Regge–Wheeler, Zerilli, and Gauge-Invariant Master Variables

For each angular harmonic with $\ell\geq2$, Schwarzschild gravitational perturbations have two physical configuration degrees of freedom: one odd-parity master field and one even-parity master field. Each satisfies a scalar wave equation, but the equation alone does not fix its physical normalization, source or boundary data. This chapter derives that complete description on the exterior $r>2GM$, then reconstructs the invariant metric and its radiation. The reduction is linear; the energy carried by the perturbation is quadratic in its amplitude.

Helpful background. Black-Hole Geometry and Mechanics fixes the Schwarzschild exterior and its null boundaries; PDEs and Green Functions for Gravitation supplies hyperbolic and distributional methods; Formulations of General Relativity separates constraints from evolution; Presymplectic Potentials and Currents prepares canonical reduction and flux.

Enter the Schwarzschild master system

Write $x=r_*$ for the tortoise coordinate and set

$$f=1-\frac{2GM}{r}, \qquad \frac{dx}{dr}=f^{-1}, \qquad \lambda=\frac{(\ell-1)(\ell+2)}2.$$

The horizon lies at $x\to-\infty$ and spatial infinity at $x\to+\infty$. The complete odd field is the Cunningham–Price–Moncrief (CPM) field; the even field is the Zerilli–Moncrief field. Both have dimension length. For either parity $P$, the common form of the equation is

$$L_P\Psi_P=S_P,\qquad L_P=-\partial_t^2+\partial_x^2-V_P(r).$$

Here $V_P$ has dimension inverse length squared and $S_P$ inverse length. The potential is $V_{\mathrm{RW}}$ in odd parity and $V_Z$ in even parity. The source is obtained from a conserved stress tensor, not chosen independently of the Einstein constraints when a matter interpretation is required.

Three distinctions organize the calculation.

• Invariant versus gauge representative. Gauge fixing chooses metric components; an invariant is unchanged under the admitted gauge transformations.
• Local inverse versus global completion. In a smooth vacuum region, a complete CPM or Zerilli–Moncrief solution determines all local invariant metric information at its fixed $\ell\geq2$. A global problem still specifies gauge regularity, boundary data and the separate $\ell=0,1$ sectors.
• Differential identity versus boundary-value problem. A transformation between equations identifies spectra only when it also maps the chosen boundary domains and remains invertible.

Detailed spectral solution methods belong to the later response chapters; generic numerical implementation belongs to Computational Relativity.

Check your preparation

Before starting, try these three short checks.

1. Which profile travels toward increasing $x$: $F(t-x)$ or $G(t+x)$? The first: a fixed phase has $dx/dt=+1$. If this is unfamiliar, review characteristics and Green functions. Travel direction and the outward normal of a boundary are different notions.
2. Does a unit-normalized scalar harmonic give a unit-normalized tensor harmonic after differentiation? No: derivatives introduce an $\ell$-dependent norm. Review tensor harmonics and parity before comparing energy coefficients.
3. Does setting one gauge amplitude to zero, by itself, remove a physical canonical pair? No: the constraint and its gauge direction must be treated together, with boundary charges handled separately. Review constraints and evolution and presymplectic currents.

Six pages in derivational order

This is the default derivational order. The qualifications in the last column are part of the results, not exceptions to discard on a first reading.

Page	Result carried forward	Qualification to retain
1. The Regge–Wheeler Equation	Complete odd master, physical action and signed energy balance.	The alternative derivative-normalized RW variable loses stationary CPM information; a finite-cut balance is not a decay theorem.
2. The Zerilli Equation	Polar constraints give a complete local vacuum master and inverse.	No extra constraint is imposed on complete master Cauchy data; low multipoles and sourced reconstruction are separate.
3. Moncrief Invariants and Hamiltonian Reduction	One physically normalized canonical pair per real harmonic and parity.	Compact-support bulk reduction does not classify boundary charges or identify every metric and scalar boundary condition.
4. Chandrasekhar Transformations and Isospectrality	Differential map, endpoint factors and Wronskian relation.	Exceptional frequencies, continued endpoint normalization and transformed domains matter. Equal poles do not determine equal waveforms.
5. Sourced Master Equations and Junction Conditions	Conserved stress projections, timelike moving jumps and retarded matching.	Master jumps do not replace shell matching or self-force regularization.
6. Metric Reconstruction and Radiative Observables	Complete vacuum masters determine invariant metrics; boundary limits determine observables.	Retain the gauge/frame, curvature tetrad, low multipoles and horizon endpoint conditions.

At each step ask: What is invariant? Which constraint was used? Which boundary orientation fixes the sign? Which information is in the master solution, and which is supplied by gauge, boundary conditions or separate low multipoles?

Two reader routes

Understand the reduction: 1 → 2 → 3 → 4. Compare both potentials, count one canonical pair per parity, and track boundary amplitudes through the first-order map. Finish able to explain conditional spectral equivalence. This is neither a computation of quasinormal frequencies nor a proof that all initial energy decays.

Compute sourced observables: 1 → 2 → 5 → 6. Consult page 3 when fixing the action normalization. Derive the source from conserved matter, compare direct jump matching with Green-function matching, and reconstruct on each smooth vacuum side. Finish by comparing the reduced current with the radiation metric, using the correct boundary limits.

One normalization from action to radiation

For one real harmonic with $\int Y^2d\Omega=1$, and one parity, define

$$N_\ell=\frac{(\ell+2)!}{(\ell-2)!},\qquad C_\ell=\frac{N_\ell}{64\pi G}.$$

The vacuum reduced action and its canonical momentum are

$$I_P^{(2)}=\frac{C_\ell}{2}\int dt\,dx \left[(\partial_t\Psi_P)^2-(\partial_x\Psi_P)^2-V_P\Psi_P^2\right], \qquad \Pi_P=C_\ell\partial_t\Psi_P.$$

Use compactly supported variations here; a problem with boundaries also specifies its boundary action and allowed variations. The factor $C_\ell$ is fixed by the gravitational action and the harmonic normalization. Multiplying the master by a constant leaves its homogeneous equation unchanged but does not leave this coefficient unchanged. The Hamiltonian reduction derives it rather than assigning a unit-normalized scalar action.

On a constant-$t$ slice between fixed tortoise radii $a\lt b$, a smooth vacuum solution has energy density $e$ and directed current $j$:

$$\begin{aligned} e&=\frac{C_\ell}{2} \left(\Psi_t^2+\Psi_x^2+V_P\Psi^2\right),& j&=-C_\ell\Psi_t\Psi_x,\\ E_{[a,b]}&=\int_a^b e\,dx,& \frac{dE_{[a,b]}}{dt}&=j(t,a)-j(t,b). \end{aligned}$$

The suppressed parity label is fixed in this equation. It follows by multiplying $L_P\Psi=0$ by $C_\ell\Psi_t$ and integrating $\partial_t e+\partial_x j=0$. A right-travelling field has positive $j$; a left-travelling field has negative $j$. Loss from the exterior at its left boundary is therefore $-j(t,a)$, not negative absorbed energy. The finite-cut example shows why boundary segments and residual energy must be retained before taking limits.

For an outgoing radiation field $\psi_P(u)$ at future null infinity, $u=t-x$, and a regular ingoing field $\psi_{P,H}(v)$ at the future horizon, $v=t+x$, the corresponding one-real-mode quadratic powers are

$$\mathcal F_\infty=C_\ell(\partial_u\psi_P)^2,\qquad \mathcal F_H=C_\ell(\partial_v\psi_{P,H})^2.$$

The first is the radiation luminosity; the second is the positive shear power on a vacuum horizon. These are limiting fields, not arbitrary finite-radius profiles. Their common coefficient agrees with the averaged fluxes in Martel and Poisson 2005, Eqs. (6.16), (7.5), pp. 104003-10–11. For complex harmonics, sum the modulus squares over both parities and the complete $m=-\ell,\ldots,\ell$ range with the reality relation $\psi_P^{\ell,-m}=(-1)^m\overline{\psi_P^{\ell m}}$. Do not treat a single unpaired complex coefficient as an independent real wave.

Positive shear power is not, on arbitrary horizon cuts, an instantaneous black-hole mass derivative. The finite-cut area balance contains an expansion endpoint term. Its settled-state mass interpretation uses stationary endpoints, or the appropriate averaging approximation; see Poisson 2004, § IV, p. 12, arXiv v2 PDF and the explicit finite-cut derivation. Likewise, positive exterior energy does not by itself prove complete decay.

A quadrupole driven by a localized source

Take one quadrupolar mode, $\ell=2$. Then $\lambda=2$. The odd potential is

$$V_{\mathrm{RW}}=f\left(\frac6{r^2}-\frac{6GM}{r^3}\right),$$

while the even potential has a different positive rational profile. Both vanish at the horizon and have leading asymptotic term $6/r^2$ at infinity; inspect the two barrier profiles. Here $C_2=3/(8\pi G)$ for each real harmonic and parity. Different potentials do not imply different numbers of physical degrees of freedom.

Now drive the equation with a localized source. A delta-prime term can jump the master field itself. For a moving interface, the time derivative of the one-sided trace contains a velocity term; inspect the moving-source construction. Reconstruction must respect these jumps and use a smooth vacuum inverse only on the appropriate sides.

Specifying the source is not enough to select the solution. Schematically,

$$\Psi_P=\Psi_{P,\mathrm{hom}}+G_{P,\mathrm{ret}}*S_P.$$

The convolution integrates over source time and tortoise position; $L_PG_{P,\mathrm{ret}}=\delta(t-t')\delta(x-x')$ with retarded support. The homogeneous term carries independent initial or incoming data. For admissible no-incoming past data and an asymptotically flat solution admitting the smooth outgoing limit used in reconstruction, retarded forcing selects the corresponding radiation field at future null infinity. Spatial separation from the source alone does not do so: a vacuum region can contain incoming waves, and its nonzero potential produces backscattering. At finite radius the solution is generally not exactly a function of $u$.

The $\ell=0,1$ modes complete this picture without joining the radiative master system. They describe changes of mass and angular momentum and choices of center of mass, subject to gauge and boundary conditions. Calling them “missing” modes is misleading; they solve different, degenerate harmonic systems.

Chapter review

Select the radiation before interpreting its power

Two solutions have the same compact source but differ by a smooth homogeneous solution. Must they have the same waveform? Does a positive outward flux prove that all initial exterior energy escapes to infinity?

Solution

No to both. Linearity gives $L_P(\Psi_1-\Psi_2)=0$, so source equality does not fix initial or incoming data. Adding a homogeneous pulse changes the waveform without changing the source. Specify a retarded prescription and the independent past or Cauchy data to obtain a unique solution in the chosen class.

On a vacuum interval, or after forcing there has ceased, integrating the balance over $t_0\leq t\leq t_1$ gives

$$E_{[a,b]}(t_1)-E_{[a,b]}(t_0) =-\int_{t_0}^{t_1}\left[j(t,b)-j(t,a)\right]dt.$$

There can be residual exterior energy and flux through both boundary portions. A whole-future statement requires controlled limits and a decay argument. The corresponding arbitrary-cut horizon mass interpretation also requires the expansion endpoint correction described above.

Change the master without changing the physics

For a fixed parity and one real unit-normalized harmonic, let $\Phi=s\Psi$, where $s$ is a real, nonzero, dimensionless constant held fixed under variations. Suppose a fixed interface at $x=x_0$ has source

$$S=F(t)\delta'(x-x_0)+B(t)\delta(x-x_0), \qquad [\Psi]=F,\qquad [\Psi_x]=B,$$

where brackets mean right minus left and the prime differentiates with respect to $x$. How do the source, jumps, action coefficient, canonical momentum, linear metric inverse, strain coefficient and unit-source Green function change? Check $s=2$. Why is $s=s(x)$ a different problem?

Solution

Since the constant commutes with $L_P$,

$$\begin{aligned} S_\Phi&=sS,& [\Phi]&=sF,&[\Phi_x]&=sB,\\ C_{\ell,\Phi}&=\frac{C_\ell}{s^2},& \Pi_\Phi&=\frac{\Pi_\Psi}{s}.&& \end{aligned}$$

Substitution in the full action leaves it unchanged. The same holds for its smooth prescribed-source term $-C_\ell\int S\Psi\,dt\,dx$; this expression is not a definition of singular-source self-work. The canonical two-form is unchanged because $\delta\Phi\wedge\delta\Pi_\Phi=\delta\Psi\wedge\delta\Pi_\Psi$. Energy and flux are unchanged as well: $C_{\ell,\Phi}(\partial_u\Phi)^2=C_\ell(\partial_u\Psi)^2$ for the corresponding radiation fields.

If the linear metric inverse is $\mathcal R$, use $\mathcal R_\Phi[\Phi]=\mathcal R[\Phi]/s$. The coefficient converting the master to strain likewise gains $1/s$. The Green kernel does not gain $s$: the operator, boundary prescription and unit delta source defining it have not changed. Instead $\Phi=G_{\mathrm{ret}}*(sS)$ for the retarded particular solution, and the homogeneous data rescale too. Using $sG_{\mathrm{ret}}$ would give $L_P(sG_{\mathrm{ret}})=s\delta(t-t')\delta(x-x')$.

For $s=2$ the source and jumps double, the physical coefficient becomes $C_\ell/4$, and the momentum and inverse/strain coefficients halve. Keeping $C_\ell$ would overestimate the power by four. This invertible change is not the vacuum relation $Q_{\mathrm{RW}}=\partial_t\Psi_{\mathrm{odd}}/2$, which annihilates stationary CPM information.

A position-dependent multiplier does not commute with the operator:

$$L_P(s\Psi)=sL_P\Psi+2s'\Psi_x+s''\Psi.$$

For example, with $V=0$, $\Psi=x$ and $s=e^{kx}$ for nonzero constant $k$, $L\Psi=0$ but $L(s\Psi)=(2k+k^2x)e^{kx}$. The extra derivative terms cannot be hidden in a constant normalization.

Map the boundary conditions

Consider a vacuum Regge–Wheeler mode with the inherited $e^{-i\omega t}$ convention and factorization data

$$A=\partial_x+W,\qquad B=-\partial_x+W,\qquad \alpha=\frac{\lambda(\lambda+1)}{3GM},\qquad W\to\alpha$$

at both endpoints. On RW solutions, $BA\psi=(\omega^2+\alpha^2)\psi$. Determine when $A$ identifies the standard quasinormal boundary lines. Why do leading exponentials alone not settle this question at damped complex frequencies? What happens to a Dirichlet wall, and does a shared pole determine a shared waveform?

Solution

First select the unit-normalized horizon and infinity solutions by their decaying behavior for $\operatorname{Im}\omega\gt0$, then analytically continue them on a specified sheet, with a specified lateral value on a cut. Their radial wave factors are $e^{-i\omega x}$ and $e^{+i\omega x}$. The map multiplies them by $\alpha-i\omega$ and $\alpha+i\omega$. With regular, nonzero endpoint normalizations and $\omega\ne\pm i\alpha$, the inverse is $B/(\omega^2+\alpha^2)$ and maps the same continued boundary lines back.

For $\operatorname{Im}\omega\lt0$ the displayed endpoint waves grow; adding a subdominant unwanted solution can leave the leading exponential unchanged. That is why continuation, rather than only the leading exponential, defines the boundary line. Normalization anomalies and both noninvertible frequencies need separate analysis; the damped one is $-i\alpha$. See Maassen van den Brink 2000, § II.A, pp. 3–4, arXiv v1 PDF for the continuation and normalization issues, and the full Chandrasekhar analysis for the exceptional solutions.

At a finite wall, $\psi(x_0)=0$ implies for $\chi=A\psi$ that $B\chi(x_0)=0$, or $\chi'(x_0)=W(x_0)\chi(x_0)$. The mapped condition is Robin, not generally Dirichlet. Keeping Dirichlet for both problems changes the domain comparison.

Finally, a shared regular pole does not fix the Green kernel’s numerator, its residue, or the projection of a physical source onto it. The differential map transforms forcing as well as solutions. Conditional equality of resonance locations therefore does not assert equal excitation amplitudes or waveforms.

Where to go next

• To follow causal forcing and resonance excitation, continue to Green Functions, Scattering, and Retarded Response.
• To compute resonances, use Quasinormal Modes, Resonances, and Pseudospectra.
• To introduce rotation, enter Kerr Perturbations and Teukolsky Theory.
• To study the mass-transfer interpretation further, read Horizon Absorption and Tidal Heating.
• For discretization, convergence and implementation, use Computational Relativity.

References

• Maassen van den Brink, A. (2000). “Analytic treatment of black-hole gravitational waves at the algebraically special frequency.” Physical Review D 62, 064009. DOI. Open PDF, arXiv v1.
• Martel, K., and Poisson, E. (2005). “Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and gauge-invariant formalism.” Physical Review D 71, 104003. DOI.
• Poisson, E. (2004). “Absorption of mass and angular momentum by a black hole: Time-domain formalisms for gravitational perturbations, and the small-hole/slow-motion approximation.” Physical Review D 70, 084044. DOI. Open PDF, arXiv v2.

Further reading

• Moncrief, V. (1974). “Gravitational perturbations of spherically symmetric systems. I. The exterior problem.” Annals of Physics 88, 323–342. DOI.
• Regge, T., and Wheeler, J. A. (1957). “Stability of a Schwarzschild Singularity.” Physical Review 108, 1063–1069. DOI.
• Zerilli, F. J. (1970). “Effective potential for even-parity Regge–Wheeler gravitational perturbation equations.” Physical Review Letters 24, 737–738. DOI.