Simpler version

Gauge Conditions & Black Hole Stability

Kartik Prabhu with Robert M. WaldAPS2013, Denver

Previously...

ADM

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

static background

\( \pi_{ab} = 0 = R\) \( N_a = 0 \) \( NR_{ab} = D_aD_bN \) any dimension, compact \(B\)

Perturbations

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

Gauge freedom

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

Inner Product

\( \langle \tilde X,X \rangle = \int\limits_\Sigma \tilde p^{ab}p_{ab} + \tilde q^{ab}q_{ab}\)

Symplectic Product

\( \langle \tilde X,SX \rangle = \int\limits_\Sigma \tilde p^{ab}q_{ab} - \tilde q^{ab}p_{ab}\)

Constraints

\( 0 = \langle X, S~G_\alpha \rangle \)

\( 0 = R^{ab}q_{ab} + \triangle q \)
\( 0 = D^b p_{ab} \)

Gauge Conditions

\( 0 = \langle X, G_\alpha \rangle \)

\( 0 = R^{ab}p_{ab} + \triangle p \)
\( 0 = D^b q_{ab} \)

Boundary Conditions

perturbed area of \(B\) \(= 0\)
perturbed expansion of \(B\) \(= 0\)
more to fix gauge

Evolution

\[\begin{split} \dot{q}_{ab} & = 2D_{(a}n_{b)} + 2N\left( p_{ab}-\tfrac{1}{d-1}p~ h_{ab} \right) \\ & \doteq Kp_{ab} \end{split}\] \[\begin{split} \dot{p}_{ab} & = D_aD_b n -h_{ab} \triangle n -nR_{ab} +\tfrac{1}{2} N \left( \triangle q_{ab} + D_aD_bq \right) \\ & \quad\, - \tfrac{1}{2}D^cN \left( D_aq_{bc} + D_bq_{ac} - D_cq_{ab} + h_{ab} D_cq \right)\\ & \quad\, - N\left( R_{c(ab)d}q^{cd} + {R^c}_{(a}q_{b)c} -h_{ab}R^{cd}q_{cd} \right)\\ & \doteq -Uq_{ab} \end{split}\]
preservation of gauge: elliptic equations for \(n\) and \(n_a\)
\( \mathscr{E} = \langle p_{ab},Kp_{ab} \rangle + \langle q_{ab},Uq_{ab} \rangle\)

Kinetic Energy

\[ \mathscr{K} \doteq \langle p_{ab},Kp_{ab} \rangle = \int\limits_\Sigma 2N\left[ (p_{ab})^2 - \tfrac{1}{d-1}p^2 \right] \]

Choose: \(\triangle\xi = -\tfrac{1}{d-1}p\) with \(\xi|_B = 0\)
\[ \chi_{ab} \doteq p_{ab} - D_aD_b \xi + R_{ab}\xi + h_{ab}\triangle \xi \]

\( \chi = 0\) traceless \( D^b\chi_{ab} = 0 \) transverse

\(\mathscr{K}\) - TT

\[ \mathscr{K} = \int\limits_\Sigma 2N(\chi_{ab})^2 \geq 0 \]

Potential Energy

\[ \mathscr{U} \doteq \langle q_{ab},U q_{ab} \rangle \]

\[ = \int\limits_\Sigma N\left[ \tfrac{1}{2}(D_cq_{ab})^2 -D_cq_{ab}D^aq^{cb} - \tfrac{3}{2}(D_aq)^2 \right] \] \[\begin{split} = & \int\limits_\Sigma N\left[ \tfrac{1}{2}(D_cq_{ab})^2 + q^{ab}R_{cabd}q^{cd} - \tfrac{3}{2}R^{ab}q_{ab}q \right] \\ & -\int\limits_B \kappa \left[ \tfrac{7}{4}(q_{rr})^2 + (q_{Ar})^2 \right] \end{split}\]

Minkowski! Rindler?

\(\mathscr{U}\) - axial

Killing field \( \phi_a \)
\( q_{ab} = 2 \phi^{-2}\alpha_{(a}\phi_{b)}\) axial perturbations
\( \phi^a\alpha_a=0 \) \( D^a\alpha_a = 0 \)

\[ \mathscr{U} = 2\int_\Sigma N\phi^2 \left[ D_{[a}\left( \phi^{-2}\alpha_{b]}\right)\right]^2 \geq 0 \]

Take Home

  • uniquely fix gauge: \( 0 = \langle X, G_\alpha \rangle\)
  • evolution: \(\dot q = Kp\) ; \( \dot p = -Uq \)
  • \( \mathscr{K} \geq 0 \)
  • \( \mathscr{U} \geq 0 \)? ; for Minkowski and axial perturbations
  • Black Holes? Stationary Case?