Simpler version

# Positive Energy & Black Hole Stability

Kartik Prabhu with Robert M. WaldAPS2014, Savannah

# 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

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

## Perturbations

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

linearised constraints & boundary conditions

$$D \geq 4$$, compact $$B$$

## 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.
unique solution due to well-posed IVF

# 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