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 productHilbert 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