Using the black string between two branes as a model of a brane-world black
hole, we compute the gravity wave perturbations and identify the features
arising from the additional polarizations of the graviton. The standard
four-dimensional gravitational wave signal acquires late-time oscillations due
to massive modes of the graviton. The Fourier transform of these oscillations
shows a series of spikes associated with the masses of the Kaluza-Klein modes,
providing in principle a spectroscopic signature of extra dimensions.Comment: 4 pages, 5 figures. Comments on the frequency and detectability of
  the massive mode signal added, and title modified. Version accepted for
  publication in Phys. Rev. Let

Chris Clarkson

R. Maartens

Roy Maartens

Sanjeev S. Seahra

English

arXiv

Seahra, S.

Clarkson, C.

Maartens, Roy

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Detecting extra dimensions with gravity wave spectroscopy:
the black string brane-world
Sanjeev S. Seahra, Chris Clarkson, Roy Maartens
Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 2EG, UK
(Dated: February 7, 2008)
Using the black string between two branes as a model of a brane-world black hole, we compute
the gravity wave perturbations and identify the features arising from the additional polarizations of
the graviton. The standard four-dimensional gravitational wave signal acquires late-time oscillations
due to massive modes of the graviton. The Fourier transform of these oscillations shows a series of
spikes associated with the masses of the Kaluza-Klein modes, providing in principle a spectroscopic
signature of extra dimensions.
Black holes are central to our understanding of gravity,
and are expected to be key sources of gravity waves that
should be detected by the current and upcoming genera-
tion of experiments. Such a detection will not only con-
firm the indirect evidence from binary pulsars for gravity
waves, but will also also allow us to probe the properties
of black holes and of gravity. In particular, this will open
up a new window for testing modifications to general rel-
ativity, such as those arising from quantum gravity the-
ories. String theory for example predicts that spacetime
has extra spatial dimensions, so that the gravitational
field propagates in higher dimensions and has extra polar-
izations. Recent developments in string theory indicate
that Standard Model fields may be confined to a four-
dimensional ‘brane’, while gravity propagates in the full
‘bulk’ spacetime. This has spurred the development of
brane-world models, such as Randall-Sundrum (RS) type
models, which can be used to explore astrophysical pre-
dictions [1]. RS type models have a five-dimensional bulk
with negative cosmological constant, so that the metric
is warped along the extra dimension. As a result, these
models provide a new approach to the hierarchy problem,
dimensional reduction and holography.
The nature of black holes that form by gravitational
collapse on an RS brane is only partly understood [1, 2],
and no exact solution is known for a black hole localized
on one brane. If there is a second ‘shadow’ brane, the
black string may be used to model large black holes on
the visible brane, when the horizon on the brane is much
greater than the extent of the horizon into the bulk [3].
The black string reproduces the Schwarzschild metric on
the visible brane but is not confined to the brane, since
there is a line singularity at r = 0 into the extra di-
mension (see Fig. 1). The shadow brane can also in-
troduce an infra-red cut-off to shut down the Gregory-
Laflamme (GL) instability of the black string at long
wavelengths [4]. If the shadow brane is close enough to
the visible brane for a given black hole mass M , or if
M is large enough for a given brane separation d, then
GMe−d/ℓ/ℓ is above a positive critical value and the GL
instability is removed (see below). This is the background
model that we perturb.
FIG. 1: Schematic of the black string
The brane separation is constrained from above by sta-
bility requirements. It is also constrained from below.
This follows since the brane separation is a massless de-
gree of freedom, felt on the visible brane as a ‘radion’
field, so that the low-energy effective theory on the visi-
ble brane is of Brans-Dicke type, with [5] ωbd = 3(e
2d/ℓ−
1)/2, where ℓ is the bulk curvature radius. The shadow
brane must be far enough away that its gravitational in-
fluence on the visible brane is within observational lim-
its. Solar system observations impose the lower limit [6]
ωbd & 4 × 10
4, so that d/ℓ & 5. The allowed region
in parameter space is shown in Fig. 2. Table-top tests of
Newton’s law impose the constraint ℓ . 0.1mm. This up-
per limit defines a mass 0.1mm/2G ∼ 10−7M⊙, so that
for astrophysical black holes it follows that 2GM ≫ ℓ is
easily satisfied: GM/ℓ & 107(M/M⊙).
The 5D black string is a solution of the Einstein equa-
tions GAB = 6ℓ
−2gAB, with metric
ds2 = a2
(
−f dt2 + f−1 dr2 + r2 dΩ2
)
+ dy2, (1)
where f(r) = 1 − 2GM/r, a(y) = e−|y|/ℓ, and the
branes are at y = 0, d. Metric perturbations satisfy
hAB+∇A∇Bh
C
C−2∇
C∇(AhB)C−8ℓ
−2hAB = 0, (2)
and gauge choices may be made to reduce the degrees
2FIG. 2: The allowed region in parameter space. The stability
boundary is well approximated by GM/ℓ = 0.1122ed/ℓ.
of freedom to 5, the number of polarizations of the 5D
graviton.
As in standard black hole perturbation theory [7], we
use spherical harmonics Ylm and their gradients to con-
struct a basis T
(j,lm)
AB for hAB, with j = 1 . . . 15. The
metric perturbation is naturally split into two decoupled
parts: polar [parity (−1)l] and axial [parity (−1)l+1].
Suppressing the (lm) indices, the expansion coefficients
are of the form C(j) = eiωτH(j)(r, y), where τ = t/GM
and ω is a dimensionless frequency. We use two gauges,
which are related by quadrature. We generalize the 4D
Regge-Wheeler (RW) gauge by setting the coefficients of
the most complicated harmonic tensors equal to zero,
which makes the remaining coefficients gauge invariant.
The RS gauge [5], given by hAA = 0 = ∇
BhAB = hAy,
has the advantage that the expansion coefficients are sep-
arable: C(j) = eiωτW (j)(r)Z(y), where [4]
a2Z ′′ − 2(aa′)′Z = −m2Z. (3)
Here, m is the effective mass on the visible brane of the
Kaluza-Klein (KK) mode of the 5D graviton. The bulk
wave functions Z(y) are the same as those for Minkowski
branes (even though the bulk is no longer anti de Sitter),
and the solution for m > 0 is Z(y) = B2(mℓ/a), where
B2 is a linear combination of Bessel functions.
Neglecting brane bending, the boundary conditions at
y = 0 , d in RS gauge are ∂y(a
−2hαβ) = 0. When m = 0,
the solution is Z ∝ a2. For this zero-mode, the met-
ric perturbations reduce to those of a 4D Schwarzschild
metric, as expected. For m 6= 0, the boundary conditions
lead to a discrete tower of KK mass eigenvalues,
mn = (zn/ℓ)e
−d/ℓ , (4)
where Y1(mnℓ)J1(zn) = J1(mnℓ)Y1(zn) [1]. The GL in-
stability exists in the range 0 < m < mcrit ≈ 0.4/GM [4].
If m1 > mcrit, then the instability is avoided, and
m1 = mcrit defines the stability curve in Fig. 2.
Radial master equations. We generalize the standard
4D analysis to find radial master equations for a reduced
set of variables, for all classes of perturbations. In the
5D case, the KK modes are governed by coupled mas-
ter equations. These can be written as matrix-valued
Schro¨dinger-like equations,
−
d2
dx2
Ψ+VΨ = ω2Ψ, (5)
where V is the potential matrix, x = ρ + 2 ln (ρ/2− 1),
and ρ = r/GM . For spherical (s-wave) polar perturba-
tions, there is only one master variable, and
Vs
f
=
µ6ρ9 + 6µ4ρ7 − 18µ4ρ6 − 24µ2ρ4 + 36µ2ρ3 + 8
ρ3 (µ2ρ3 + 2)
2 ,
(6)
where µ = GMm. For l ≥ 2 axial perturbations, there
are two master variables and the potential matrix in the
RW gauge is
Va
f
= µ2I + ρ−3
[
l(l+ 1)ρ− 6 µ2
4ρ3 l(l + 1)ρ
]
. (7)
The diagonal elements of Va are identical to the
RW potentials for spin-2 and spin-1 excitations of 4D
Schwarzschild, plus a mass term. Massive KK modes
of the 5D graviton include spin-1 perturbations. (For
µ = 0, the spin-1 contribution is pure gauge.) For the
other types of perturbations, we find that the l = 1 and
l ≥ 2 polar perturbations are governed by two and three
master variables respectively, while there is only one mas-
ter variable for the axial p-wave.
Stability. All master equations can be cast in the form
L(µ, ω)Ψ = 0, where L is a linear differential operator.
Stability holds if L is positive definite when the bound-
ary conditions Ψ = 0 are imposed at x = ±∞, and
ω2 < 0 [7]. We have verified the positivity of L an-
alytically for all classes of axial perturbation. For the
l ≥ 1 polar case we performed a numerical search for so-
lutions satisfying the above boundary conditions in the
relevant region of (µ, ω) parameter space. No solutions
were found, suggesting that no instability exists. The op-
posite is true for the s-wave case. The reason lies in the
potential, which is shown in Fig. 3 for different values of
µ. For small µ, there is a potential well that can support
a normalizable bound state with ω2 < 0, which implies
that L is not positive definite. This well vanishes for large
µ, suggesting that an instability exists for all modes with
0 < µ < µcrit. We used a standard WKB-inspired phase
integral analysis [8] to develop a new improved deriva-
tion of the GL instability. Our approach leads to an ac-
curate determination, µcrit = 0.4301, in agreement with
recent results from full numerical relativity [9]. Stability
is achieved if the mass of the first KK mode is such that
GMm1 > µcrit (see Fig. 2).
Gravity wave signals. Perturbations of a stable black
string describe gravitational radiation that may in prin-
ciple be observable by current or future detectors. The
3total gravity wave signal at the observer (x = xobs) is
a superposition of the waveforms ψn(τ) associated with
the mass eigenvalues mn of Eq. (3). In Fig. 4, we
present signals associated with the four lowest masses for
a marginally stable black string, for l = 2 axial modes.
These are obtained by re-introducing time dependence
into the Schro¨dinger equation (5), via ω → −i ∂/∂τ , and
numerically integrating the resulting PDE with V = Va.
We assume static Gaussian initial data centered about
x = 50 and an observer at xobs = 100. Because the mas-
ter equations reduce to the 4D RW equation when µ = 0,
the bottom curve exhibits the damped single-frequency
(quasi-normal) ringing followed by a power-law tail famil-
iar from the 4D analysis [7]. The massive mode signals
are quite different [10], showing much less damping than
the massless mode, and late-time monochromatic oscil-
lations instead of a featureless power-law tail. This is
reminiscent of the behaviour of massive scalar fields in
4D spherically symmetric spacetimes, where a detailed
Green’s function analysis shows [11] that the late-time
signal is ∝ t−5/6 sin mt, irrespective of l.
To reconstruct the total waveform ψtot for the l = 2
axial case, as measured by an observer on the visible
brane, we need to specify an initial profile F for the per-
turbation in the bulk. Using a conformal bulk coordinate
ξ = a−1, and assuming that F and Zn are appropriately
normalized, we have
ψtot(τ) =
∑
n
cnZn(0)ψn(τ), cn =
∫ ed/ℓ
1
ξFZndξ, (8)
where
∑
n c
2
n = 1. Hence, c
2
n is the fractional energy in
the n-mode. Massive KK modes introduce striking new
features in the gravity wave signal that are potentially
observable, as shown in Fig. 5. The strength of the new
features is sensitive to the initial data F (ξ). Here we
simply illustrate the possibilities via two forms of F (ξ)
that correspond to very different situations.
In Fig. 5, the upper signal has d/ℓ = 6 [12], and the
initial data F (ξ) corresponds to the zero-mode 1/ξ2 with
µx
V
s
/µ
2
–20
–10 0 10 20 30 40 50 0.4
0.6 0.8
1
–2
–1
0
1
2
FIG. 3: The s-wave potential for varying µ. The critical po-
tential is indicated by the heavy line. For µ < µcrit, there is
a bound state with ω2 < 0, indicating an instability.
FIG. 4: Gravity wave signals associated with 3 massive KK
modes and the 0-mode of l = 2 axial perturbations, for a
marginally stable black string. The inset shows the zero-mode
signal on a logarithmic scale.
a cut-off applied one eighth of the distance to the shadow
brane. This data is motivated by the results of numerical
calculations of the bulk gravitational field around a small
brane-localized black hole [2]. If the black hole mass is
small enough to be in the unstable region of Fig. 2, we
may think of the black hole as the endpoint of a GL in-
stability [2, 9]. The initial data is then thought of as
arising from an encounter between this black hole and
the much heavier black string. We find that in this case,
the coefficients of the massive mode signals are of order
10−4, which suggests that the total waveform will only
exhibit minor deviations from the GR prediction. This is
certainly true for early times, where the signal is domi-
nated by ordinary 4D Schwarzschild quasi-normal ringing
(cf. the inset of Fig. 4). However, as can be inferred from
the individual massive waveforms in Fig. 4, the late-time
signal is very different from the 4D case–it is both os-
cillatory and very lightly damped. The inset shows the
Fourier transform of the signal for late times. There are
eight discrete peaks in the spectrum, corresponding to
each of the eight non-zero mass modes.
In the lower panel, we take d/ℓ = 20 and Gaussian ini-
tial data in the bulk centered halfway between the branes
(if the center is moved closer to the shadow brane, the
results are qualitatively similar). This could correspond
to an event which ‘mainly takes place in the bulk’, such
as the merger of two black strings. In this case, the zero
mode waveform is dominated by the massive mode sig-
nals, and the waveform is very different from the 4D case.
The contributions from the odd modes m3, m5 and m7
are suppressed, which results in a late-time transform
4FIG. 5: Composite l = 2 axial gravity wave signal (normal-
ized) from the 9 lowest mass modes of a marginally stable
black string. The upper panel corresponds to ‘truncated zero-
mode’ bulk initial data with d/ℓ = 6, the lower panel to Gaus-
sian bulk initial data with d/ℓ = 20. Insets show the Fourier
transform of the respective waveforms for 300 ≤ τ ≤ 1000.
The percentage of energy in the 4D zero-mode part of the
signal is 99.96 % (upper) and 2.281 × 10−13 % (lower).
with five principal peaks in the inset. These peaks in
fact display a degree of fine structure, as shown in the
blow-up, that deserves further investigation.
Conclusions. Will the consideration of other types of
perturbations or different multipoles in some way con-
taminate the spectroscopic signal? We find that in all
cases, as r → ∞, master variables with µ 6= 0 behave as
massive fields propagating on Schwarzschild spacetime,
which is enough to guarantee the slowly-decaying oscil-
lating tail. Massless modes for l > 2 are known to be
subject to more damping than for l = 2, so if the mas-
sive signal dominates the latter for late times, it will also
dominate the former.
Can the massive signal be resolved by realistic grav-
ity wave detectors? There are two separate issues: the
frequency of the massive modes and their relative am-
plitude. From the formulas above, one finds that the
discrete frequencies in the late-time tail are
fn = zne
26.9−d/ℓ(0.1mm/ℓ) Hz. (9)
For d/ℓ & 5, zn is well approximated by J1(zn) = 0.
Note that unlike 4D black hole quasinormal modes, these
frequencies are independent of the mass M . Taking ℓ =
0.1mm and d/ℓ = 24, results in f1 ∼ 100Hz, which is
ideal for a LIGO detection of extra dimensions. For d/ℓ =
33, we find an optimal LISA signal, with f1 ∼ 0.01Hz.
For these parameters, the lower limits on black string
masses are M > 100M⊙ and M > 10
6M⊙, respectively.
In regard to the amplitudes, the most optimistic type
of events involve bulk-based initial data, i.e., black string
mergers. Since the zero-mode energy is small in these
situations, we expect the strength of the massive mode
oscillations to be comparable to the quasinormal ring-
ing amplitude in the analogous 4D case. The situation
is more delicate for brane based initial data, where the
zero mode-tends to dominate the signal. We find that
the relative massive amplitude tends to increase with de-
creasing d/ℓ. However, small d/ℓ implies large fn, so the
best prospect of seeing this type of signal lies with high
frequency detectors.
The discrete nature of the late-time Fourier transform
of the black string waveform is the most important ob-
servable feature of this model. Its detection would pro-
vide clear evidence of extra dimensions. It would also give
the direct spectroscopic measurement of the KK masses
mn, which in turn provides information about d and ℓ.
Although we have analyzed a specific model of brane-
world black holes, we expect that qualitatively similar
features will arise for other models with compact extra
dimensions, since they all have a discrete tower of mas-
sive KK modes. Furthermore, since the massive modes
travel below light-speed, there will be potentially observ-
able time-delay in their arrival, which could be of or-
der seconds or longer for distant sources. Also, the light
damping of KK modes means that they may have a sig-
nificant integrated contribution to the stochastic gravity
wave background.
Acknowledgements SSS is supported by NSERC, CC
and RM by PPARC. We thank C Kiefer, B Kol, K
Koyama, D Langlois and J Soda for discussions.
[1] See e.g., R. Maartens, Liv. Rev. Rel. 7, 1 (2004).
[2] H. Kudoh, T. Tanaka, and T. Nakamura, Phys. Rev.
D68, 024035 (2003).
[3] S. Kanno, and J. Soda, Class. Quant. Grav. 21, 1915
(2004).
[4] R. Gregory, Class. Quant. Grav. 17, L125 (2000).
[5] J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778
(2000).
[6] B. Bertotti, L. Iess, and P. Tortora, Nature 425, 374
(2003); C. M. Will and N. Yunes, gr-qc/0403100.
[7] See, e.g., H. Nollert, Class. Quant. Grav. 16, R159
(1999).
[8] N. Froeman, P. O. Froeman, N. Andersson, and A. Hoek-
back, Phys. Rev. D45, 2609 (1992).
[9] M. W. Choptuik, et al., Phys. Rev. D68, 044001 (2003).
[10] Amplitude modulations in the massive mode signals are
also found in the case of Minkowski branes: A. O. Barvin-
sky, A. Y. Kamenshchik, A. Rathke, and C. Kiefer, An-
nalen Phys. 12, 343 (2003).
[11] H. Koyama and A. Tomimatsu, Phys. Rev. D65, 084031
(2002).
[12] For numerical convenience we choose the marginally sta-
ble case, so that M ≪ M⊙ for d/ℓ = 6, but the results
will be qualitatively similar for M & M⊙.


Detecting extra dimensions with gravity-wave spectroscopy: the black-string brane world

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Detecting extra dimensions with gravity wave spectroscopy: the black
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Detecting extra dimensions with gravity wave spectroscopy: the black string brane-world

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