Canonical energy &
linear stability of Schwarzschild
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
- remain uniformly bounded on the exterior
- 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
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)
- \(\mathscr E_{\mathrm{can}}(\tilde\gamma) \sim (\nabla^1\tilde\gamma)^2 \) (positivity unknown, even in Schwarzschild)
- \(\mathscr E_{\mathrm{DHR}} \sim (\nabla^5\tilde\gamma)^2 \)
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
Note \(\gamma \sim \nabla^4 \tilde\gamma\)
Canonical energy
Canonical energy of \(\gamma_{ab}\)
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!
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 \(\psi\)
- \(\psi\) → DHR's Regge-Wheeler-like variable \(\Psi\) → \(\mathscr E_{\mathrm{DHR}}\)
- \(\psi\) 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,
- no Regge-Wheeler variable, so DHR method does not easily generalise
- Hertz potential construction still works; \(\mathscr E_{\mathrm{can}} \geq 0\)?
Do electromagnetism in Kerr first.
- Fackerell-Ipser equation has complex potential so no real, conserved energy.
- 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)_/¯