Simpler version

Gauge, Energy & Black Hole Stability

with GR20, Warszawa

Recap...

• 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 $$\Sigma$$ & $$N, N_a$$

Perturbations

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

Linearised Constraints

$\begin{split} 0 = & \tfrac{2}{\sqrt{h}}\left( \pi^{ab} - \tfrac{1}{d-1}\pi~ h^{ab} \right)p_{ab} + \tfrac{2}{h}\left( \pi^{ac} {\pi_c}^b - \tfrac{1}{d-1} \pi\pi^{ab} \right) q_{ab} \\ & -\tfrac{1}{h}\left[ \pi^{ab}\pi_{ab} - \tfrac{1}{d-1}\pi^2 \right]q - D^aD^bq_{ab} + \triangle q + R^{ab}q_{ab} \\ 0 = & 2D^bp_{ab} + \tfrac{\pi^{bc}}{\sqrt{h}} \left ( D_c q_{ba} + D_b q_{ca} - D_a q_{bc} \right) \end{split}$

any dimension, compact $$B$$

static background

$$\pi_{ab} = 0 = R$$ $$N_a = 0$$ $$NR_{ab} = D_aD_bN$$

Gauge freedom

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

Symplectic Product

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

Inner Product

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

Constraints

$$0 = \langle X, S~G_\alpha \rangle$$

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

Gauge Condns

$$0 = \langle X, G_\alpha \rangle$$

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

Boundary Conditions

perturbed area of $$B$$ $$= 0$$
perturbed expansion of $$B$$ $$= 0$$
more to fix gauge

Kinetic Energy

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

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

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

Minkowski is (linearly) stable.

Axial perturbations

Killing field $$\phi_a$$
$$q_{ab} = 2 \phi^{-2}\alpha_{(a}\phi_{b)}$$
$$\phi^a\alpha_a=0 = \mathscr{L}_\phi \alpha$$ $$D^a\alpha_a = 0$$

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

Rindler spacetime

$$Riem = 0$$; $$B \neq 0$$

$\begin{split} \mathscr{U} & = \frac{1}{2}\int_\Sigma N (D_cq_{ab})^2 - \kappa \int_B\left[ \frac{7}{4}(q_{rr})^2 + (q_{Ar})^2 \right] \\ & = \int_\Sigma N \left( D_{[c} \zeta_{a]b} \right)^2 - \kappa \int_B \left[ \frac{1}{2} \left( \mathscr D_A x \right)^2 + \frac{5d-6}{4(d-1)} \left( \mathscr D^2\rho \right)^2 \right] \end{split}$

$\begin{split} \mathscr D_A x &\doteq q_{Ar}\big\vert_B \\ - (d-1)\triangle \rho &\doteq q \end{split}$
boundary contributions vanish for $$q=0$$

Moral of the Story

• uniquely fix gauge: $$0 = \langle X, G_\alpha \rangle$$
• $$\mathscr{E = K + U}$$ static background
• $$\mathscr{K} \geq 0$$
• $$\mathscr{U} \geq 0$$? ; for Minkowski and axial perturbations
• Rindler? Black Holes? Stationary Case?