detections by aLIGO
[41], aVIRGO [42], and KAGRA [43] should focus on
extracting the late-time ringdown signal, where the actual
QNMs of the final object are eventually excited. Even in the
absence of a horizon, these modes are expected to be in the
same frequency range of the BH QNMs and therefore,
might be detectable with advanced GW interferometers.
Furthermore, their extremely long damping time (cf. Fig. 2)
might be used to enhance the signal through long-time
integrations, even if the energy contained in these mode is
weak. Estimating the signal-to-noise ratio required for
such precise measurements is an important extension of
our work.
Horizonless compact objects require exotic matter con-
figurations and almost inevitably possess a stable light ring
at r<3M [29]. The latter might be associated with various
instabilities, including fragmentation and collapse [29] and
the ergoregion instability [44–47] when the object rotates
sufficiently fast. While our results are generic, the viability
of a BH mimicker depends on the specific model, espe-
cially on its compactness and spin [48].
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The recent GW detection by aLIGO [1] enormously
strengthens the evidence for stellar-mass BHs, whose
existence is already supported by various indirect obser-
vations in the electromagnetic band (cf. e.g. Refs. [49,50]).
While BHs remain the most convincing Occam’s razor
hypothesis, it is important to bear in mind the elusive nature
of an event horizon and the challenges associated with its
direct detection.
The postmerger signal detected by aLIGO has been
recently investigated in the context of tests of gravity
(cf., e.g., Refs. [51,52]). Our results show that only late-
time ringdown detections might be used to rule out exotic
alternatives to BHs and to test quantum effects at the
horizon scale. As it stands, the single event GW150914 [1]
does not provide the final evidence for horizons, but
strongly supports the existence of light rings, itself a
genuinely general-relativistic effect.
We thank Emanuele Berti, Valeria Ferrari, and Leonardo
Gualtieri for interesting comments on a draft of this Letter.
V. C. acknowledges financial support provided under the
European Union’s H2020 ERC Consolidator Grant “Matter
and strong-field gravity: New frontiers in Einstein’s theory”
Grant No. MaGRaTh–646597. Research at Perimeter
Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario
through the Ministry of Economic Development and
Innovation. This work was partially supported by FCT-
Portugal through the Project No. IF/00293/2013, by the
H2020-MSCA-RISE-2015 Grant No. StronGrHEP-
690904, and by “NewCompstar” (COST action MP1304).
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PRL 116, 171101 (2016)
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