Simpler version

Positive Energy & Black Hole Stability

Kartik Prabhu with Robert M. WaldAPS2014, Savannah

Negative Energy & Black Hole Instability


  • 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)


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


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


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

linearised constraints & boundary conditions

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

Gauge freedom

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


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


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

Kinetic Energy

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


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


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


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