
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 ⇔ positive canonical energy for perturbations- Gaussian null coordinates
- boundary conditions
- fix gauge near horizon
ADM
Static Background
(πab,hab)
Gauge freedom
Gα=(DaDbα−hab△α−Rabα2D(aαb))
infinitesimal
Perturbations
X=(pab,qab)
Inner Product
⟨˜X,X⟩=∫Σ˜pabpab+˜qabqabSymplectic Product
⟨˜X,SX⟩=∫Σ˜pabqab−˜qabpabConstraints
0=⟨X,S Gα⟩∀α∈C∞0
0=Rabqab+△q Hamiltonian
0=Dbpab diffeomorphism
Gα satisfies constraints!
Gauge Conditions
0=⟨X,Gα⟩∀α∈C∞0
0=Rabpab+△p
0=Dbqab
perturbed area of B =0
perturbed expansion of B =0
Boundary terms
Evolution
˙qab=2D(anb)+2N(pab−1d−1p hab)≐Kpab˙pab=DaDbn−hab△n−nRab+≐−Uqab
preservation of gauge: elliptic equations for n and na
Kinetic Energy
K≐⟨pab,Kpab⟩=∫Σ2N[(pab)2−1d−1p2]
TT-decomposition
Choose: △ξ=−1d−1p with ξ|B=0
χab≐pab−DaDbξ+Rabξ+hab△ξ
- χ=0 traceless
- constraint : Dbχab=0 transverse
Potential Energy
U≐⟨qab,Uqab⟩=∫ΣN[12(Dcqab)2−32(Daq)2+qabRcabdqcd]−∫Bκ(qar)2Will TT help?
Take Home
- uniquely fix gauge: 0=⟨X,Gα⟩
- evolution: ˙q=Kp ; ˙p=−Uq
- K≥0
- U≥0? ; stationary case?
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