Notes by Willie Wong (wongwwy -at- member.ams.org). Comments and corrections welcome.
Motivation: instability of higher dimensional vacuum black holes (when they rotate sufficiently fast)
Gregory-Laflamme instability. Black brane solution in $D+p$ dimensions is a $D$-Schwarzschild cross $\mathbb{R}^p$. Cross section of the horizon has topology sphere cross plane. GL: perturbations in the flat directions can grow exponentially in time, if it has sufficiently long wave lengths.
Two families of explicit AF black holes in higher dimensions: Myers-Perry black holes: similar to Kerr, uniquely parametrised by mass $M$ and angular momenta $J_i$, $i = 1 \ldots \lfloor (d-1)/2 \rfloor$.
Black rings in $d = 5$. Horizon has $S^1 \times S^2$ topology. (Perhaps numerically obsered in $d > 5$ also.)
Singly spinnin MP BH are those where $J = J_1$ and $J_2 = \cdots = J_i = 0$. For $d \geq 6$ there are no upper bound on angular momentum (unlike Kerr). These "ultraspinning" ones are conjectured to be unstable (Eparan & Myers 2003): locally on the axis of rotation it looks really flat and like a blackbrane since spinning causes the hole to flatten.
Threshold of rotationally symmetric instability found by linearized analysis (Dias, Figueras, Monteiro, Santos, Emparan 2009). "Bar mode" instability found by nonlinear numerical solution for $d = 5$ (Shibata & Yoshino 2010) Found that lumpiness are "shed" by radiation, and the spin drops and get to a lower spin MP.
Cohomogneity-1 MP: Odd dimension $d$, set all angular momenta the same. Depends only on radial coordinate (even simpler than Kerr!) Perturbation equations are ODEs.
This family has an upper bound on $J$, and saturation gives extreme BH. $d = 5$ has no sign of instability (Murata & Soda 2008). $d > 5$ exponentially growing modes close to extremality (Dias, Figueras, Monteiro, HSR, Santos 2010) at the linearised level.
Linearised gravitational perturbations of higher-dimensional black holes give a large set of coupled PDEs and requires numerical solution. Are there easier methods?
What didn't work:
Penrose inequality: area of the outer most trapped surface should be less than the final Bondi/Hawking mass, which by mass loss should be bounded by initial ADM mass. Proven in the time-symmetric case at the initial data level by Huisken & Ilmanen 2001.
Local Penrose inequality: consider a $d$-dim black hole parametrised by mass and angular momenta (e.g. MP). If BH stable, perturbations radiate and solution settle down to a stationary BH. Assuming rotational symmetric perturbations, we have conservation of angular momenta. So we have that at the end of the day we should have that the area should be bounded by the area of the black hole corresponding to the initial data's mass and angular momenta.
Local in the sense that it only applies to initial data that is closed to a stationary BH.
So if this inequality is violated, the original assumption that BH is stable is false.
NB by the above, a local penrose inequality implies that an asymptotically stable stattionary BH is a local maximum of area for fixed mass and angular momentum in the space of rotationally symmetric initial data.
So now we need to construct initial data. Use conformal method and solve assuming small data perturbation in series expansion. Assume one can change the signs simultaneously of $t$ and $\phi$ (the angular rotation coordinates). This implies the horizon is minimal surface. We expand to second order since first order is satisfied by first law of black hole mechanics.
For black strings this predicts instability at a threshold comparable to Gregory Laflamme. For MP, another prediction, gives worse threshold than untraspinning, but comparable.
Black rings: instability for all "fat" rings and for sufficiently "thin" rings. Don't know about the middle.
Some stuff about "non uniform black strings".
Consider $D$-dimensional GR. Vacuum without cosmological constant. Methods should apply to other theories of gravity. Also interesting to consider black branes in $D+p$ dimensions.
Talk about linearised stability: write down perturbation equations and study modes.
The computation is non-trivial: Schwarzschild was done by Regge-Wheeler, and then Zerilli. Schwarzschild black string was Gregory-Laflamme. Difficulty: decoupling the equations and fixing the gauge to make stability issues more apparent.
Other methods? In addition to previous talk, consider Thermodynamic stability.
Write entropy $S$ as a function of energy $E$ and other extensive state parameters $\Xi$. Condition for thermodynamic instability is that the Hessian $\nabla^2 S$ admits a positive eigenvalue. If this happens, one can increase total entropy by exchanging $E$ and or $X_i$ between different parts of the system. For the case of $E$ being the eigenvector, this corresponds to a negative heat capacity.
Suffices to investigate
($T$ is the temperature, $\delta^2$ denotes second variation). First variation is always okay by first law of thermodynamics.
We can use $M$ for energy, area as entropy, and angular momenta as state variables. (Gubser-Mitra) Problem: Schwarzschild has negative heat capacity. But seen as a sign for Schwarzschild-strings to be unstable.
A new fundamental stability criterion. It in particular implies that a version of (local) Penrose inequality is necessary and sufficient for black hole stability with respect to axi-symmetric perturbations.
Result: Consider perturbations of a static or stationary-axisymmetric black hole or black brane with bifurcate Killing horizon. The canonical energy of the perturbation $\gamma$ is defined to be (the integration of the symplectic current $\omega$ over a slice that intersects the horizon). A necessary and sufficient condition for stability is positivity of $\mathcal{E}$ over perturbations preserving mass and both linear and angular momenta.
Can show that
$\kappa$ is surface gravity.
The constraint on mass/linear/angular momenta rules out the "instability" of Schwarzschild.
But if this negativity (that $\mathcal{E}$ can be negative when not restricting to mass-momenta-preserving perturbations) is available, a corresponding black brane can be made unstable (for mass-momenta-preserving perturbations) by inserting a long wave-length oscillation on the brane. This proves the Gubser-Mitra conjecture.
So how does this work? Take Lagrangian for the vacuum GR. First variation is where $E$ is the Einstein and is the boundary term.
The symplectic current is
Noether current where $C$ is some constraints, which can be taken to vanish for solutions.
Variational identity:
The first two terms vanish is the background is a solution and if the variation satisfy constraints (linearised equations). ADM conserved quantities (Hamiltonians) are those whose variation are integrals (and spatial finity) of the interior of that last bracket.
Integration the identity using $X$ the Killing field null on the horizon, this gives the first law of blackhole mechanics for stationary axisymmetric black holes.
Now, choose perturbation of a black hole. Choose gauge in which that the position of the horizon (bifurcate sphere) does not change to first order. Now we take another variation of the variational identity, this gives the entropy-like formula on $\mathcal{E} = \delta^2 M\cdots$ from before.
One can show that for "gauge transformations" (infinitesimal diffeomorphisms that preserves asymptotic flatness, ADM mass and momentum and angular momentum, and fixes the boundary gauge) the form $\mathcal{E}$ doesn't change. This can be further refined to a characterisation of degeneracy of $\mathcal{E}$ as precisely those perturbations that goes toward another stationary axi-symmetric black hole.
That negative modes to $\mathcal{E}$ corresponds to instability is argued via considering monotonicity laws for $\mathcal{E}$ by studying the Bondi news function at Scri.
Modes: special solutions of wave equations that are periodic in time $\Psi = e^{i\omega t}\phi(x) $ and in particular does not decay.
Modes can be shown not to exist under certain assumptions, but we can construct quasimodes, which doesn't exactly solve the wave equation, but only up to an error.
Joint work with G. Holzegel.
Remark: not to be confused with quasinormal modes.
AdS: take $l > 0$ and $R^4$, take metric to be
Ricci is $-3/l^2 g$. Drew Penrose diagram.
Note: not globally hyperbolic. In ADS there exists modes with finite energy. $\|\phi\|{H^1{ADS}} < \infty$ (weighted $L^2$ norm on first derivative on constant $t$ slice).
On bounded domains, we can easily get modes (confined waves); nondecay on ADS. On open systems we can add a black hole to kill it. By putting in "non radiation" boundary conditions at the timelike boundary you can keep energy.
Schwarzschild ADS the metric is given by adding a $-2m/r$ term into the $dt^2$ and $dr^2$ terms of the metric.
Theorem (GH, JS 2011) On Kerr-ADS if the angular momentum $a$ is bounded by $l$ and horizon radius is bigger than $\sqrt{|a|l}$, then there exists no periodic in $t$ solutions to the wave equation and there is decay: energy on slices intersection both Scri and horizon decays like $1/\sqrt{\log t}$ if one gives up one derivative.
Remark: study actually Klein-Gordon for all masses strictly greater than the BF bound $-c(\ell) = - \frac{9}{4l^2}$.
Remark: well-posedness requires boundary conditions on Scri, which are incorporated in the form of the norms we allow (which is essentially Dirichlet). (Sometimes Neumann is also okay.)
Theorem (GH 2010) Well-posedness holds in $H^2_{ADS}$ assuming the BF bound for any asymptotically ADS background.
(Related work: Ishibashi-Wald, Bachelot, Vasy, Claude Warnick.)
Remark: boundeness was shown by Holzegel 2009. For the range of coefficients in the Theorem, we can use the Hawking Reall vector field (which is causal globaly) to get boundedness without understanding decay (circumventing the superradiance problem).
Remark: Loss is classical; due to trapping. One can gain decay by sacrificing more derivatives.
Remark: log decay is expected to be sharp or close to sharp. Expected not to be able to get polynomial. Evidence: (1) quasinormal modes in Schwarzschild-ADS (there exists numerical work) (2) New trapping phenomenon: stable trapping unlike the AF case.
Theorem (GH, JS) Sch-ADS, Klein-Gordon mass bounded below by $-2/l^2$ the conformal mass. $\exists c > 0$ such that
In other words, we cannot do better than $o(1) / \log(2 + \tau^*)$. Observe that there is gap between $o(1)/\log t$ and $1 / \sqrt{\log t}$.
Remark: proof by construction of quasimodes. One can construct a sequence of quasimodes supported on ${r \geq 3M - \delta}$ such that the error is supported in $3M \pm \delta$ and the size of the error term drops exponentially in $n$ (the index of the quasimodes). Also can make the frequency roughly $n^2$. The actual solution will be approximately the quasimode on exponential time before it decays.
The construction of these quasimodes (estimating the error) uses "Agmon estimates" for solutions to elliptic operators.
Proof of First Theorem: separation of variables and prove decay for individual frequencies. Stable trapping causes loss in high frequencies, which when glued make the loss.
What is stable trapping? It is an artifact of the timelike boundary condition on non-radiating. (In coming waves of insufficient energy bounces back to timelike boundary, which "reflects" back.)
