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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

image/svg+xml Σ~ Σ B

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=(DaDbαhabαRabα2D(aαb))


infinitesimal

Perturbations

X=(pab,qab)


Inner Product

˜X,X=Σ˜pabpab+˜qabqab

Symplectic Product

˜X,SX=Σ˜pabqab˜qabpab

Constraints

0=X,S GααC0

0=Rabqab+q Hamiltonian
0=Dbpab diffeomorphism

Gα satisfies constraints!

Gauge Conditions

0=X,GααC0

0=Rabpab+p
0=Dbqab

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

Boundary terms

Evolution

˙qab=2D(anb)+2N(pab1d1p hab)Kpab˙pab=DaDbnhabnnRab+Uqab
preservation of gauge: elliptic equations for n and na
canonical energy =p,Kp+q,Uq

Kinetic Energy

Kpab,Kpab=Σ2N[(pab)21d1p2]

TT-decomposition

Choose: ξ=1d1p with ξ|B=0
χabpabDaDbξ+Rabξ+habξ

  • χ=0 traceless
  • constraint : Dbχab=0 transverse

Potential Energy

Uqab,Uqab=ΣN[12(Dcqab)232(Daq)2+qabRcabdqcd]Bκ(qar)2
Will TT help?

Take Home

  • uniquely fix gauge: 0=X,Gα
  • evolution: ˙q=Kp ; ˙p=Uq
  • K0
  • U0? ; stationary case?

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