# Growth rate of Black Hole Instabilities

# Perspective

- Hollands and Wald [arXiv:1201.0463]
- for axisymmetric perturbations, \(\mathscr{E} > 0\implies\) mode stability.
- Negative energy perturbation
**can not**go to stationary solution. (“instability”)

- show negative energy\(\implies\)unbounded growth. Find the growth rate.

# Answer

perturbations \(Q = \mathcal{K}P\) define: \[ \omega^2 = \frac{ \langle Q, \mathcal A Q \rangle_{\mathscr H} }{ \langle Q, Q \rangle_{\mathscr H} } = \frac{\mathscr U(Q, Q)}{\mathscr K (P,P)} \]

\(\omega^2 < 0 \) for some \(Q\) \(\implies\) exponential growth;

at least \(\sim e^{|\omega|t}\)

# ADM

\( (\pi^{ab},h_{ab})\) on \(\Sigma\)

## Perturbations

\(( p_{ab},q_{ab})\)

linearised constraints & boundary conditions

\(D \geq 4\), compact bifurcation surface \(B\)

restrict to static or stationary-axisymmetric with \(t\phi\)-reflection isometry.

## static

\(P \equiv p_{ab}\) (\(t\)-odd) \(Q \equiv q_{ab}\) (\(t\)-even)

## stationary-axisymmetric

\(P \equiv (p_{polar}, q_{axial})\) (\(t\phi\)-odd)
\(Q \equiv (p_{axial}, q_{polar})\) (\(t\phi\)-even)

canonical transformation

# Canonical Energy

\(\mathscr E = \mathscr K(P,P) + \mathscr U(Q,Q) \)

# Theorem

For perturbations around a static background or axisymmetric perturbations around stationary-axisymmetric background, the kinetic energy is a symmetric bilinear form on \(P\), gauge-invariant and:

- \(\mathscr K \geq 0 \)
- \(\mathscr K = 0 \) iff \(P\) is a pure gauge

# Evolution

Fix a gauge which preserve the reflection symmetry.

\[ \dot Q = \mathcal K P \quad \dot P = -\mathcal U Q \] gauge-fixed ADM evolution equations.

\(\mathscr E = \int P\dot Q - Q\dot P = \langle P, \mathcal KP \rangle + \langle Q, \mathcal UQ \rangle \)

\( \langle -,- \rangle \) is \(L^2\) inner product# Hilbert space

dynamical evolution \(\ddot Q = -\mathcal A Q\) where \(\mathcal A = \mathcal{KU} \)Define Hilbert space \(\mathscr H\) of \(Q\)s so that: \(\langle Q', Q\rangle_{\mathscr H} = \langle Q', \mathcal K^{-1} Q\rangle \) “inverse KE”

On this we have \(\langle Q', \mathcal AQ\rangle_{\mathscr H} = \langle Q', \mathcal U Q\rangle \)

\(\mathcal A\) is symmetric in \(\mathscr H\)# Spectral Solution

Using the spectral theorem

\[\begin{split} Q_t & = \cos(t{\mathcal A}_+^{1/2})\Pi_+ Q_0 + \sin(t{\mathcal A}_+^{1/2}){\mathcal A}_+^{-1/2} \Pi_+\dot Q_0 \\ & \quad + \Pi_0Q_0 + t \Pi_0 \dot Q_0 \\ & \quad + \cosh(t{\mathcal A}_-^{1/2})\Pi_- Q_0 + \sinh(t{\mathcal A}_-^{1/2}){\mathcal A}_-^{-1/2} \Pi_-\dot Q_0 \end{split}\]

- \({\mathcal A}_+ \) and \( -{\mathcal A}_- \) are the positive and negative parts of \( {\mathcal A}\)
- \(\Pi_+, \Pi_0, \Pi_-\) are projections onto the spectrum of \( {\mathcal A} \) respectively.

# Theorem

If the potential energy \(\mathscr U < 0\) on \(\mathscr H\), then the black hole background is unstable in the sense that there exist solutions to the linearised evolution equations which grow without bound (at least exponentially) in time.

Local gauge invariant quantities also grow without bound.

# \(\mathscr{H}\) caveat

- Perturbation in \(\mathscr{H}\) if \(\mathcal{K}\) can be inverted
- Perturbation \(X = \mathcal{L}_t X'\) i.e. \(Q = \mathcal K P'\)
- Similar to Wald’s 1979-result for scalar fields in Schwarzschild \(\psi\vert_B = 0\)

# Moral of the Story

- \( \mathscr{E = K + U} \) static/stationary-axisymmetric
- \( \mathscr{K} \geq 0 \) zero on gauge
- \(\mathscr{H}\) with “inverse KE norm”
- \( \mathscr{U} < 0 \) on \(\mathscr{H}\) \(\Rightarrow\) exponential growth instability