Simpler version

# Gauge Conditions&Black HoleStability

Kartik PrabhuMWRM2012, Chicago

# Previously on...

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

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

Static Background
$$(\pi^{ab},h_{ab})$$

# Gauge freedom

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

infinitesimal

# Perturbations

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

## 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

$$0 = \langle X, S~G_\alpha \rangle \quad \forall \alpha \in C_0^\infty$$

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

$$G_\alpha$$ satisfies constraints!

# Gauge Conditions

$$0 = \langle X, G_\alpha \rangle \quad \forall \alpha \in C_0^\infty$$

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

perturbed area of $$B$$ $$= 0$$
perturbed expansion of $$B$$ $$= 0$$

# Evolution

$\begin{split} \dot{q}_{ab} & = 2D_{(a}n_{b)} + 2N\left( p_{ab}-\tfrac{1}{d-1}p~ h_{ab} \right) \\ & \doteq Kp_{ab}\\ \dot{p}_{ab} &= D_aD_b n -h_{ab} \triangle n -nR_{ab}+ \\ & \doteq -Uq_{ab} \end{split}$
preservation of gauge: elliptic equations for $$n$$ and $$n_a$$
canonical energy $$= \langle p, Kp\rangle + \langle q, Uq \rangle$$

# Kinetic Energy

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

# TT-decomposition

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
• constraint : $$D^b\chi_{ab} = 0$$ transverse

# Potential Energy

$\begin{split} \mathscr{U} & \doteq \langle q_{ab},U q_{ab} \rangle \\ & = \int\limits_\Sigma N\left[ \tfrac{1}{2}(D_cq_{ab})^2 - \tfrac{3}{2}(D_aq)^2 \right.\\ & \qquad \left. + q^{ab}R_{cabd}q^{cd} \right] -\int\limits_B \kappa (q_{ar})^2 \end{split}$
Will TT help?

# Take Home

• uniquely fix gauge: $$0 = \langle X, G_\alpha \rangle$$
• evolution: $$\dot q = Kp$$ ; $$\dot p = -Uq$$
• $$\mathscr{K} \geq 0$$
• $$\mathscr{U} \geq 0$$? ; stationary case?