Canonical energy &
linear stability of Schwarzschild

Kartik Prabhu with Robert M. Wald APS 2017, Washington D.C.

Linear stability of Schwarzschild

Dafermos, Holzegel, Rodnianski (arXiv:1601.06467)

Solutions to linearised Einstein equations around Schwarzschild with regular asymptotically flat initial data

  1. remain uniformly bounded on the exterior
  2. decay inverse polynomially to linearised Kerr

Teukolsky variable

\(\tilde\gamma_{ab}\) be linearised perturbation of Schwarzschild

  • \(\psi = r^4\psi_4\) (rescaled Weyl component) satisfies the Teukolsky equation \[ \left[ (þ' + \rho)(þ + 3\rho ) - ð'ð - 3 \Psi_2 \right]\psi = 0 \]

\(þ, þ', ð, ð'\) are GHP derivatives, \(\rho\) is a GHP spin-coefficient, and \(\Psi_2 = - \frac{M}{r^3}\) .

DHR variable

  • DHR construct new variable \[ \Psi := (þ + \rho)(þ + 3\rho)\psi \sim \nabla^4\tilde\gamma \] which satisfies a wave-like Regge-Wheeler equation.

DHR energy

Conserved and positive energy

\[ \mathscr E_{\mathrm{DHR}} = \int N \left[ \tfrac{1}{2}\left\vert (þ + þ') \Psi \right\vert^2 + \tfrac{1}{2} \left\vert (D_r - 2\rho)\Psi \right\vert^2 + \left\vert ð\Psi \right\vert^2 + \left\vert ð'\Psi \right\vert^2 + \left( \frac{4}{r^2} - \frac{6M}{r^3} \right) \left\vert \Psi \right\vert^2 \right] \]

use \(\mathscr E_{\mathrm{DHR}} \sim (\nabla^3\psi)^2 \sim (\nabla^5\tilde\gamma)^2 \) to get decay estimates on \(\Psi\), descend down to \(\tilde\gamma_{ab}\).

Kerr?

no Regge-Wheeler variable, so DHR method does not easily generalise

Use canonical energy method of Hollands and Wald (arXiv:1201.0463)

Can we use canonical energy method to get something \(\sim (\nabla^5\tilde\gamma)^2\) and positive?

Yes! Let's go back to Schwarzschild…

Hertz potential

Use \(\psi\) as a Hertz potential to generate a complex solution to Einstein's equation \[ \gamma_{ab} = -l_al_b U + l_{(a}m_{b)} V - m_am_b W \] with

\[ U = ð^2\psi \\ V = \left[ þ ð + ð ( þ + 3\rho ) \right]\psi \\ W = ( þ - \rho ) ( þ + 3\rho )\psi \\ \]

Note \(\gamma \sim \nabla^4 \tilde\gamma\)

Canonical energy

Canonical energy of \(\gamma_{ab}\)

\[ \mathscr E_{\mathrm{can}} = \frac{1}{4}\int N \left[ \tfrac{1}{4} \left\vert (þ + þ' + D_r + 2\rho)V \right\vert^2 + \left\vert (þ + þ')W - ð' V \right\vert^2 + \left\vert D_r W + ð' V \right\vert^2 \right] \]

so \(\mathscr E_{\mathrm{can}} \sim (\nabla^3\psi)^2 \sim (\nabla^5\tilde\gamma)^2 \) and positive.

So we have two conserved, positive energies at the same order in \(\tilde\gamma\). Are they the same?

Equality!

\[ \mathscr E_{\mathrm{DHR}} = \int N \left[ \tfrac{1}{2}\left\vert (þ + þ') \Psi \right\vert^2 + \tfrac{1}{2} \left\vert (D_r - 2\rho)\Psi \right\vert^2 + \left\vert ð\Psi \right\vert^2 + \left\vert ð'\Psi \right\vert^2 + \left( \frac{4}{r^2} - \frac{6M}{r^3} \right) \left\vert \Psi \right\vert^2 \right] \]
\[ \mathscr E_{\mathrm{can}} = \frac{1}{4}\int N \left[ \tfrac{1}{4} \left\vert (þ + þ' + D_r + 2\rho)V \right\vert^2 + \left\vert (þ + þ')W - ð' V \right\vert^2 + \left\vert D_r W + ð' V \right\vert^2 \right] \]

Repeatedly use Teukolsky equation for \(\psi\), background Schwarzschild identities, integration-by-parts to get \(\mathscr E_{\mathrm{can}} = 4 \mathscr E_{\mathrm{DHR}}\).

Two roads diverged in a wood …

metric perturbation \(\tilde\gamma_{ab}\) → Teukolsky variable \(\psi\)

  1. \(\psi\) → DHR's Regge-Wheeler-like variable \(\Psi\) → \(\mathscr E_{\mathrm{DHR}}\)
  2. \(\psi\) as Hertz potential → complex metric \(\gamma_{ab}\) → canonical energy \(\mathscr E_{\mathrm{can}}\)

In Schwarzschild, both are equivalent.

Also, works for electromagnetism in Schwarzschild using Fackerell-Isper-like equation (see Pasqualotto arXiv:1612.07244)

… but only one leads to Kerr

In Kerr,

  1. no Regge-Wheeler variable, so DHR method does not easily generalise
  2. Hertz potential construction still works; \(\mathscr E_{\mathrm{can}} \geq 0\)?

Do electromagnetism in Kerr first.

  1. Fackerell-Ipser equation has complex potential so no real, conserved energy.
  2. use Hertz potential + canonical energy method: on going

And both that morning equally lay

neither generalises easily to higher dimensions; curvature components do not decouple so no single Teukolsky variable

What happens? ¯\_(ツ)_/¯