Simpler version

image/svg+xml Σ~ Σ B

Gauge Conditions
&Black Hole Stability

Kartik PrabhuMWRM2012, Chicago

Previously on...

ADM

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 \(B\) \(= 0\)

Boundary terms

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?