Odd-Parity Metric Perturbations

Odd-parity, or axial, perturbations of a Schwarzschild black hole look like three independent metric amplitudes. In linearized vacuum gravity, however, their gauge-invariant content at each $\ell\ge2$ is determined by one Cunningham–Price–Moncrief master field. This page derives the invariant, the Einstein constraints and the local reconstruction of the metric. The reconstruction includes stationary perturbations: it is a different master variable, obtained by differentiating in time, that loses stationary information. The angular-momentum dipole must be treated separately.

Required background. PDEs and Green Functions for Gravitation supplies hyperbolic equations, Cauchy data and the distinction between an equation and its boundary conditions.

Helpful background. Black-Hole Geometry and Mechanics fixes the Schwarzschild horizon; Formulations of General Relativity separates constraints, gauge and propagation; Killing Fields and Isometry Groups explains the exceptional rotational dipole.

Axial metric amplitudes

Consider the first-order vacuum perturbation $g_{\mu\nu}(\eta)=g_{\mu\nu}^{(0)}+\eta p_{\mu\nu}+O(\eta^2)$, where $\eta$ is a dimensionless expansion parameter and $\delta G_{\mu\nu}[p]=0$. The background is the four-dimensional Schwarzschild solution of mass $M$, with $\mu=GM\gt0$. In the exterior $r\gt2\mu$,

$$ds_0^2=g_{ab}\,dx^a dx^b+r^2\Omega_{AB}\,d\theta^A d\theta^B,$$ $$g_{ab}\,dx^a dx^b=-f\,dt^2+f^{-1}dr^2, \qquad f=1-\frac{2\mu}{r}.$$

Lowercase indices $a,b$ run over $t,r$; uppercase indices $A,B$ run over the unit sphere. Write $\nabla_a$ for the orbit-space derivative, $D_A$ for the unit-sphere derivative, and $r_a=\nabla_a r$. The two-dimensional orbit space is the space left after the symmetry spheres have been factored out; its metric, not the full four-metric, raises lowercase indices.

For one $(\ell,m)$ mode, suppress the harmonic labels and set $L=\ell(\ell+1)$ and $q=L-2=(\ell-1)(\ell+2)$. The conventions of Tensor Harmonics and Parity Decomposition give

$$D^A D_A Y=-LY,\qquad X_A=-\epsilon_A{}^B D_B Y,\qquad X_{AB}=D_{(A}X_{B)},\qquad \epsilon_{\theta\phi}=\sin\theta.$$

Parentheses have weight one-half. The scalar harmonics have unit integral norm; the vector and tensor harmonics retain the derivative normalization in this definition. In particular, $X_{AB}$ is trace-free. An odd metric mode has

$$p_{ab}=0,\qquad p_{aB}=h_a(t,r)X_B,\qquad p_{AB}=h_2(t,r)X_{AB}.$$

The coefficient $h_2$ here does not include an extra extracted factor $r^2$. With $t,r$ measured in length and angles dimensionless, $h_a$ has dimension length and $h_2$ has dimension length squared. This distinction matters in the gauge transformation.

Odd parity means the angular tensor transforms with $(-1)^{\ell+1}$ under the antipodal map; it does not mean that only odd values of $\ell$ are allowed. There is no odd monopole because $X_A^{00}=0$. At $\ell=1$, $X_A$ survives but $X_{AB}=0$. Until the dipole section, assume $\ell\ge2$, so that $q\gt0$ and all three amplitudes exist. These definitions match Martel and Poisson 2005, §§ III and V.A, pp. 104003-3 and 104003-6–7, PDF; their geometrized mass is the length $\mu$ used here.

Removing the gauge dependence

An odd gauge generator has covariant components $\Xi_a=0$ and $\Xi_A=\xi(t,r)X_A$. Use the inverse-identification generator, so that $p'=p-\mathcal L_\Xi g^{(0)}$. This is the same transformation as Chapter 1’s active-plus convention with the generator replaced by $\Xi=-\xi_{\mathrm{Ch.\,1}}$; it is not a different physical gauge freedom. The four-dimensional Lie derivative gives

$$h_a'=h_a-\nabla_a\xi+\frac{2r_a}{r}\xi, \qquad h_2'=h_2-2\xi.$$

The radial term comes from the warped-product connection $\Gamma^A{}_{aB}=r_a\delta^A_B/r$. Both covariant derivatives in $2\nabla_{(a}\Xi_{B)}$ contribute one such term. Omitting one of them would spoil gauge invariance.

The invariant orbit-space one-form is

$$k_a\equiv\widetilde h_a =h_a-\frac12\nabla_a h_2+\frac{r_a}{r}h_2 =h_a-\frac{r^2}{2}\nabla_a\!\left(\frac{h_2}{r^2}\right).$$

Indeed, the change in $h_a$ is canceled by the changes in the two $h_2$ terms, without using any field equation. Choosing $\xi=h_2/2$ gives the odd Regge–Wheeler gauge $h_2'=0$ and $h_a'=k_a$. This is a convenient representative, not the definition of the physical field. For a global boundary-value problem, the generator must also be regular and preserve the chosen boundary conditions. The transformation and invariant are Martel and Poisson 2005, § V.A, Eqs. (5.5)–(5.7), p. 104003-7, PDF.

The axial Einstein equations

Fix the orbit volume form by $\epsilon_{tr}=+1$. Raising both indices with the Lorentzian orbit metric gives $\epsilon^{tr}=-1$, not $+1$. Define the invariant curl

$$C=\epsilon^{ab}\left(\nabla_a k_b-\frac{2r_a}{r}k_b\right) =\partial_r k_t-\partial_t k_r-\frac{2k_t}{r}.$$

It is the curl of the radially rescaled one-form: equivalently, $C=r^2\epsilon^{ab}\nabla_a(k_b/r^2)$. Thus it is not enough to take the ordinary curl of $k_a$ without the $2k_t/r$ term.

The odd projections of the linearized Einstein tensor can be written

$$\delta G_{ab}=0,\qquad \delta G_{aB}=\frac12 E_a X_B,\qquad \delta G_{AB}=E X_{AB},$$ $$E_a=\frac{qk_a-\epsilon_{ab}\nabla^b(r^2C)}{r^2}, \qquad E=\nabla_a k^a.$$

These are the complete vacuum equations $E_a=0$ and $E=0$ in a compact form. One way to obtain them is to compute the linearized Ricci tensor in $h_2=0$ gauge, where $h_a=k_a$, and project onto $X_B$ and $X_{AB}$. The Ricci-flat background makes the resulting linearized Einstein tensor gauge invariant, so the same formulas in $k_a$ hold in every odd gauge. Expanding the curl gives the projected operator and scalar equation of Martel and Poisson 2005, § V.B, Eqs. (5.8)–(5.12), p. 104003-7, PDF.

For an explicit calculation, the three expressions are

$$\begin{aligned} E_t&=\frac{q}{r^2}k_t-\frac{f}{r^2}\partial_r(r^2C),\\ E_r&=\frac{q}{r^2}k_r-\frac{1}{fr^2}\partial_t(r^2C),\\ E&=-\frac{1}{f}\partial_t k_t+\partial_r(fk_r). \end{aligned}$$

They are not three independent evolution equations. The off-shell Bianchi identity is

$$\nabla_a E^a+\frac{2r_a}{r}E^a-\frac{q}{r^2}E=0.$$

Here “off shell” means before imposing $E_a=0$ or $E=0$. It follows that, for $q\ne0$, the vector equations imply the scalar constraint. This identity explains why the apparently separate metric components cannot be assigned arbitrary independent dynamics.

The master field and the local inverse

The Cunningham–Price–Moncrief field is

$$\Psi_{\mathrm{odd}} =\frac{2r}{q}C =\frac{2r}{q}\epsilon^{ab} \left(\nabla_a k_b-\frac{2r_a}{r}k_b\right).$$

It is a gauge-invariant orbit scalar of dimension length. Its normalization agrees with Martel and Poisson 2005, § V.C, Eq. (5.13), p. 104003-7, PDF. The decisive simplification is that $r^2C=qr\Psi_{\mathrm{odd}}/2$. Substitute this into $E_a=0$, cancel $q$, and obtain

$$k_a=\frac12\epsilon_{ab}\nabla^b(r\Psi_{\mathrm{odd}}),$$ $$k_t=\frac f2\partial_r(r\Psi_{\mathrm{odd}}), \qquad k_r=\frac{r}{2f}\partial_t\Psi_{\mathrm{odd}}.$$

Thus the full master field determines the invariant metric locally in vacuum at fixed $\ell\ge2$. There is no additional stationary invariant piece at fixed $\Psi_{\mathrm{odd}}$: if the master field is zero, the vector equations force $k_a=0$. No division by frequency or integration in time has been used. A metric representative follows by choosing $h_2$ and using

$$h_a=k_a+\frac12\nabla_a h_2-\frac{r_a}{r}h_2.$$

What equation must the master field satisfy? Insert the reconstructed $k_a[\psi]$ into the definition of $C$. A direct differentiation gives

$$\frac{2C[k[\psi]]}{r} =-\frac{1}{f}\partial_t^2\psi+\partial_r(f\partial_r\psi) +\left(\frac{f'}{r}-\frac{2f}{r^2}\right)\psi, \qquad f'=\frac{df}{dr}.$$

For consistency with $\psi=2rC/q$, the left side must equal $q\psi/r^2$. Use $f'=2\mu/r^2$ and $dr_*/dr=f^{-1}$, and multiply by $f$. This yields

$$\left[-\partial_t^2+\partial_{r_*}^2-V_{\mathrm{RW}}(r)\right]\psi=0, \qquad V_{\mathrm{RW}}(r)=f\left[\frac{\ell(\ell+1)}{r^2}-\frac{6\mu}{r^3}\right].$$

Conversely, any smooth solution of this equation reconstructs a one-form whose curl returns that same $\psi$. Its vector equations then vanish, and its scalar constraint vanishes directly because

$$\nabla_a k^a =-\frac12\partial_t\partial_r(r\psi) +\frac12\partial_r\partial_t(r\psi)=0.$$

This proves both directions of the vacuum reduction, rather than merely assigning a known wave equation to a convenient combination of components. The operator agrees with Martel and Poisson 2005, § V.C, Eqs. (5.14)–(5.15), p. 104003-8, PDF. The curvature term $-6\mu/r^3$ comes from the Einstein reduction. A constant rescaling of the harmonics or master field cannot change this homogeneous potential; it changes the conversion between amplitudes and observables.

The potential has dimension length$^{-2}$, tends to the centrifugal potential at large radius, and vanishes at the horizon. These are useful checks, not a specification of a scattering or quasinormal-mode problem. One dynamical configuration variable requires two initial-data functions, $\psi(t_0,r_*)$ and $\partial_t\psi(t_0,r_*)$. Their regularity and any boundary or asymptotic conditions still have to be chosen.

Horizon coordinates and stationary information

The factors $1/f$ in Schwarzschild coordinates do not themselves establish a singular perturbation. Use advanced time $v=t+r_*$, for which the orbit metric is $-f\,dv^2+2\,dv\,dr$. The one-form transformation is $k=k_t\,dv+(k_r-k_t/f)\,dr$. For the same scalar $\psi$, the reconstruction takes the following forms; radial derivatives in each row hold that row’s time coordinate fixed.

Orbit coordinates	Time component	Radial component
$(t,r)$	$k_t=\tfrac f2\partial_r(r\psi)$	$k_r=\tfrac r{2f}\partial_t\psi$
$(v,r)$	$k_v=\tfrac f2\partial_r(r\psi)+\tfrac r2\partial_v\psi$	$k_r^{(v,r)}=-\tfrac12\partial_r(r\psi)$

The apparently singular terms cancel in the second row. If $\psi(v,r)$ is smooth through the future horizon, these components are smooth there and give a regular local representative with $h_2=0$. The field equation alone does not guarantee that an arbitrarily chosen solution has that smoothness.

A second invariant often called the Regge–Wheeler function is

$$Q_{\mathrm{RW}}\equiv\frac{f k_r}{r} =\frac12\partial_t\Psi_{\mathrm{odd}}.$$

The equality uses the vacuum Einstein equations, not just the definition of the invariants. It is the exact factor in Martel and Poisson 2005, § V.D, Eqs. (5.17)–(5.18), p. 104003-8, PDF. For a Fourier mode, $Q_{\mathrm{RW}}=-i\omega\Psi_{\mathrm{odd}}/2$. At nonzero frequency the two variables are interchangeable. At zero frequency that inverse does not exist: a stationary nonzero $\Psi_{\mathrm{odd}}$ can have $Q_{\mathrm{RW}}=0$. The stationary quadrupole in the exercises makes the distinction explicit.

With asymptotically outgoing, finite radiative data, this CPM normalization also gives

$$p_{AB}^{\mathrm{rad,odd}} =r\sum_{\ell m}\Psi_{\mathrm{odd}}^{\ell m}(u,\infty)X_{AB}^{\ell m}+O(1), \qquad u=t-r_*.$$

This is a large-radius radiation-gauge statement, not the angular metric in Regge–Wheeler gauge. The derivation and observable conversion are developed in Metric Reconstruction and Radiative Observables and in Martel and Poisson 2005, § VI, Eqs. (6.9)–(6.13), p. 104003-9, PDF.

The rotational dipole

At $\ell=1$, $X_{AB}=0$ and $q=0$. There is no amplitude $h_2$ and no CPM definition obtained by dividing by $q$. The remaining $h_a$ are gauge dependent, but their rescaled curl is still invariant:

$$\mathcal F_{tr} =\partial_t\!\left(\frac{h_r}{r^2}\right) -\partial_r\!\left(\frac{h_t}{r^2}\right), \qquad \left(\frac{h_a}{r^2}\right)' =\frac{h_a}{r^2}-\partial_a\!\left(\frac{\xi}{r^2}\right).$$

This $\mathcal F$ is a metric-perturbation curl, not an electromagnetic field. The vector Einstein equations remain defined, with $q=0$ and $C=-r^2\mathcal F_{tr}$. They imply

$$\partial_r(r^4\mathcal F_{tr})=0, \qquad \partial_t(r^4\mathcal F_{tr})=0.$$

Thus there is a conserved dipole coefficient, not a propagating odd spin-two wave. Both equations matter: radial constancy alone would allow that coefficient to vary in time. There is no independent $E X_{AB}$ equation when the tensor harmonic vanishes. This is the exceptional sector described in Martel and Poisson 2005, § V.E, Eqs. (5.21)–(5.22), p. 104003-8, PDF.

For example, choose the rotation axis so that $Y_{10}=N\cos\theta$, with $N=\sqrt{3/(4\pi)}$. Then $X_\phi=-N\sin^2\theta$. Differentiating the Kerr metric at zero angular momentum gives $p_{t\phi}=-2G\delta J\sin^2\theta/r$, hence

$$h_t=\frac{2G\delta J}{Nr},\qquad h_r=0,\qquad r^4\mathcal F_{tr}=\frac{6G\delta J}{N}.$$

Here $\delta J$ is the physical angular-momentum variation. These are exterior Schwarzschild-coordinate components, not a claimed regular horizon chart for the dipole. A rotating coordinate frame instead gives $h_t=Ar^2$ and zero curl, as the exercise shows. Distinguishing the physical angular-momentum change from an allowed frame change requires the asymptotic frame and admissible gauge transformations to be fixed; see Low Multipoles, Parameter Shifts, and Gauge Modes.

The result of this page is therefore one gauge-invariant master field for each vacuum odd $\ell\ge2$, together with a separate nonradiative dipole. Matter sources require the conserved source projections and their jumps; they cannot be inserted into the vacuum inverse without modification. The Regge–Wheeler Equation develops that next stage. Rotating-background separability belongs to the Kerr chapters, where this spherical parity split is not the full reduction.

Exercises

A pure-gauge metric with nonzero components. Let $\chi(t,r)$ be smooth and set $h_2=2\chi$ and $h_a=\nabla_a\chi-2r_a\chi/r$. Compute $k_a$ and $\Psi_{\mathrm{odd}}$. Can this perturbation always be removed in a prescribed global problem?

Solution

Substitution gives

$$k_a=\nabla_a\chi-\frac{2r_a}{r}\chi -\frac12\nabla_a(2\chi)+\frac{r_a}{r}(2\chi)=0.$$

Consequently $C=0$ and $\Psi_{\mathrm{odd}}=0$. The generator $\xi=\chi$ sets both $h_2'$ and $h_a'$ to zero. This proves local pure-gauge form. Global removability additionally requires that this generator obey the problem’s regularity and boundary conditions; nonzero metric components alone establish neither radiation nor a physical charge.

A stationary field lost by time differentiation. For $\ell=2$, test $\psi=cr^3$, where $c$ is a constant of dimension length$^{-2}$. Reconstruct $k_a$, verify the static master equation, and compute $Q_{\mathrm{RW}}$. Is this an asymptotically flat black-hole deformation?

Solution

The reconstruction gives

$$k_t=2cfr^3,\qquad k_r=0,\qquad Q_{\mathrm{RW}}=0.$$

Since $\partial_{r_*}=f\partial_r$,

$$\partial_{r_*}^2(cr^3) =3cf(f'r^2+2fr) =6cf(r-\mu) =V_{\mathrm{RW}}(r)\,cr^3 \qquad(\ell=2).$$

The master equation is therefore satisfied, and the scalar constraint vanishes because the metric is stationary and $k_r=0$. The ingoing components are $k_v=2cfr^3$ and $k_r^{(v,r)}=-2cr^3$, which are smooth at the future horizon. But the field grows at large radius; it represents a local growing tidal solution, not an isolated asymptotically flat deformation. For nonzero $\eta c$, its perturbative use must also be restricted to radii where the metric correction remains small. A static nonzero invariant is not synonymous with radiation or new isolated black-hole hair.

Angular momentum or a rotating frame? In the stationary dipole sector, take $h_r=0$ and solve the conserved-curl equation for $h_t(r)$. Identify which integration constant changes the angular momentum.

Solution

Writing $r^4\mathcal F_{tr}=K$ gives $-\partial_r(h_t/r^2)=K/r^4$, so

$$h_t=Ar^2+\frac{B}{r},\qquad K=3B.$$

The term $Ar^2$ has zero curl. The generator $\xi=At r^2$ removes it: $h_t'=h_t-Ar^2$ and $h_r'=-\partial_r(At r^2)+2At r=0$. It is a coordinate rotation; whether this generator is allowed is fixed by the boundary-frame convention. The $B/r$ term has nonzero invariant curl and corresponds to $B=2G\delta J/N$ in the normalization above. It cannot be removed by an odd gauge transformation, whose change in $\mathcal F_{tr}$ is identically zero.

References

• Martel, Karl, and Eric Poisson. “Gravitational Perturbations of the Schwarzschild Spacetime: A Practical Covariant and Gauge-Invariant Formalism.” Physical Review D 71 (2005): 104003. DOI. Open PDF.