Linearized Gravitational Perturbations

• Linearized gravitational perturbations are small deviations from exact solutions used to probe the stability and spectrum of black hole spacetimes.
• The approach employs separable mode analysis via the Teukolsky equation and Hertz potential formalism to reconstruct full metric perturbations.
• Nonlinear and second order effects may challenge strict boundary conditions, highlighting limitations in maintaining Kerr/CFT correspondence.

Linearized gravitational perturbations refer to infinitesimal deviations from an exact gravitational background solution, studied at first order in perturbation amplitude. Within the context of general relativity and particularly for black hole spacetimes, linearized perturbations probe the stability, spectrum of excitations, and conserved quantities of the underlying geometry. Such studies are central to the analysis of dynamics near black holes, tests of the Kerr/CFT correspondence, and the understanding of gravitational wave emission from perturbed compact objects.

1. Teukolsky Master Equation and Mode Structure

The primary framework for linearized gravitational (and more generally, massless field) perturbations of the near-horizon extreme Kerr (NHEK) geometry is the Teukolsky master equation. For a massless field of arbitrary spin $s$ ($s=0,\pm\frac12,\pm1,\pm2$), the equation is separable on this Petrov type D background and can be written schematically as

$\mathcal{T}^{(s)} \Psi^{(s)} = T_{(s)}$

where $\Psi^{(s)}$ is a field of spin weight $s$ and $T_{(s)}$ is a possible source. For the NHEK geometry, an ansatz of the form

$$\Psi^{(s)}(t,r,\theta,\phi) = e^{-i\omega t} e^{im\phi} R^{(s)}_{lm\omega}(r) S^{(s)}_{lm}(\theta)$$

enables separation into angular and radial equations. The angular part is governed by a spin-weighted spheroidal harmonic equation whose separation constant $\Lambda^{(s)}_{lm}$ is determined numerically or by asymptotic expansion.

The radial equation for $\Phi^{(s)}_{lm\omega}(r)$ (related to $R^{(s)}$ by $R^{(s)} = (1 + r^2)^{-s/2} \Phi^{(s)}$) becomes

$$\frac{d}{dr}\left[(1 + r^2) \frac{d}{dr} \Phi\right] - \left[\mu^2 - \frac{(\omega + q r)^2}{1 + r^2}\right]\Phi = 0$$

with $q = m - is$ and $\mu^2 = q^2 + \widetilde{\Lambda}^{(s)}_{lm}$, where $\widetilde{\Lambda}^{(s)}_{lm}$ is a shifted separation constant. This radial equation is reminiscent of a charged massive scalar in $AdS_2$ with a homogeneous electric field and admits hypergeometric solutions, fully specified by boundary conditions at infinity.

2. Classification: Normal Modes vs. Traveling Waves and Stability

The radial solutions, through their asymptotic behavior $\Phi \sim |r|^{-1/2\pm\eta/2}$ with

$\eta = \sqrt{1 + 4\mu^2 - q^2}$

are naturally classified by the reality or imaginary character of $\eta$:

• Normal modes ($\eta$ real): These correspond generally to axisymmetric or low $|m|$ perturbations. The solutions decay at large $r$ as power laws, and imposing normalizability quantizes the frequency:

$$\omega = \pm\left(n + \frac{1}{2} + \frac{\eta}{2}\right),\quad n=0,1,2,\ldots$$

These normal modes are regular and obey nearly all the fall-off conditions (e.g., GHSS/Kerr–CFT boundary conditions) and compose a discrete spectrum without instabilities.

• Traveling waves ($\eta$ imaginary): For $|m|$ near $l$, the solutions exhibit oscillatory asymptotic behavior, associated with quasinormal (damped) modes. Outgoing boundary conditions select a discrete set of complex frequencies (with negative imaginary part), corresponding to damped quasinormal ringing. While traveling waves can carry energy and angular momentum to infinity, proper outgoing conditions eliminate any growing (unstable) modes.

The analysis determines that, after all physical boundary conditions are imposed, NHEK is linearly stable to gravitational perturbations in both sectors.

3. Reconstruction of Metric Perturbations: Hertz Potential Formalism

The gauge-invariant quantities governed by the Teukolsky equation (notably the NP scalars $\Psi_0$ and $\Psi_4$ for gravity) do not by themselves describe the full spacetime geometry; the full metric perturbation $h_{\mu\nu}$ is needed for analysis of conserved quantities and boundary condition compatibility.

The Hertz potential formalism provides a map from solutions to the Teukolsky equation back to metric perturbations. Introducing a Hertz potential $\Psi_H$ of opposite spin weight (solving the adjoint Teukolsky equation), the IRG (ingoing radiation gauge) metric perturbation is constructed as

$$h_{\mu\nu}^{\mathrm{IRG}} = \text{Re}\left\{ \ldots \Psi_H \right\}$$

with a specific second-order differential operator acting on $\Psi_H$ (involving tetrad derivatives and spin coefficients). This procedure—due to Cohen–Kegeles, Chrzanowski, and Wald—enables explicit metric reconstruction necessary for the analysis of conserved charges and detailed comparison to asymptotic symmetry (GHSS/Kerr–CFT) conditions.

4. Computation of Energy and Angular Momentum

Energy ($\mathcal{E}$) and angular momentum ($\mathcal{J}$) for both massless scalar and gravitational perturbations are computed by constructing conserved currents associated with spacetime Killing vectors and integrating the corresponding stress–energy pseudotensors.

• For a massless scalar, the canonical stress–energy tensor provides

$$Q_\xi[\Phi] = \int_\Sigma d^3x\, \sqrt{-\gamma} T^{\mu\nu} n_\mu \xi_\nu$$

for a slice $\Sigma$ and Killing vector $\xi^\mu$. Due to orthogonality of the separated modes, energy decomposes as a sum over mode amplitudes.

• For gravitational perturbations (Landau–Lifshitz pseudotensor), the metric perturbation's higher-derivative structure is taken into account. Numerical integration over allowed normal modes demonstrates positive-definite energy, with consistent ratios $$\mathcal{J}_{nlm}/\mathcal{E}_{nlm} = m/\omega_{nlm}$$.

The energy positivity of all normal modes confirms the physical regularity of the perturbation spectrum and compliance with the "zero-energy" boundary condition central to the Kerr–CFT conjecture.

5. Second Order Perturbations and Nonlinear Effects

Second-order corrections are governed by an Einstein equation of the form: $$G^{(1)}_{\mu\nu}[h^{(2)}] = -G^{(2)}_{\mu\nu}[h^{(1)}]$$ with source quadratic in the first-order metric perturbations. If initial data excite only normal modes, the second-order metric must reconstruct a solution with the corresponding energy as a boundary integral.

Normal modes decay sufficiently rapidly such that their second-order contributions may violate GHSS/Kerr–CFT fall-off requirements—unless one finely restricts to a subset of "large-gauge" or isometric configurations. Furthermore, mass-changing (zero-mode) perturbations, which move the solution infinitesimally away from extremality, are not captured by the Teukolsky/Hertz framework and, in NHEK, would violate the strict boundary conditions if associated energy is nonzero.

This underscores a key issue: even if linear perturbations strictly obey GHSS/Kerr–CFT asymptotics, nonlinear effects may generically excite modes incompatible with these strict asymptotic structures.

6. Implications for Kerr/CFT and Linear Stability

The analysis directly supports the conjectured linear stability of the NHEK geometry under arbitrary spin perturbations. The explicit computation of the full normal mode spectrum and demonstration of energy positivity solidify the claim that no linear instabilities arise, consistent with the requirements of the Kerr–CFT correspondence.

However, the stringent fall-off conditions required by the dual CFT severely constrain the permissible nontrivial nonlinear dynamics. Only configurations related by large gauge transformations (isometries) are necessarily compatible with the required asymptotic structure, suggesting severe restrictions on the allowed dynamics in a true gravity/CFT duality in this setting.

Table: Key Structures in Linearized Perturbations of NHEK

Structure	Mathematical Description	Physical or Dynamical Role
Teukolsky Eq.	$\mathcal{T}^{(s)}\Psi^{(s)}=T_{(s)}$	Governs gauge-invariant NP scalars
Mode solutions	Normal: $\omega = \pm(n+\frac12+\frac{\eta}{2})$; Traveling: oscillatory asymptotics	Spectrum and classification
Hertz potential	$\mathcal{O}^\dagger\Psi_H=0$, $h_{\mu\nu} = \mathcal{D}[\Psi_H]$	Metric reconstruction
Energy formula	$\mathcal{E}_{nlm} \propto \omega_{nlm} \ldots$	Positivity, stability analysis
Fall-off conditions	GHSS/Kerr–CFT boundary conditions on $h_{\mu\nu}$	Consistency with Kerr/CFT correspondence
Second order source	$G^{(2)}_{\mu\nu}[h^{(1)}]$	Nonlinear backreaction

Conclusion

Linearized gravitational perturbations of the NHEK geometry are governed by a separable Teukolsky master equation, admit a discrete spectrum of normal modes (with power-law decay and no instabilities), and require Hertz potential machinery for full metric reconstruction. The computations of energy and angular momentum demonstrate physical viability and compatibility with expected boundary conditions. At second order, nonlinear perturbations pose potential difficulties for maintaining the strict fall-off requirements demanded by the Kerr–CFT conjecture, generically suggesting that only a narrowly restricted set of perturbations preserve the duality structure. This body of analysis underpins the stability of the near-horizon extremal Kerr geometry in the linear regime and clarifies the scope and limitations of nontrivial dynamics in attempts to realize gravity/CFT holography in this context (0906.2380).