Flat space inside, Schwarzschild outside. How much of a wave the shell sends back, how it rings, how it changes
the polarization, and why it does not want to stay still.
Juan Pablo Elia
Question
Reflection and transmission $\mathcal R(\omega),\mathcal T(\omega)$, quasinormal modes and
polarization ($h_+$, $h_\times$) for odd and even gravitational perturbations of the simplest horizonless compact object.
Obstacle
A domain-wall shell always collapses. Which shell can be static, and is it stable?
Answer in one line
A pressure-supported shell: a δ-barrier for axial waves, black-hole-like when compact, ringing with
echoes, acting as a polarizing mirror, and slowly unstable.
~30 s. The simplest black-hole mimicker: a membrane of matter with nothing inside. Units G = c = 1;
frequencies in units of 1/M. The talk follows a single line of thought: background, junction, scattering,
ringing, stability, back to the background, and finally what the two polarizations see.
The line of thought each answer raises the next question
#
question
answer
1
Can the shell be static?
Only with pressure $p=\kappa(R)\sigma$ (fluid shell)
all at low $\omega$, none at high $\omega$, BH-like as $R\to2M$
5
How does it ring?
cavity, trapped (echo) and matter modes
6
Is the static shell stable?
No: one matter mode grows
7
So is step 1 consistent?
fluid shell: yes (slow growth); frozen wall: no
8
What happens to $h_+$ and $h_\times$?
parity is conserved, polarization is not: a birefringent mirror
9
Is one parity more important?
equal in energy; even carries the matter physics
~30 s. This is the map of the talk. Step 7 closes the loop: the frequency-domain calculation
assumes a static background, and we check whether that was legitimate. Steps 8 and 9 translate everything into what a
detector measures, the two polarizations.
Toolbox: perturbing a spherical spacetime the ideas used throughout
Split the metric: $g=g^{(0)}+h$, with $h$ small. On a spherical background every
perturbation is a sum of spherical-harmonic multipoles $(\ell,m)$; waves need $\ell\ge2$.
Two parities. Under inversion $\vec x\to-\vec x$, even (polar) pieces pick up $(-1)^\ell$,
odd (axial) pieces $(-1)^{\ell+1}$. On a spherical background they never mix.
Master functions. For each $(\ell,m,{\rm parity})$ all metric components combine into one
gauge-invariant scalar $\Psi(t,r)$ that obeys a 1D wave equation with an effective potential $V(r)$.
Tortoise coordinate $r_*$: light moves as $dr_*=\pm dt$, and the horizon sits at $r_*\to-\infty$.
Frequency domain: $\Psi\propto e^{-i\omega t}$; frequencies in units of $1/M$, $G=c=1$.
Scattering
Real $\omega$. Send in $e^{-i\omega r_*}$ from infinity and measure what comes back:
$\Psi\to B_{\rm in}e^{-i\omega r_*}+B_{\rm out}e^{+i\omega r_*}$. Reflection amplitude $S=B_{\rm out}/B_{\rm in}$, reflectivity
$\mathcal R=|S|^2$.
Quasinormal modes (QNMs)
Complex $\omega$ at which a solution is purely outgoing ($B_{\rm in}=0$): the free ringing
of the object. ${\rm Re}\,\omega$ is the frequency, $-{\rm Im}\,\omega$ the damping rate. ${\rm Im}\,\omega>0$ would mean a
growing mode, an instability. QNMs are the poles of $S(\omega)$.
Black-hole reference
Schwarzschild, $\ell=2$: $\omega M=0.3737-0.0890i$. Waves are reflected by the potential
barrier near the photon sphere ($r=3M$) and absorbed by the horizon.
~1.5 min. Everything later uses these words: multipole, parity, master function, tortoise coordinate,
reflection amplitude, QNM. Stress that parity is a symmetry of any spherical object, so the two parities are two
independent problems. The black-hole numbers are the benchmark we compare to.
The setup one shell, two spacetimes
Questions:
Scattering: reflection $\mathcal R(\omega)$ and transmission $\mathcal T(\omega)$ of a gravitational
wave, for odd (axial) and even (polar) parity.
Ringing: the quasinormal modes (QNMs), complex frequencies with outgoing waves only.
Ingredients:
Birkhoff's theorem (every spherical vacuum region is a piece of Schwarzschild) plus regularity at the centre
(interior mass $=0$) force a flat interior.
Perturbations obey Regge–Wheeler / Zerilli equations outside and the flat wave equation inside.
The shell glues the two with junction conditions. That is the hard part.
~1 min. The picture: the shell is a membrane with energy density and pressure. Outside it looks
like a black hole of mass M; inside it is empty flat space. Emphasize the time-dilation factor: the same wave has
a higher frequency inside, nu = omega/sqrt(f(R)).
Thin shells in general relativity Israel junction conditions
A shell is a hypersurface $\Sigma$ carrying a surface stress tensor $S_{ab}$. Across it:
So a static shell needs positive surface pressure, fixed by $R/M$. Dust ($p=0$) and a
domain wall ($p=-\sigma$) cannot be static.
~1 min. No need to derive; just the logic: two junction conditions, then the equation of motion. The
punchline is the sign: gravity + tension both pull inward.
The collapse problem model A (domain wall) vs model C (fluid shell)
Left: a domain wall ($S_{ab}=-\sigma h_{ab}$) released from rest at its turning
point. It collapses on the light-crossing time, $\omega_{\rm dyn}\approx\sqrt2/R$, and forms a black hole. Right: the static
fluid shell ($p=\kappa\sigma$). It is in exact equilibrium, but an even-parity $\ell=2$ perturbation grows as
$e^{\gamma t}$ with $\gamma\simeq0.6\,(M/R^3)^{1/2}$ (deformation exaggerated).
~1.5 min. Run the animation. Point out the two clocks: the wall's own clock reaches the centre in
finite proper time; a distant observer sees it freeze at r = 2M. Then switch to the fluid shell: static, but the
l = 2 deformation grows, which is the instability found by Pitre, Schneider & Poisson (2026). Compare the growth time with
the wall's collapse time.
Which shell? the options for a static background
option
shell
verdict
A
domain wall frozen at its turning point, $\dot R=0$, $\ddot R=-\frac{1+3\sqrt{f_R}}{2R}$
adiabatic only
B
static dust, $p=0$
does not exist
C
static perfect fluid, $p=\kappa\sigma$, $\kappa$ fixed by $R/M$
exact (but unstable)
D
de Sitter / Schwarzschild interior (gravastar)
not treated here
Model C is valid for $R\ge 25M/12$ (energy condition $p\le\sigma$, causality $v_s^2\le1$).
The perturbed shell needs an equation of state: $\delta p=\Gamma\frac{p}{\sigma+p}\delta\sigma$, with $\Gamma=2$.
Inside, frequencies are blueshifted: $\nu=\omega/\sqrt{f_R}$.
The frozen snapshot of model A is sensible only if a wave period is much shorter
than the collapse time. The curve is the wall's radius (exterior time); the bars are the periods of its lowest QNMs.
They are comparable, so model A is at best marginal.
~1 min. B is impossible. C is the primary model: truly static, one-parameter family (compactness).
A is kept as a frozen snapshot, and the timeline shows why that is dangerous: the collapse is not slow compared with
the waves.
Gauge-invariant master functions (Martel & Poisson 2005): $\Psi_{\rm odd}$ is built from the axial
metric components, $\Psi_{\rm even}$ from the polar ones.
even (Zerilli–Moncrief): Zerilli potential $V_{\rm Z}$ (Zerilli 1970). For a black hole both give the same QNMs and reflectivity (isospectrality).
inside: $\Psi''+[\nu^2-\ell(\ell+1)/r^2]\Psi=0$, so $\Psi=\hat\jmath_\ell(\nu r)$ (Riccati–Bessel)
The interior clock runs slow
$dT=\sqrt{f_R}\,dt$ on the shell, so the same wave has frequency $\nu=\omega/\sqrt{f_R}$ inside:
it is blueshifted.
~1 min. In a global tortoise coordinate the whole problem is a 1D wave equation: the flat centrifugal
barrier inside, the Schwarzschild barrier outside, and the shell between them. Dotted: what a black hole would have instead.
Everything about the shell is in the junction conditions.
How the shell enters the wave equation linearized Israel conditions
The same for wall and fluid, and independent of $\omega$ and of the equation of state. The
background term $[K]^{(0)}$ cancels against the pressure, and odd waves do not move the matter.
Even waves move the shell. Unknowns: radial displacement $\xi_\pm$, surface-density change
$\delta\sigma$ (with $\delta p=v_s^2\delta\sigma$) and tangential flow $U$. They are solved together with the wave from 4 metric-continuity
and 4 Israel equations. So $\Psi$ jumps, and $\mathsf T$ depends on $\omega,\ell,R/M,\Gamma$. $\det\mathsf T$ is exactly the
flux-conservation condition.
Independent test: take a smooth shell of thickness ε (anisotropic fluid,
$V\supset4\pi\rho\,e^{2\Phi}$) and let $\varepsilon\to0$.
It converges linearly to the derived $\Delta$, and to no alternative.
~1.5 min. Method: expand the Israel conditions to first order around the static background. Each side is
described in its own Regge–Wheeler gauge, so the shell sits at different perturbed positions seen from inside and outside;
a perturbed embedding with displacements xi+ and xi- takes care of this. Symbolic algebra throughout. Odd parity is very simple: a repulsive delta potential
whose strength is the surface density times the redshift. Even parity is a genuine 2x2 matching.
How much does the shell reflect?
With a regular interior nothing is absorbed, so everything comes back: $|S|=1$ and only
the phase of $S$ carries information. To measure how much the object reflects, we open the interior: whatever
crosses the shell is let go, as a horizon would.
$\omega\to0$: $\mathcal R\to1$. Long waves are reflected.
Compact shells: the transition sits at the barrier top, as for a black hole.
Dilute shells: the transition sits at $\omega R\sim\ell$ (flat centrifugal barrier).
~1 min. Emphasize the definition: with the interior opened, the black hole becomes a limit of the same
problem. Energy conservation R + T = 1 holds to 1e-10 and is a real test of the junction matrix.
Two limits transparent at high frequency · black-hole-like when compact
The shell sinks into the redshifted throat. The approach is slow, $\propto(R-2M)^{1/2}$.
~1 min. Left: log-log, both parities collapse onto the delta-barrier law; the dip in the odd curve is an
interference between the barrier and the shell reflections. Right: the reflectivity tends to the black-hole one. Keep this
in mind: the reflectivity has a black-hole limit, but the ringing does not.
How the shell rings quasinormal modes
QNMs: the regular interior solution, carried through the shell, is purely outgoing at infinity.
Cavity modes, $\omega\sim1/R$. For dilute shells, $\omega R\approx n\pi-\tfrac{i}{2}\ln[2\ell(\ell+1)R/M]$: a leaky box.
Trapped modes for $R<3M$: caught between the interior and the photon-sphere barrier.
${\rm Im}\,\omega M\approx-10^{-6}$ at $R=2.01M$. These are the echoes.
Matter modes (even): oscillations of the shell itself, $\omega\sim(M/R^3)^{1/2}$.
None of them tends to the black-hole QNMs (stars) as $R\to2M$. The
black hole rings because waves leak into the horizon; the shell sends them back.
~1 min. Contrast with the previous slide: the reflectivity is continuous in the black-hole limit, the
spectrum is not. Trapped modes become exponentially long-lived as the shell approaches 2M.
The same pulse on a black hole and on a shell time-domain simulation
object
late-time fit
QNM (frequency domain)
black hole
$0.3731-0.0885i$
$0.3737-0.0890i$
shell $R=2.2M$
$0.4215-0.0352i$
$0.4211-0.0350i$
The black hole rings down in $\sim30M$. The compact shell keeps ringing at its trapped mode.
All three share the exterior power-law tail.
~1 min. Leapfrog evolution of the odd l = 2 equation with the delta barrier, independent of the
frequency-domain code. The fitted frequencies agree to about 1e-3, a direct confirmation of the spectrum. Next slide:
run it live.
A wave packet hits the shell live simulation, odd parity, ℓ = 2
$\partial_t^2\Psi=\partial_x^2\Psi-V(x)\Psi-\Delta_{\rm odd}\delta(x-x_R)\Psi$ in the global tortoise coordinate
(leapfrog, absorbing outer boundary). Top: $\Psi(x)$ with the potential (gray) and the flat interior (blue band).
Bottom: $\log_{10}|\Psi|$ recorded far away, showing the direct reflection, then trapped-mode ringing and echoes.
~1.5 min. Try R = 3M at omega ~ 0.45: part of the packet is reflected by the barrier, part enters
the flat interior, bounces off the centrifugal barrier near the centre and leaks out in a train of echoes. At
R = 2.2M the leakage is slower (long-lived trapped modes). Turning the delta-shell off shows that its effect is small
at high frequency (the shell becomes transparent). For R = 2.2M the late signal decays like exp(-0.035 t), the imaginary part of the trapped QNM 0.4211 - 0.0350i.
The static shell is unstable even-parity matter modes
Four matter modes per $\ell$. One is purely imaginary with ${\rm Im}\,\omega>0$:
It agrees with Pitre, Schneider & Poisson (2026), who used a completely different method, at every compactness.
Newtonian origin: a pressure-supported self-gravitating membrane buckles.
$\varsigma^4-S\varsigma^2+P=0$ gives $1.505$ and $0.515\,i$ (stars).
The growth time is $\simeq1.6\,(R^3/M)^{1/2}$: slow compared with the waves.
~1 min. The shell is held up by tangential pressure against its own gravity; l >= 2 deformations run away.
This is ordinary Newtonian physics, and GR shifts the numbers by O(M/R). A by-product: the constant term of PSP26's quartic
(7.6) contains a misprint; their closed form (7.8) is right.
Back to step 1: is the background slow enough?
fluid shell Instability rate
$0.6(M/R^3)^{1/2}\ll$ wave frequencies. Scattering and QNMs are a good adiabatic idealization.
frozen wall Collapses on $R/\sqrt2$, as fast
as the waves. Its even junction is inconsistent (8 equations, 6 unknowns). Which 2 equations you drop changes $\mathcal R$ at $O(1)$.
~1 min. This closes the loop. For the frozen wall the odd sector is identical to the fluid shell (the
delta barrier), but the even sector is undetermined by the approximation: from total reflection to the delta law
depending on the prescription. It would need a genuinely time-dependent treatment.
From $\Psi_{\rm even}$, $\Psi_{\rm odd}$ to $h_+$, $h_\times$ what a detector measures
Parity is not polarization: each parity produces both $h_+$ and $h_\times$.
Duality: the factor $-i$ means the odd field is the even one with the polarization rotated by
$45^\circ$. They are $E$- and $B$-type patterns (gradient and curl).
Only for $m=0$ is even $=h_+$ and odd $=h_\times$ literally.
Both parities carry energy with the same weight.
~1 min. ${}_{-2}Y$ are the spin-weighted spherical harmonics, the natural basis for a spin-2 field on
the sky. For the l = 2, m = 2 mode of a binary, the even mode gives the familiar (1 + cos^2 theta) h+ and 2 cos theta hx; the odd
mode has them swapped. At the equator even is pure plus and odd is pure cross.
Can reflection turn $h_\times$ into $h_+$? a selection rule and a birefringent mirror
Selection rule
The object is spherical and parity invariant, so even and odd never mix. An axisymmetric
pure-$h_\times$ (odd) wave always comes back as pure $h_\times$, for a shell, a star or a black hole.
For one $(\ell,m)$, reflection acts on the $(E,B)$ components as a Jones matrix:
A wave with equal $E$ and $B$ content (e.g. $h_+=h_\times$ at $m=0$) comes back with a fraction
$\sin^2(\Phi/2)$ of its power in the orthogonal polarization. The same fraction flips the helicity.
~1.5 min. Parity is conserved, polarization is not. The object acts on the two parities like a
birefringent crystal: same or different amplitude, different phase. For the black hole the ratio follows exactly from the
Chandrasekhar transformation, which maps Regge–Wheeler solutions into Zerilli solutions. Isospectrality means equal
moduli, not equal phases: about 21 degrees of retardance at the l = 2 QNM frequency. Numerically verified to 5e-8.
Polarization of the reflected wave interactive, $\ell=2$, $m=0$
Incident: linear polarization with $h_+=h_\times$ (equal even and odd content; dashed line). Reflected:
$h_+\propto\cos(\omega t-\Phi)$, $h_\times\propto\cos\omega t$, an ellipse. Left: the retardance $\Phi(\omega)$ from the computed $S_{\rm even}$ and
$S_{\rm odd}$. Shells return all the energy; the black hole returns only the fraction $\mathcal R$.
~1.5 min. Move the slider. At low frequency all objects give the same small ellipse (the universal
black-hole value). For the shells the phase of S_even winds by 2 pi at the even-only matter resonance: the ellipse opens
into a circle and then into the orthogonal line, i.e. complete conversion h+ = hx -> h+ = -hx. Use the fine slider:
the resonance is only ~1e-4 wide.
How much polarization changes shells vs black hole
Low frequency: universal. Long waves only see the exterior field, so every
object has the black-hole retardance. At the even-only matter resonance conversion is complete; elsewhere a few per cent
(up to 19% at $R=2.2M$).
Helicity-reversing cross section of a circularly polarized plane wave: the same for any spherical
object at long wavelengths, and equal to the classic Schwarzschild result.
~1 min. The sum over multipoles is exactly 1/108, which gives 4 pi M^2 / 3, the helicity-reversing part
of the long-wavelength cross section M^2 [cos^8 + sin^8](theta/2) / sin^4(theta/2). Numerically: 4.188 M^2 for all objects at omega M = 0.02.
The same incident $h$ pulse, carried by $\Psi_{\rm even}$ or by $\Psi_{\rm odd}$
For one $(\ell,m)$, $h_+(t)$ and $h_\times(t)$ in any direction are a fixed angular factor
times $\Psi(u)$. So these curves are the reflected $h$. The prompt reflections almost coincide; the trapped modes differ
($0.421-0.035i$ odd, $0.447-0.044i$ even) and dephase.
Only the even channel keeps ringing at the matter frequency (decay time
$\sim3000M$ for $R=3M$) and excites the unstable mode, added causally from the pole of $S_{\rm even}$. It is weakly
excited but takes over after $\sim74$–$440M$.
~1 min. Waveforms are built by Fourier synthesis of S(omega) times the pulse spectrum, on adaptive frequency
grids that resolve the narrow resonances. The odd channel agrees with an independent time-domain evolution to better than 1%.
The growing mode requires moving the Fourier contour above its pole at omega = i gamma.
Is one parity more important?
Energy: neither
Equal weights in the flux; a plane wave contains both equally; a non-absorbing object
returns all the energy in both ($|S_{\rm even}|=|S_{\rm odd}|=1$).
Physics: even is richer
Only even waves move the shell and compress its matter: matter resonances, the instability,
and the dependence on the equation of state. Odd waves see a universal δ-barrier.
Sources: even dominates
For slow sources even $\ell=2$ is the mass quadrupole. The odd current quadrupole carries
$\frac{1}{36}\big(\frac{m_1-m_2}{m}\big)^2v^2$ of the power for a binary; radial infall excites only even modes.
~1 min. With an absorbing interior the two parities also differ in strength: near the barrier top the
even one is reflected up to about 18 times more (R = 6M). Practical rule: keep both for scattering of external waves; for
slow sources even dominates; the odd sector is the simple and universal one, a good first test of any code.
Why we trust the numbers
check
result
energy $|\mathcal R+\mathcal T-1|$
$\lesssim6\times10^{-10}$
transfer matrix vs shooting, $\mathcal R,\mathcal T$
$\lesssim10^{-9}$
same, 397 QNMs
median $4\times10^{-12}$, worst $6\times10^{-5}$
Schwarzschild $\ell=2,n=0$
$0.3736716844-0.0889623157i$
BH reflectivity (Vishveshwara)
$1.55\times10^{-2}$ at $(2M\omega)^2=1$
WKB orders 1, 3, 6
rel. error $7\%$, $0.15\%$, $0.02\%$
even junction $\det\mathsf T\sqrt{f_R}$
$1\pm10^{-12}$
matter modes vs Pitre et al. 2026 · Newtonian limit
agree
BH $S_{\rm even}/S_{\rm odd}$ vs Chandrasekhar
$5\times10^{-8}$
$\sigma_{\rm flip}(\omega\to0)$ vs $4\pi M^2/3$
$4.188$ vs $4.1888\,M^2$
Fourier synthesis vs time-domain evolution
$<1\%$ of the peak
~45 s. Two independent methods: continued-fraction exterior plus transfer matrix, and direct shooting.
Black-hole benchmarks from Leaver and Berti et al. Every limit is reproduced.
Take-home messages
A thin shell with flat interior is static only with positive pressure $p=\sigma(1-\sqrt{f_R})/4\sqrt{f_R}$. Domain walls and dust collapse.
For odd waves the shell is a repulsive δ-potential, $\Delta=\sqrt{f_R}(1-\sqrt{f_R})/R$. For even waves it is a
flux-conserving 2×2 matrix that carries the shell's own dynamics.
$\mathcal R(\omega)$: opaque at low frequency, transparent at high ($\propto\omega^{-2}$), black-hole-like as $R\to2M$.
The ringing is not black-hole-like: cavity modes, long-lived trapped modes (echoes), and matter modes.
The static shell is unstable (a Newtonian buckling mode) but slowly, so the results stand. The frozen domain wall is not a controlled model.
Parity is conserved, polarization is not: any spherical object is a birefringent mirror. Universal at long
wavelengths ($\sigma_{\rm flip}=4\pi M^2/3$), complete at the shell's matter resonance.
Neither parity dominates the energy; the even one carries all the matter physics and dominates slow sources.
Methods
Linearized Israel junction conditions derived symbolically; transfer-matrix and shooting
solvers (continued fraction, DOP853); Leaver and WKB benchmarks; time-domain leapfrog evolution; Fourier synthesis with
causal treatment of the unstable mode.
Key references
Regge & Wheeler 1957 · Zerilli 1970 · Vishveshwara 1970 · Israel 1966 · Ipser & Sikivie 1984 ·
Poisson 2004 · Martel & Poisson 2005 · Leaver 1985 · Iyer & Will 1987 · Pani et al. 2009 ·
Boyanov et al. 2025 · Yang, Bonga & Pan 2023 · Pitre, Schneider & Poisson 2026 · Chandrasekhar 1983 ·
Peters 1976 · Dolan 2008 · Blanchet 2014
~30 s. Wrap up.
Backup: multipoles, parity and equation of state
Higher $\ell$ see a higher barrier. Even and odd differ at intermediate frequencies, so the
shell breaks the isospectrality of Schwarzschild. $\Gamma$ matters only near the narrow matter resonances.
Backup slide for questions.
Backup: rotated polarization in the time domain, and a check
Left: for a pulse with equal $E$ and $B$ content, the part returning in the orthogonal polarization,
$(\Psi^{\rm even}_{\rm out}-\Psi^{\rm odd}_{\rm out})/2$: 3.7% of the reflected energy for the black hole, 1% for $R=3M$. Right: the
frequency-domain synthesis agrees with a direct time-domain evolution.
