Black hole perturbations and master equations

On this page

• counting physical degrees of freedom (dofs) in a $D$-dimensional Einstein–Maxwell theory,
• organizing perturbations by symmetry (tensor/vector/scalar under a residual rotation group),
• understanding why the system reduces to master scalars obeying decoupled second-order ODEs,
• what is special about AdS$_4$ vs AdS$_5$ (parity vs helicity, tensor channel existence, and when “isospectrality” does or does not hold).
• generalizing to Einstein–Maxwell–dilaton (EMD) systems

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1. Einstein–Maxwell theory in $D$ dimensions

We start with perturbations of Reissner–Nordström–AdS (RN–AdS) black holes / branes, but the dof counting and sector decomposition are much more general.

We consider Einstein–Maxwell with a cosmological constant (AdS):

$$S=\frac{1}{2\kappa^2}\int d^D x\,\sqrt{-g}\left(R-2\Lambda\right) -\frac{1}{4g_F^2}\int d^D x\,\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}, \qquad \Lambda<0.$$

A standard RN–AdS background (static, “maximally symmetric horizon”) can be written in the Kodama–Ishibashi (KI) form

$$ds^2 = -f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Sigma_{n,K}^2, \qquad n\equiv D-2,$$

where $d\Sigma_{n,K}^2$ is an $n$-dimensional space of constant curvature $K\in\{1,0,-1\}$ (sphere/plane/hyperboloid). A convenient parameterization is

$$f(r)=K-\lambda r^2-\frac{2M}{r^{n-1}}+\frac{Q^2}{r^{2n-2}}, \qquad \lambda=-\frac{1}{L^2}\ \text{for AdS}.$$

For the planar brane relevant to AdS/CFT, take $K=0$ and Fourier-expand along the horizon directions.

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2. Physical degrees of freedom in $D$ dimensions

Counting dofs of the fields (See more)

These are on-shell dofs (polarizations) of massless fields:

• Graviton in $D$: $$N_g(D)=\frac{D(D+1)}{2}-2D=\frac{D(D-3)}{2}.$$
• Maxwell field in $D$: $$N_A(D)=D-2.$$

So Einstein–Maxwell has

$$N_{\text{EM}}(D)=\frac{D(D-3)}{2}+(D-2)=\frac{D^2-D-4}{2}.$$

Concrete checks:

• $D=4$: $N_g=2$, $N_A=2$, total $4$.
• $D=5$: $N_g=5$, $N_A=3$, total $8$.

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3. Spin channels under $SO(D-3)$

On a planar AdS black brane, it is natural to Fourier expand along the boundary directions and (for $k\neq 0$) choose momentum along one axis, say $\delta\Psi(t,r,\vec x)\sim e^{-i\omega t+i k z}$. This picks out the $z$ direction and leaves a residual rotation symmetry acting on the transverse directions $x^i$:

$$SO(D-3)\equiv SO(N),\qquad N=D-3.$$

Perturbations decompose into irreducible representations of $SO(N)$:

• Scalar channel ($s=0$): $SO(N)$ singlets, representation dimension: 1,
• Vector channel ($s=1$): $SO(N)$ vectors (one transverse index $i$), representation dimension: $N$,
• Tensor channel ($s=2$): traceless symmetric tensors $TT_{ij}$ under $SO(N)$ (exists only for $N\ge 2$, i.e. $D\ge 5$), representation dimension: $$\dim(TT)=\frac{N(N+1)}{2}-1=\frac{(N+2)(N-1)}{2} =\frac{(D-1)(D-4)}{2}.$$

The number of independent copies in each channel is the dimension of the corresponding $SO(N)$ representation. This “multiplicity bookkeeping’’ matters because each master equation typically comes with that many polarization copies. For example, in $D=5$ ($N=2$) one finds

$$\dim(TT)=2,\qquad \dim(V)=2,\qquad \dim(S)=1,$$

which matches the familiar helicity language under $SO(2)$: two helicity-$\pm2$ modes, two helicity-$\pm1$ modes, and one helicity-$0$ mode. In $D=4$ the transverse group is trivial ($SO(1)$), so this helicity classification degenerates; on spherical horizons one instead uses parity (odd/even) as the natural organizing principle.

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4. Decomposing physical dofs by channel

A clean way to see the channel decomposition is to restrict the little-group representations from $SO(D-2)=SO(N+1)$ down to $SO(N)$.

4.1 Graviton channel split

The graviton dofs satisfy the identity

$$\frac{D(D-3)}{2} = \underbrace{\frac{(D-1)(D-4)}{2}}_{\text{tensor}} +\underbrace{(D-3)}_{\text{vector}} +\underbrace{1}_{\text{scalar}}.$$

Interpretation:

• tensor channel: the “genuine” helicity-2/tensor polarizations,
• vector channel: momentum-flux–type polarizations,
• scalar channel: energy-density–type polarization.

4.2 Maxwell channel split

The Maxwell dofs satisfy

$$D-2=\underbrace{(D-3)}_{\text{vector}}+\underbrace{1}_{\text{scalar}}.$$

There is no tensor piece for a Maxwell field.

4.3 Combined Einstein–Maxwell

So the physical dofs per channel are

• tensor: $\frac{(D-1)(D-4)}{2}$ (graviton only),
• vector: $2(D-3)$ (graviton + Maxwell),
• scalar: $2$ (graviton + Maxwell).

This already tells you how many independent master scalars are needed.

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5. Master scalars: how many and why?

A useful “rule of thumb” that is actually quite robust on symmetric backgrounds:

In a given symmetry channel, the number of independent master scalars equals the number of field species that contribute to that channel.

For Einstein–Maxwell:

• tensor channel: only the graviton contributes
  $\Rightarrow$ 1 master scalar $\Phi_T$.

• vector channel: graviton + Maxwell contribute
  $\Rightarrow$ 2 master scalars, often diagonalized to $\Phi_{V\pm}$ on RN backgrounds.

• scalar channel: graviton + Maxwell contribute
  $\Rightarrow$ 2 master scalars, often diagonalized to $\Phi_{S\pm}$ on RN backgrounds.

So for $D\ge 5$:

$$\text{\# master equations} = 1\ (\text{tensor}) + 2\ (\text{vector}) + 2\ (\text{scalar}) = 5.$$

The total dof count is then

$$1\cdot\frac{(D-1)(D-4)}{2} + 2\cdot(D-3) + 2\cdot 1 = \frac{D(D-3)}{2}+(D-2),$$

matching the field-theory result exactly.

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6. A table summary

Decomposition of perturbations into tensor / vector / scalar channels under SO(N), N = D−3.

6.1 Field dofs

Dimension	graviton dofs	Maxwell dofs	total
$D$	$\frac{D(D-3)}{2}$	$D-2$	$\frac{D^2-D-4}{2}$
$4$	$2$	$2$	$4$
$5$	$5$	$3$	$8$

6.2 Channel dofs and master equations (planar/helicity viewpoint)

Channel (spin under $SO(N)$)	rep dimension (“copies”)	dofs in channel	# master scalars
tensor	$\frac{(D-1)(D-4)}{2}$	$\frac{(D-1)(D-4)}{2}$	$1$
vector	$D-3$	$2(D-3)$	$2$
scalar	$1$	$2$	$2$

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7. Gauge invariance and what “master scalar” means

7.1 The gauge redundancies

Linear perturbations involve:

• diffeomorphisms: $\delta h_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu$,
• $U(1)$ gauge transformations: $\delta a_\mu=\partial_\mu \lambda$.

A master variable is a combination of $(h_{\mu\nu},a_\mu)$ that is gauge invariant under both, and for which the linearized equations reduce to a decoupled 2nd-order equation.

7.2 Two common formalisms

• Kodama–Ishibashi (KI): harmonic decomposition on $d\Sigma_{n,K}^2$ and covariant 2D orbit-space equations. Works for any $D=n+2$ and $K=\pm1,0$.
• Kovtun–Starinets (KS): helicity channels for planar branes in radial gauge (common in holography). Equivalent in content.

This page uses KI notation for general $D$ and occasionally translates intuition into helicity language.

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8. Master equations in the KI formalism (general $D$)

In all sectors, after Fourier transforming $\sim e^{-i\omega t}$ the master equation takes the universal “Schrödinger/Sturm–Liouville” form

$$f(r)\,\frac{d}{dr}\!\left(f(r)\frac{d\Phi}{dr}\right)+\left(\omega^2-V(r)\right)\Phi=0, \qquad \text{or}\qquad \frac{d^2\Phi}{dr_*^2}+\left(\omega^2-V\right)\Phi=0,$$

with $dr_*/dr=1/f$.

8.1 Tensor channel (exists only for $D\ge 5$)

• No Maxwell tensor-type perturbations exist, so this sector is purely gravitational.
• The tensor master equation is equivalent to the wave equation for a minimally coupled massless scalar on the same background (after appropriate rescaling and harmonic separation).

Intuition: there is no other field content in Einstein–Maxwell that can transform as a TT rank-2 tensor under the horizon symmetry, so no mixing is possible.

8.2 Vector channel (2 master fields $\Phi_{V\pm}$)

Vector harmonics have eigenvalue $k_V^2$ on $d\Sigma_{n,K}^2$, and define

$$m_V \equiv k_V^2-(n-1)K.$$

The vector sector mixes gravitational and Maxwell perturbations but can be diagonalized into two decoupled master fields $\Phi_{V\pm}$, each obeying

$$f(f\Phi_{V\pm}')'+(\omega^2-V_{V\pm})\Phi_{V\pm}=0.$$

A standard closed-form expression for the KI vector potentials is

$$V_{V\pm}(r)=\frac{f(r)}{r^2}\left[ k_V^2+\frac{(n^2-2n+4)K}{4}-\frac{n(n-2)}{4}\lambda r^2 +\frac{n(5n-2)}{4}\frac{Q^2}{r^{2n-2}} +\frac{\mu^{(V)}_\pm}{r^{n-1}} \right],$$

where

$$\mu^{(V)}_\pm=-\frac{n^2+2}{2}M\pm\Delta, \qquad \Delta=\sqrt{(n^2-1)^2M^2+2n(n-1)m_VQ^2}.$$

Planar branes and momentum

For planar horizons ($K=0$), the harmonic eigenvalues reduce to $k_V^2=k^2$ for a plane wave with spatial momentum $k$ along $z$. For example, for planar RN–AdS$_5$, set $n=3$, $K=0$, $\lambda=-1/L^2$ and take $k_V^2=k^2$.

8.3 Scalar channel (2 master fields $\Phi_{S\pm}$)

Scalar harmonics have eigenvalue $k^2$ and define

$$m\equiv k^2-nK.$$

The scalar sector again mixes gravitational and Maxwell perturbations. KI show it can be diagonalized into two decoupled master fields $\Phi_{S\pm}$ obeying

$$f(f\Phi_{S\pm}')'+(\omega^2-V_{S\pm})\Phi_{S\pm}=0.$$

The scalar potentials admit a compact representation in terms of auxiliary variables

$$x=\frac{2M}{r^{n-1}},\qquad y=\lambda r^2,\qquad z=\frac{Q^2}{r^{2n-2}}, \qquad H=m+\frac{n(n+1)}{2}x-n^2 z,$$

and

$$V_{S\pm}(r)=\frac{f(r)\,U_\pm(r)}{64\,r^2\,H_\pm(r)^2}.$$

The explicit polynomials $U_\pm$ and functions $H_\pm$ are known in closed form but are lengthy; for practical work, it is often best to implement them symbolically once and reuse them for numerics.

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9. Specialization: AdS$_4$ (parity and isospectrality)

9.1 No tensor channel in $D=4$

For $D=4$ we have $n=2$ and there are no TT tensor harmonics on $S^2$. So there is no “tensor-type” sector.

Perturbations are usually organized instead into:

• odd (axial) parity,
• even (polar) parity.

In Einstein–Maxwell, each parity sector contains coupled gravito-electromagnetic perturbations. After building gauge invariants, each parity typically yields two decoupled master equations (often described as “gravitational-led” and “electromagnetic-led”).

9.2 About isospectrality

In 4D asymptotically flat RN, the axial and polar potentials are related by a Chandrasekhar/Darboux transformation, which implies isospectrality under appropriately matched boundary conditions.

In AdS, the bulk differential operators can still be related in a similar way, but spectral equivalence can depend on the AdS boundary conditions (because the Darboux map can send one boundary condition to a different Robin condition in the partner problem). So:

• the equations can be “partnered,”
• the spectra can coincide or not depending on which AdS boundary conditions you impose.

This is a key conceptual difference between “isospectral potentials” and “isospectral QNM spectra.”

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10. Specialization: AdS$_5$ (helicity channels and isospectrality breaking down)

For AdS$_5$ we have $n=3$ and for planar branes the transverse group is $SO(2)$:

• tensor channel = helicity-2 (2 copies),
• vector channel = helicity-1 (2 copies),
• scalar channel = helicity-0 (1 copy).

Einstein–Maxwell therefore gives:

• 1 tensor master equation (2 polarizations),
• 2 vector master equations (each with 2 polarizations),
• 2 scalar master equations (each with 1 polarization),

total $2+4+2=8$ dofs.

Unlike $D=4$, there is no generic Chandrasekhar-type partnering between the distinct helicity channels, and even within a fixed channel the “$\pm$” potentials are not typically supersymmetric partners. So there is no generic isospectrality tying together the AdS$_5$ channels.

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11. Summary checklist

When you perturb an RN–AdS black hole/brane in Einstein–Maxwell:

1. Count dofs: $N_g=D(D-3)/2$, $N_A=D-2$.
2. Choose symmetry basis:
   • planar: classify under $SO(D-3)$ (tensor/vector/scalar),
   • spherical in $D=4$: parity (odd/even),
   • KI: tensor/vector/scalar harmonics on $K_n$.
3. Build gauge-invariant variables.
4. Reduce to master scalars: In total there are 5 distinct master equations in $D\ge 5$ (tensor + 2 vector + 2 scalar), with multiplicities determined by symmetry. In $D=4$ there is no tensor channel, and the system is often described as odd/even parity (axial/polar) instead.
5. Impose physically appropriate boundary conditions (ingoing at the horizon; AdS boundary conditions consistent with your dual CFT sources/vevs).

Adding a scalar field: Einstein–Maxwell–dilaton (EMD)

Einstein–Maxwell is the simplest charged matter system. In holography and string-inspired models one often adds a neutral scalar (a “dilaton” or modulus), with a scalar potential and a scalar-dependent gauge kinetic function. This turns the decoupled RN story into a matrix-valued master problem: the tensor channel remains a single equation, the vector channel becomes a coupled $2\times 2$ system, and the scalar channel becomes a coupled $3\times 3$ system in general. This structure is systematic and holds in any dimension $D$ for backgrounds with a maximally symmetric $(D-2)$-dimensional spatial part. See [7] for more details.

Action and background ansatz

A very general EMD model (covering most “bottom-up” holographic setups) is

$$S=\frac{1}{2\kappa^2}\int d^D x\,\sqrt{-g}\left( R-2\Lambda -\eta\,(\partial\phi)^2 -\frac{1}{4}Z(\phi)\,F_{\mu\nu}F^{\mu\nu} -V(\phi) \right),$$

where $Z(\phi)$ is the gauge kinetic function and $V(\phi)$ is the scalar potential. (Different normalizations are common; the only essential points are that $Z(\phi)$ and $V(\phi)$ are arbitrary functions and $\phi$ can have a nontrivial radial profile.)

A time-independent background with a maximally symmetric $(D-2)$-dimensional spatial part can be written as

$$ds^2=-f(r)\,dt^2+\frac{\zeta(r)^2}{f(r)}\,dr^2+S(r)^2\,dX^2_{(n,K)}, \qquad A=a(r)\,dt, \qquad \phi=\phi(r),$$

with $n=D-2$ and $K\in\{1,0,-1\}$ labeling spherical/planar/hyperbolic horizons. In EMD, $f,\zeta,S,a,\phi$ are all dynamical and typically need to be obtained numerically for a given choice of $Z(\phi),V(\phi)$.

Degrees of freedom: what changes when a scalar is added?

A single real scalar field contributes one physical degree of freedom in any $D$.

So the total on-shell dof count becomes

$$N_{\text{EMD}}(D)=\underbrace{\frac{D(D-3)}{2}}_{\text{graviton}} +\underbrace{(D-2)}_{\text{Maxwell}} +\underbrace{1}_{\text{scalar}} =\frac{D^2-D-2}{2}.$$

For AdS$_4$ and AdS$_5$:

• $D=4$: $2+2+1=5$ dofs,
• $D=5$: $5+3+1=9$ dofs.

Channel decomposition under $SO(D-3)$ (planar/helicity viewpoint)

Fix momentum along one direction. The residual transverse rotation group is $SO(D-3)$. The new scalar field is a singlet, so it contributes only to the scalar channel.

Let $N=D-3$.

• Tensor channel dofs (graviton only): $\frac{(D-1)(D-4)}{2}$.
• Vector channel dofs (graviton + Maxwell): $2(D-3)$.
• Scalar channel dofs (graviton + Maxwell + scalar): $3$.

This implies the following master-field counting for generic modes:

Channel	# master scalars	copies per master scalar	total dofs carried
tensor	$1$	$\frac{(D-1)(D-4)}{2}$	$\frac{(D-1)(D-4)}{2}$
vector	$2$	$D-3$	$2(D-3)$
scalar	$3$	$1$	$3$

As a check:

$$1\cdot\frac{(D-1)(D-4)}{2}+2(D-3)+3=\frac{D(D-3)}{2}+(D-2)+1.$$

Special cases

• AdS$_4$ ($D=4$): no tensor channel. You have a vector/odd sector with 2 master scalars and a scalar/even sector with 3 master scalars (gravity + gauge + scalar), totaling 5.
• AdS$_5$ ($D=5$): tensor channel exists and has 2 polarizations; the scalar channel carries 3 master scalars (one copy), so total dofs are $2 + 4 + 3 = 9$.

Master equations become a coupled Klein–Gordon system (potential matrices)

A powerful general result is that you can express all gauge-invariant perturbations in terms of master scalars that satisfy a matrix-valued Klein–Gordon-type system.

For each helicity/channel $h\in\{2,1,0\}$ (tensor/vector/scalar), introduce a vector of master scalars

$$\vec{\Phi}^{(h)}(t,r)=\begin{pmatrix}\Phi^{(h)}_1 \\ \vdots \\ \Phi^{(h)}_{N_h}\end{pmatrix}, \qquad N_2=1,\;N_1=2,\;N_0=3,$$

and a symmetric potential matrix $W^{(h)}(r)$ of size $N_h\times N_h$.

Then the master system takes the universal form

$$\square\,\Phi^{(h)}_s-\sum_{s'=1}^{N_h}W^{(h)}_{s s'}(r)\,\Phi^{(h)}_{s'}=0,$$

where $\square$ is the wave operator on the background metric and $W^{(h)}$ contains non-derivative couplings only (no derivative mixing). The matrices are symmetric, and positivity of eigenvalues of an appropriate (possibly deformed) potential matrix gives a sufficient stability criterion. In general, these coupled systems do not fully diagonalize into independent ODEs. Only in special cases (notably Reissner–Nordström, where the scalar is absent/constant) do you recover the fully decoupled KI equations.

Why this is the right generalization of KI

In Einstein–Maxwell (no scalar), the vector and scalar channels each yield a $2\times 2$ system; on RN backgrounds these can be diagonalized to the familiar $\pm$ master equations. In EMD, the scalar channel gains one more dynamical field, so it becomes $3\times 3$ in general.

When can you further decouple (diagonalize) the coupled systems? (See more)

Even though the master equations are “only” coupled by a potential matrix, you still cannot usually diagonalize them by a constant change of basis, because the eigenvectors typically depend on $r$. Generically one can treat the vector/scalar channel as a genuinely coupled system.

• In the vector channel, a necessary and sufficient condition for full decoupling into two independent equations is that the eigenvectors of the $2\times2$ potential matrix are $r$-independent. A simple equivalent condition is that

$$\frac{W^{(1)}_{11}(r)-W^{(1)}_{22}(r)}{W^{(1)}_{12}(r)}=\text{const}.$$

If this holds, the two decoupled potentials are just the two eigenvalues of $W^{(1)}(r)$.

• In the scalar channel, decoupling would require an $r$-independent eigenbasis for the $3\times3$ matrix, which is rarer.

Practical workflow: perturbations in EMD models

Most holographic EMD backgrounds are not analytic, so a robust workflow is:

1. Background: solve the background ODEs for $f,\zeta,S,a,\phi$ numerically for your chosen $Z(\phi),V(\phi)$.
2. Choose a channel (tensor/vector/scalar) using the residual symmetry (helicity under $SO(D-3)$ for planar branes; tensor/vector/scalar harmonics for KI on $S^{D-2}$).
3. Build gauge invariants:
   • tensor: already gauge invariant,
   • vector: gauge invariants from $(h_{t\alpha},h_{r\alpha},a_\alpha)$,
   • scalar: gauge invariants from $(h_{tt},h_{tr},h_{rr},h_{t x},h_{r x},h_{xx},h_{\perp\perp},a_t,a_r,a_x,\varphi)$ (schematically).
4. Solve the master system:
   • tensor: one ODE,
   • vector: coupled $2\times2$ ODE system,
   • scalar: coupled $3\times3$ ODE system.
5. Boundary conditions:
   • impose ingoing conditions at the horizon (matrix Frobenius expansion),
   • impose the AdS boundary conditions appropriate to your quantization (Dirichlet/Neumann/mixed for $\phi$; standard source/vev conditions for metric and gauge field).
6. Spectra/Green’s functions: compute quasinormal modes or retarded correlators by solving the coupled boundary value problem (e.g. determinant method: build a basis of ingoing solutions and look for vanishing sources).

Final remarks: Exceptional modes and degenerations

The sector decomposition and master-field diagonalizations described above are for generic modes. At special momenta/harmonics, some of the usual constructions degenerate. This reflects symmetry enhancement and constraints: some modes become non‑radiative (pure gauge or parameter shifts), even though the underlying linearized equations remain regular.

Planar branes: why $k=0$ is special

For a planar brane, perturbations are usually expanded as $e^{-i\omega t+i k z}$. For $k\neq 0$, the momentum selects a direction and the little group is $SO(D-3)$, giving the familiar tensor/vector/scalar channels. At strictly $k=0$ there is no preferred direction and the symmetry enlarges to $SO(D-2)$, so the distinction between “sound/shear/tensor’’ channels ceases to be intrinsic; different finite‑$k$ channels can coincide by symmetry.

Technically, many convenient gauge‑invariant variables (and some diagonalization steps) are constructed assuming $k\neq 0$ and contain explicit divisions by $k$ (or by harmonic combinations that reduce to $k$ in the planar case), so taking $k\to 0$ can make the chosen variables ill‑defined even when the physics is smooth.

Spherical horizons: lowest harmonics ($\ell=0,1$) are exceptional

On compact horizons (e.g. $S^{D-2}$), the harmonic decomposition introduces discrete modes. The lowest harmonics are special:

• For vector-type perturbations, the exceptional case $m_V=0$ (including the $\ell=1$ vector harmonic on $S^n$) carries no gravitational dynamical degree of freedom in the source-free problem; it corresponds physically to adding infinitesimal rotation.
• For scalar-type perturbations, the $\ell=0$ mode changes the available gauge generators and harmonic building blocks; the generic gauge-invariant variables of the master-equation construction are not defined, and the physical system reduces (often to a parameter-shift mode plus possibly matter fluctuations).

EMD systems: couplings can vanish and the effective rank can drop

In EMD theories the master equations are typically coupled (vector: $2\times2$, scalar: $3\times3$), and full decoupling is not generic. But the potential-matrix entries often carry explicit factors such as $\sqrt{k^2-nK}$ and background-profile-dependent terms (from $A_t'(r)$, $\phi'(r)$, $Z(\phi)$, $Z'(\phi)$, etc.), so mixing can disappear in special limits (including planar $k=0$) or for special backgrounds.

Practical rule

When taking limits such as $k\to 0$, low harmonics, extremality, or special background profiles, verify that the master-variable map remains invertible and that the coupled system has the expected rank. If a diagonalization degenerates, switch to a basis of gauge invariants that is well-defined in that limit and recompute the master system there.

References

[1] S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford; New York, 1983).

[2] H. Kodama and A. Ishibashi, A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions, Prog. Theor. Phys. 110, 701–722 (2003) [arXiv:hep-th/0305147].

[3] H. Kodama and A. Ishibashi, Master Equations for Perturbations of Generalised Static Black Holes with Charge in Higher Dimensions, Prog. Theor. Phys. 111, 29–73 (2004) [arXiv:hep-th/0308128].

[4] A. Ishibashi and H. Kodama, Perturbations and Stability of Static Black Holes in Higher Dimensions, Prog. Theor. Phys. Suppl. 189, 165–209 (2011) [arXiv:1103.6148].

[5] P. K. Kovtun and A. O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72, 086009 (2005) [arXiv:hep-th/0506184].

[6] Y. Matsuo, S.-J. Sin, S. Takeuchi, T. Tsukioka and C.-M. Yoo, Sound Modes in Holographic Hydrodynamics for Charged AdS Black Hole, Nucl. Phys. B 820, 593–619 (2009) [arXiv:0901.0610].

[7] A. Jansen, A. Rostworowski and M. Rutkowski, Master equations and stability of Einstein–Maxwell–scalar black holes, JHEP 12, 036 (2019) [arXiv:1909.04049].