# Positive Energy & Black Hole Stability

# Negative Energy & Black Hole Instability

# Perspective

- Hollands and Wald [arXiv:1201.0463]
- Dynamic stability \(\Leftrightarrow\) positive canonical energy for axisymmetric perturbations.
- Negative energy perturbation
**can not**go to stationary solution. (“instability”)

- show negative energy \(\Rightarrow\) unbounded growth

(for some perturbations)

# Strategy

- \( \mathscr{E = K + U} \)
- \( \mathscr{K} \geq 0 \)kinetic energy
- \(\mathscr{H}\) with “inverse KE norm”
- \( \mathscr{U} < 0 \) on \(\mathscr{H}\) \(\Rightarrow\) exponential growth instability

# ADM

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

## Perturbations

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

linearised constraints & boundary conditions

## Gauge freedom

\( G_\alpha = \begin{pmatrix} D_aD_b\alpha - h_{ab}\triangle\alpha - R_{ab}\alpha + \ldots \\ 2D_{(a}\alpha_{b)} + \ldots \end{pmatrix} \)

## static

\(P\) (\(t\)-odd) \(Q\) (\(t\)-even)

## stationary-axisymmetric

\(P\) (\(t\phi\)-odd) \(Q\) (\(t\phi\)-even)

# Kinetic Energy

\(\mathscr{K}\): part of canonical energy with only \(P\)

# 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 preserving the reflection symmetry.

\[ \dot Q = \mathcal K P \qquad \dot P = -\mathcal U Q \qquad \ddot Q = -\mathcal A Q \] where \( \mathcal A = \mathcal{KU} \)

\(\mathscr K = \langle P, \mathcal KP \rangle \) and \(\mathscr U = \langle Q, \mathcal UQ \rangle \)

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

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

with dynamical evolution \(\ddot Q = -\mathcal A Q\).

“mode solutions” need not be physical

# Spectral Solution

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

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

# \(\mathscr{H}\) caveat

- Perturbation in \(\mathscr{H}\) if \(\mathcal{K}\) can be inverted
- Perturbation \(X = \mathcal{L}_t X'\)
- Similar to Wald’s 1979-result for scalar fields in Schwarzschild \(\psi = 0\vert_B\)
- Possible(?) to improve following Kay-Wald (1987) or better
*Lectures on black holes and linear waves*—Dafermos & Rodnianski

# 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