Simpler version

Gauge Conditions & Black Hole Stability

with APS2013, Denver

Previously...

• Hollands and Wald [arXiv:1201.0463]
Dynamic stability $$\Leftrightarrow$$ positive canonical energy for perturbations

• Gaussian null coordinates
• boundary conditions
• fix gauge near horizon

$$(\pi^{ab},h_{ab})$$ 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?