My presentation on gauge conditions for perturbations in GR and their use in blackhole stability proofs, given at the Midwest Relativity Meeting 2012 held at the University of Chicago.

Gauge Conditions & Stability in GR

Gauge Conditions
&Black Hole Stability

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

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

\[ \begin{split} \dot{q}_{ab} & = 2D_{(a}n_{b)} + 2N\left( p_{ab}-\tfrac{1}{d-1}p~ h_{ab} \right) \\ & \doteq Kp_{ab}\\ \dot{p}_{ab} &= D_aD_b n -h_{ab} \triangle n -nR_{ab}+ \\ & \doteq -Uq_{ab} \end{split} \]

\[\begin{split} \dot{p}_{ab} & = D_aD_b n -h_{ab} \triangle n -nR_{ab} +\tfrac{1}{2} N \left( \triangle q_{ab} + D_aD_bq \right) \\ & \quad\, - \tfrac{1}{2}D^cN \left( D_aq_{bc} + D_bq_{ac} - D_cq_{ab} + h_{ab} D_cq \right)\\ & \quad\, - N\left( R_{c(ab)d}q^{cd} + {R^c}_{(a}q_{b)c} -h_{ab}R^{cd}q_{cd} \right) \end{split}\]

preservation of gauge: elliptic equations for \(n\) and \(n_a\)

\[\begin{split} 0 & = 2 D^bD_{(a}n_{b)} - \tfrac{2}{d-1}D_a( Np ) + 2 D^b N p_{ab} \\ 0 & = - (d-1) \triangle^2 n + R_{ab} D^aD^bn - R_{ab}R^{ab}n + NR^{ab} \left[ (d-\tfrac{1}{2})\triangle q_{ab} - (d-\tfrac{3}{2}) D_aD_bq \right]\\ & \quad + \left( \tfrac{5d-4}{2} D^aN R^{bc} - D^cN R^{ba} + 2(d-1)ND^aR^{bc} \right)D_aq_{bc} - (d-1)D^aNR_{ab} D^bq \\ & \quad + \left ((d-1)N\triangle R^{ab} +\tfrac{5}{2}(d-1)D_cN D^cR^{ab} + NR_{cd}R^{acdb} - NR^{ac}R^b_c \right)q_{ab} \end{split}\]

canonical energy \(= \langle p, Kp\rangle + \langle q, Uq \rangle \)

Kinetic Energy

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

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

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?

• Presentation given at the 22^nd Midwest Relativity Meeting at the University of Chicago
• based on the work done in collaboration with Robert. M. Wald (University of Chicago) and Stefan Hollands (Cardiff University).
• Impress.js by Bartek Szopka.
• Typeface: Holtwood One SC