Notes by Willie Wong (wongwwy -at- member.ams.org). Comments and corrections welcome.
Establishes the topological simplicity at the fundamental group of DOC. Results on censorship are generally spacetime results: involve assumptions that are global in time. Goal: to distill away the evolutionary difficulties and to get a censorship result from initial data.
Joint work with Eichmair and Dan Pollack. arXiv:1204.0278.
Review: Gannon-Lee singularity theorem. If space-time has null energy condition with Cauchy surface $S$ regular near infinity and $\pi_1(S) \neq 0$ then $M$ is future null geodesically incomplete.
I.e. nontrivial topology leads to gravitational collapse. If WCC is true you get event horizon. Hence outside should have simple topology. $\implies$ Friedman, Schleich, and Witt.
Theorem: $(M,g)$ GH, AF, NEC. Then every causal curve from $\mathscr{I}^\pm$ to the other can be deformed with fixed enpoints to a curve lying in a simply connected neighborhood $U$ of null infinity (which we assume to exist).
"Proof": pass to universal cover, if not deformable then the curve must lift to something that goes from one end to another end, which will pass through trapped surfaces necessarily.
Technically FSW is only about causal curve and not about the full topology (any curve). Chrusciel+Wald (in stationary case) and then Galloway (1995 general) that topology can be shown using this technology.
Various extensions: Asym. ADS (Schleich Witt Woogar 1999) and Kaluza-Klein (Chrusciel Solis 2009).
Toward the initial data level.
Step 1: initial data version of Gannon-Lee. Question: What is an initial data singularity theorem? $\implies$ Condition tha imply existence of trapped surface should be okay. (Can use Penrose singularity etc.)
$(M,g)$ the space-time. $(V,h,K)$ the data. $\Sigma$ a closed 2-sided surface. $\ell_+ = u+\nu$ outward null normal and $\ell_-$ inward null normal (future directed as $u$ is the unit time like and $\nu$ the space-like normal).
. Remark: where $H$is the mean curvature of $\Sigma$ within $V$. Trapped surface is when $\theta_\pm < 0$. Define outer trapped and MOTS.
State Penrose Singularity. In terms of initial data singularity theorem, c.f. work of Beig and O'Murchadha.
Schoen and Yau have given criteria for existence of MOTS in initial data set $\implies$ so that is an initial data singularity theorem!
Theorem (Penrose for Mots): spacetime is GH, noncompact $V$, NEC, MOTS $\Sigma$, "generic condition" holds on each future and past inextendible null normal geodesic to $\Sigma$. Then null normal geodesics must be future or past incomplete.
Comment: first do the case where $\Sigma$ separates $V$. This is almost analogous to the classical Penrose. For the case where $\Sigma$ does not separate $V$, lift using a covering space argument and regain a separating case.
So a initial data result that implies existence of MOTs should be regarded as a singularity theorem.
Immersed MOTS (to make things more general [and also necessary]). Example: $\mathbb{R}P^3$ geon (obtained by folding Schwarzschild where the radial coordinate gets identified with its "negative" and the spherical coordinates get send to antipodes). Schwarzschild double covers the geon. The bifurcate "$\mathbb{R}P^2$" is not orientable, but we call it an immersed MOTS (since it is double covered by a sphere).
Defn: an immersed MOTS if $\exists$ a finite cover of $V$ in which the lift of $\Sigma$ is a bona fide MOTS.
Observation: Penrose type theorem for MOTS above extends to immersed MOTS. (Incompleteness upstairs must mean incompleteness downstairs.) So existence of immersed MOTS is also a sufficient statement of initial data singularity.
Theorem (Main; Gannon-Lee type). If the AF initial data set $(V,h,K)$ is such that $V$ is not diffeo to $\mathbb{R}^3$ then $V$ contains an immersed MOTS.
Comments: AF requires $V$ to have finitely many ends each tends to flat. If one also assume DEC, then the conclusion can be refined: if $V$ is not diffeo then it contains an immersed spherical MOTS. Can be viewed as a non-time-symmetric version of work of Meeks-Simon-Yau (Annals, 1982). The proof relies on recent existence results for MOTS.
Note: actually uses (full) geometricization of 3 manifolds!
Theorem (Existence of MOTS): if $W$ is a 3D connected compact manifold with boundary inside the initial data set. Suppose the boundary has two parts: one part is outer-trapped, one part is outer-untrapped. Then there must exists a smooth compact MOTS in $W$ that separarates the two parts of the the boundary. (Eichmair for $d\geq 3$).
Sketch of proof of main theorem: Assume not immersed MOTS and show it is diffeo to $\mathbb{R}^3$.
Now, towards an initial data topological censorship. One should think of $V$ as a slice of DOC whose boundary corresponds to a cross section of event horizon. We represent this cross section by a MOTS. Assume further there are no immersed MOTS in $V\setminus \partial V$ (so that it looks like a slice of DOC).
Theorem (Main 2): let $V$ be AF data such that $V$ is manifold-with-boundary. Assume $\partial V$ is compact MOTS. If $\partial V$ is composed of disjoint union of spheres, and if there are no immersed MOTS in $V\setminus \partial V$, then $V$ is diffeo to $\mathbb{R}^3\setminus$ finitely many balls.
(If we assume DEC, we don't need to assume that $\partial V$ are spherical by Galloway-Schoen.)
Proof is similar to the proof for the Gannon-Lee analogue theorem, just more care required to deal with the MOTS boundary.
Question: can the assumption on immersed MOTS be weaked to just honest-to-god MOTS. No! the Geon is a counterexample. But is it the only type of counterexample?
Joint work with Dafermos and Rodnianski
Start with the problem of black hole stability. Is Kerr stable? That is, do sufficiently small perturbations of Kerr initial data converge to a black hole solution and in particular to another Kerr solution on the domain of outer communications.
Difficulties:
So far almost all progress has been made for the linear wave equation on a fixed Schwarzschild and Kerr background. The philosophy is to understand the decay mechanisms and obstructions in a robust manner before tackling a non-linear problem.
Complete understanding of linear homogeneous wave on sub-extremal Kerr due to Mihalis and Igor. (Also Tataru-Tohaneaunu, Andersson-Blue for decay in slow rotation; Metcalfe-Tataru-Tohaneanu, Tataru, Donninger-Schlag-Soffer for optimal decay rate [in terms of application to nonlinear problem]; Luk, Marzuola-Metcalfe-Tataru-Tohaneanu for nonlinear problem; Blue and GH for higher spin.)
Full problem: use Demetri-Sergiu approach. Ricci vanishes. Riemannn equals Weyl. Equations to consider are Bianchi $DW = D*W = 0$ + null-structure equations $\nabla \Gamma + \Gamma\Gamma = W$
Linearisation around Minkowski background the $\Gamma$ and $W$ equations decouple. So Chris-Kla can just study the spin 2 equations without thinking about $\Gamma$. On curved backgrounds however, the linearisation still has coupling since, for example, $DW = \partial W + \Gamma W$ and the linearisation of $\Gamma W$ gives linear contributions.
A first step to takle the full problem is studying the "ultimately Schwarzschildean spacetimes" Roughly speaking assume derivatives of curvature converge to their fixed Schwarzschild values measured with appropriate energy and spacetime norms on the curvature.
Theorem: derivative gain: if ultimately Sch. with $n$ derivatives assumed, can prove also convergence in $n+1$ derivatives.
Comment: it is a conditional result. In particular, existence of non-trivial ult. Sch. spacetimes is assumed, but not known to contain more than just Sch. The rates of approach are the ones expected from Price's law (actuall, much weaker).
The result adapts wave equation to spin 2 + null structure. It also provides a new method to obtain "peeling" estimates.
Kerr is much harder although some of the techniques carry through.
Theorem (Main): the set of ultimately Sch spacetimes is quite large. In particular the previous Theorem is non-empty.
Remarks:
Solving backwards, the red-shift which gives exponential decay becomes blue-shift and gives exponential growth along the horizon, which means that the assumption of exponential convergence is pretty much necessary for the construction.
Toy problem: prescribe characteristic initial data for linear wave on horizon on null infinity. Solve backwards.
Goal: truncate near $i_0$ and show uniform energy bounds irrespective of how far the initial truncation is. For the uniform bound we need to bootstrap exponential decay.
Note that since we allow exponential stuff, we don't need the multiplier vector field to be particularly good: since Gronwall just gives something like it anyway.
Full problem: We fix a differentiable structure from Schwarzschild. Assume a metric that looks something like a Kerr (that this form of metric is available is due to Rendall, Christodoulou). Null decompose w.r.t. foliation and renormalize relative to Schwarzschild and get a system of hyperbolic and transport equations for decaying quantities.
Roughely the proof goes like:
There is an important distinct in the transport equation depending on whether one starts from null infinity or horizon, rooted in the radial decay of the components. Integrating from horizon you do not lose in $r$. But integrating from null infinity, you need to gain smallness by an additional $1/r^2$ decay because $r$ is large. When there is a borderline term, however, it would be a null structure: one term in the product can be integrated from horizon and you can gain in that direction.
Conjecture: solving backwards requires exponential decay; generic polynomial decay will lead to a singular horizon.
Joint work with Jan Metzger
First some basic definitions. Riemannian three manifold $(M,g)$ is AF (up to second derivatives), complete. Recall that this implies the scalar curvature is integrable.
Assume that either $\partial M = \emptyset$ or $\partial M$ is the outer most minimal surface.
Example: time-symmetric slice of Schwarzschild.
Defn: $(M,g)$ AF is $C^k$-asymptotic to Schwarzschild $g_m$ with $m > 0$ if metric
Defn: $(M,g)$ AF, $V > 0$ a number. Let
Defn: a minimiser of the above is called an isoperimetric region.
In general for non compact manifolds, minimisers need not exist. (Compact case existence due to geometric measure theory.) Note that minimisers are critical points, and such that $\partial \Omega$ is CMC (not minimal because of the Lagrange multiplier from $V > 0$).
A critical point for the variation problem is said to be stable if $\forall u \in C^1(\partial\Omega)$ with $\int u = 0$ we have
$k$ above is the second fundamental form of $\partial\Omega$ and $\nu$ is the outward normal. Basically this is a constraint on second variation of $A_g(V)$.
Look at Schwarzschild, the mean curvature starts at 0 at the throat, increases to a maximum, and drop down and decays like $2/r$. So for every mean curvature less than the maximum there are exactly two CMC surfaces with the mean curvature. (Simon Brendle 2011)
Bray 1998: isoperimetric surfaces in Schwarzschild exist, and are precisely the constant radius spheres.
Theorem: (Huisken, unpublished) Assume $(M,g)$ AF (order 2), $R \geq 0$. The isoperimetric mass is non-negative. Equality holds iff $(M,g)$ is Euclidean plane.
Fan-Shi-Tam 2009: $m_{\text{iso}} \geq m_{\text{ADM}}$
This talk: if $(M,g)$ is $C^0$ asymptotic to Schwarzschild with $m > 0$ we have $m = m_{\text{iso}}$.
A bit of history: Huisken-Yau 1996 Theorem: $C^4$ asymptotic to $g_m$ then there exists a compact set $K\subset M$ s.t. $M\setminus K$ is foliated by stable CMC spheres ${\Sigma_H}{0 < H < H_0}$ with MC $H$ s.t. $\Sigma_H$ becomes "rounder and rounder" as $H\searrow 0$; the center of mass of these spheres converges to the "Huisken-Yau geometric center of mass" (in AF coordinates); and that $\Sigma_H$ are the unique stable spheres of CMC $H$ containing $B{H^{-q}}$ for some $q \in (1/2,1]$.
Theorem (Qing- Tian 2007): $\Sigma_H$ is unique ... containing $B_R$ for some large $R$ independent of $H$.
Comment: analytically delicate, uses perturbation from Schwarzschild strongly.
Theorem (Main 1; E-Metzger 2010): $(M,g)$ is $C^2$ asymptotically to Schwarzschild, $R > 0$, then $\Sigma_H$ are the unique large (in terms of area) connected stable CMC surfaces in $(M,g)$ which contains the horizon $\partial M$.
Comment: in the proof, the $R>0$ condition is strongly used: cannot relax to $R \geq 0$. But can possibly relax to $R > 0$ at "sufficiently many points". The proof is a delicate modification of the Schoen-Yau proof of the PMT.
Theorem (Main 2: E-M 2010, 2012): $\Sigma_H$ of Huisken-Yau are uniquely isoperimetric.
Joint work with Igor Rodnianski
Introduction: Penrose solution (1972) an explicit solution to EVE. Metric is where $\Theta$ is the Heaviside step function. The Riemann curvature tensor has $\delta$ singularity.
Khan-Penrose later discovered an explicit solution to EVE representing the interaction of two plane implulsive gravitational waves. After the two waves interact, there is an eventual singularity whre the space-time cannot be extended in $C^0$.
Besides being explicit, they are both in plane symmetry. So must have non-AF. Goal: find analogue with finite extent and non-planar symmetry.
Study impulsive gravitational waves formulated as a characteristic IVP on the intersection of two null hypersurfaces. Prescribe initial data such that the Riemann is singular on the hypersurfaces.
The free data on null hypersurface is tracesless part of null second fundamental forms. Prescribe such that $\hat{\chi}$ has a jump at $S_{o,u_o}$ and $\hat{\bar{\chi}}$ is smooth, but not necessarily smooth.
Result: can show that there exists a unique local solution, the curvature has a $\delta$ singularity across the incoming null hypersurface emanating from the initial singularity, and the spacetime is smooth away from the hypersurface.
Remarks:
Consider the Bianchi equations $\Box R = R*R$. The RHS doesn't necessarily make sense since potentially there could be two delta functions multiplying each other. Need to use structure!
Formulate in double null. $e_3, e_4$ are null and $e_1, e_2$ tangent to spheres. Ricci coefficients are $\psi$ and curvature are $\Psi$.
Strategy: combine $L^2$ energy estimates for $\Psi$ with transport equations for $\psi$. Problem: curvature is not in $L^2$.
Solution: revisit the energy estimates using Bianchi equations directly and not Bel-Robinson tensor. Trick: replace $\alpha$ by $-\nabla_4\hat{\chi} + \psi\psi$ everywhere. Now we have no $\alpha$s!
A priori, when taking limits, one has to look at the difference between two space-times, and the structure may not still have this nice structure! It is a miracle that the equations of the difference of space-times does not depend on the $\alpha$ of either. In fact now we see that the theorem can be written even when $\alpha$ is just a distribution! (Doesn't have to be a measure even.)
Next: what happens when the two impuleses interact? Let $\hat{\chi}$ and $\hat{\bar{\chi}}$ both have jump discontinuities across the same two sphere.
Theorem: With the given initial data, a unique local solution to the vacuum Einstein equations exists. Moreover, the curvature tensor of the solution has a delta singularity across the null hypersurface emanating from each of the singular 2-spheres and the solution is smooth away from the union of these hypersurfaces.
Epilogue: Formation of trapped surface. A consequence of the above theorems is that we get an improved theorem on the formation of trapped surfaces. The trick is to build a semi-global solution for a large-data problem. Christodoulou used the short-pulse method.
In particular, it uses that the data on the incoming surface is trivial (small). Using the above, we can put data that is not necessarily small, but just regular. Christodoulou uses the large data $\|\hat{\chi}\|\infty \approx \delta^{-1/2}$ and $\|\alpha\|\infty \approx \delta^{-3/2}$. But in this formulation we don't care about $\alpha$ and can put $\|\hat{\chi}\|_2 \approx 1$. (The regularity is given by an assumption on an inequality between certain geometric quantities along the incoming boundary.)
(Jonathan went really fast; so these notes are not as complete as the previous ones.)
Question: how does angular momentum affect the shape and size of the apparent horizon in a black hole space-time.
Consider maximal slice of an axi-symmetric space-time. Not necessarily vacuum. Study area-angular-momentum inequality.
(Argh... too many pictures and equations to try to copy down on computer.)
Result with Dain: Let $(\Sigma, g,K)$ be a smooth vacuum axisymmetric maximal data set with possibly many AF ends $E_1, \ldots, E_n$. Let $S$ be any compact oriented surface. Then area of $S$ is $> 8\pi J(S)$, where $J$ is the angular momentum obtained as an integral of the second fundamental form over the surface $S$. Equality is not possible (in the case of extremal Kerr, one of the ends is not AF!)
Proof by contradiction. Suppose we have two ends separated by a saturating sphere. From results in near-horizon geometry, we know the geometry around this sphere. Use local existence in maximal gauge. Note that this sphere is minimal. Now use the second variation formula for area. This implies that the second time derivative of the area of this sphere is negative, which implies that the area will drop below $8\pi J$ (since the Komar angular momentum is conserved), which is known not to be possible.
(There's more, but I couldn't get it all down.)
Construct initial data for spherically symmetric Einstein-Vlasov system such that in the future a black hole forms and that in the past the spacetime is geodesically coplete, and thus has no white hole.
Note that in numerical relativity it is often the case that black hole data have an incomplete past. First example of an initial data with the above properties for "realistic" (in the sense that Vlasov is used by astrophysicists) matter model.
Dafermos (2010) showed that this type of data exists for Einstein-scalar field.
Review Einstein-Vlasov system.
Mathematical formulation of the system; easier to look up on the Living Reviews article.
Theorem: there exists a class of initial data for the spherically symmetric Einstein-Vlasov system with the property that black holes form in the future time direction and in the past time direction the spacetime is geodesically complete.
Corollary: Arbitrary small mass-radius ratio is allowed.
Corollary: there exists a class such that the corresponding solutions exist for all Schwarzschild time.
Take initial data as constructed in previous work (HA, Kunze, Rein 2011) on black hole formation. Globally existence in the past does not follow from HA Kunze Rein 2007. Some new ideas needed.
Use ideas from HA 2010 "Regularity..." to overcome.
Work in Schwarzschild coordinates. The constructed solution has the particles moving outwards (in reverse time) all the way to time infinity. Construct an auxillary function for which one has roughly an (almost) monotonicity law for the log derivative. By Gronwall if the log derivative is integrable then this auxillary function which measures the velocity (say) of the matter is bounded from below, which implies that the matter stays outgoing.
