Notes by Willie Wong (wongwwy -at- member.ams.org). Comments and corrections welcome.
Separate the piece of "scaling" from the "conformal" part of the data.
Background: obtain sufficient and necessary conditions for AF vacuum data to develop near spacelike infinity into solutions of the Einstein equations with smooth conformal structure at null infinity. (In data existence of smooth conformal compactifications at spacelike infinity.)
Two infinities: $i$ is the space-like infinity wr.t. the initial data slice $\tilde{S}$ and can have smooth conformal extension of data on $\tilde{S}$ to $i$. $i^0$ space like infinity with respect to the space-time, where $m_{ADM} \neq 0$ and cannot have smooth conformal space-time extension to $i^0$.
So we blow-up the point $i$ and $i^0$. $i\to I^0 \sim \mathbb{S}^2$. But $i^0\to I \sim (-1,1) \times I^0$, a cylinder. The initial data slice becomes $i \to {0} \times I^0 \subset I$.
So near $I^0$ we can study an initial boundary value problem for Einstein's equations. From considerations of the hyperbolicity of this problem, we see that generically initial data jets cannot be solved smoothly up to where $I$ intersects $\mathscr{I}^\pm$: we obtain a logarithmic divergence there. On the other hand, there could be cancellations: for example the static solutions to EVE.
Question: what are the necessary or sufficient conditions for this smooth extension?
Friedrich 1998: at the critical sets $I \cap \mathscr{I}^\pm$ the jets develop logarithmic singularities for $k \geq 2$ if the dual Cotton tensor of the metric $h$, $B_{ab} = \frac12 D_c(R_{da} - \frac14 R h_{da})\epsilon_b{}^{cd}$ does not satisfy certain differential-algebraic conditions.
Also Valiente Kroon (2011): if $\theta^2 h$ defines a time-reflection symmetric vacuum data near space-like infinity conformal to $h$, then the solution jets of $\theta^2 h$ extend smoothly if and only if $\theta - 1$ vanishes to all orders in at space-like infinity.
What is static vacuum data: AF, mass $m$. Take positive function $v$ and Riemannian metric $h$ on a manifold diffeo to $R^3$ minus a ball so that the static vacuum equations and on $S$.
Beig Simon (1981) analytic conformal extension at space-like infinity. (There exists a conformal factor that converts a neighborhood of space-like infinity [compactified] to a disc with a real analytic Riemannian metric.)
Question: what is the condition between Friedrich 1998 and "asymptotically static vaccum initial data set"? (One is necessary, one is sufficient.)
Conjecture? The data are conformal to static data with analytic conformal factor if and only if Friedrich's condition is satisfied.
Essentially a study of jets/taylor expansions.
The understanding of the conjecture asks what is required for analytic data to be conormal to static. To do so we end up with whether there exists a scalar function satisfying an a priori highly singular, highly overdetermined system of equations. (Though uniqueness and analyticity is automatic if solution exists).
To guarantee regularity we need full understanding of Taylor coefficients at infinity. In other words, we need conditions on asymptotic expansion coefficients to all orders. By holomorphically extending the (real) analytic background to $\mathbb{C}^3$, we can analyse the problem instead using complex geometry. The condition on the Cotton tensor is then necessary and sufficient to guarantee the regularity problem.
Taking antisymmetrised derivative of the overdetermined equation we derive further consistency relations. These can be used to show that in a large class the derivative of the overdetermined equation can be inverted. (Certain matrix of coefficients need to have unique eigenvalues and diagonalizable.)
Conclusion: the AF metric which satisfy after a conformal rescaling the static field equations and for which $D^bB_{abc}(i)$ diagonalizes with three different eigenvalues are precisely
And there exists a corresponding generalization to the smooth case.
Two methods to constraint such equations: Holst-Nagy-Tsogtgerel-Maxwell and Dahl-Gicquaud-Humbert.
First recall the constraint equations and the conformal method. $(\mathcal{M},h)$ Lorentzian, EVE. $M$ is a slice, with normal timelike $\nu$. Define $g, K$ on $M$.
Gauss and Codazzi equations gives the two constraints. and
Questions: construct solutions to the constraint problems (physically reasonable ones in particular); parametrize the set of solutions of the constraint equations.
Now use $\tilde{g}$ and $\tilde{k}$ for the physical induced metric and second fundamental form...
Methods: gluing method (underdetermined elliptic system) or conformal method and variants. [See review article by Bartnik-Isenberg]
What is the conformal method? Principle: fix part of the initial data and solve for the other part. Given:
Solve for
such that $\tilde{g} = \phi^{N-2} g$ and $\tilde{K} = \tau \tilde{g} + \phi^{-2}( \sigma + LW)$ where $(LW){ij} = \nabla_i W_j + \nabla_j W_i - \frac{2}{n} \nabla^k W_k g{ij}$ is the conformal Killing operator.
Plug into the constraint equations, and the Hamiltonian constraint becomes the "Lichnerowicz equation", which is a elliptic equation for $\phi$. The momentum constraint is $- \frac12 L^L W = (n-1) \phi^N d\tau$. ($N = n^ = \frac{2n}{n-3}$ the $L^2$ Sobolev exponent)
Difficulties:
Previous results were mostly about when $\tau$ is constant or "almost" constant. (The RHS to the vector constraint either vanishes, or is "small".)
Strategy: assume the background $(M,g)$ has no conformally Killing vector fields (genericity).
What happens if $d\tau$ is large? Strategy of HNTM method)
Answer to the third requirement is scalar curvature.
Theorem of Maxwell: Assume $(M,g)$ has positive Yamabe class and that $\sigma \not\equiv 0$ but small. Then there exists at least one solution $(\phi, W)$ to the equations of the conformal method.
Comments on HNTM:
Prop: Assume that $\tau$ vanishes nowhere, and the operator $-\frac{3n-2}{n-1}\triangle + R$ is non-negative. Then if $\sigma \equiv 0$ and $\|d\tau / \tau\|_{L^n}$ is bounded by some fixed constant depending on $g$, then there is no nontrivial solution to the system. (And trivial solutions are unphysical)
Now move on to the DGH method.
The Heuristics is almost true: Theorem (DGH) If $\tau > 0$ (and in non-negative Yamabe class, non-trivial $\sigma$), Then if all the modified limit equation admits no non-trivial solutions (modification: multiply RHS by any constant $\in (0,1]$) then the constraint equations have a solution.
Method: break criticality by changing the RHS of vector constraint $\phi^N \to \phi^{N-\epsilon}$. Under the assumptions this system can be solved, and has uniform a priori bound independent of $\epsilon$. Then take convergence using compactness ("escaping" sequences must tend to solutions of the modified limit equation, which we assume not to exist...) The modifying coefficient is related to the $\epsilon$.
Application: by Bochner for $L^*L$, if $(M,g)$ has $Ric \leq -\theta g$ and $\| \frac{d\tau}{\tau}\|_\infty \leq \sqrt{ \frac{n\theta}{n-1}}$ then there is no non-zero solution to the limit equations.
Another application, similar result holds if $\|\frac{d\tau}{\tau}\|_n$ is bounded by $\frac12 \sqrt\frac{n}{n-1}$ times the Sobolev constant.
For non-closed case, need one more step: show $\sup \phi$ attained in a compact subset. Works for asym. hyp or AF. Even compact with boundary of Dirichlet and Neumann boundary conditions. Now working on compact manifolds with MOTS boundary .
One can also find examples of triples $(M,g,\tau)$ where the limit equation admits solutions.
The assumption that $\tau$ is signed appears crucially! When $\tau$ becomes zero at a point, the "subcriticality" modification no longer singles out the correct term in the Lichnerowicz equation, so perhaps expect concentration phenomenon.
Consider a dynamical system (i.e. ODE) $\dot{x} = f(x)$ where $x\in \mathbb{R}^n$. Particular solutions:
Would like to characterise the set of points near the cycle which converge to the cycle at large times.
Note that if such a solution exists, (stability of the cycle), expect the limiting solution to spend a lot of time near the fixed points. This would mean that the stability of the fixed points play important roles.
One way to capture the latter is the linearise near the fixed point and consider the eigenvalues. But of course one needs to be careful about whether nonlinear evolution and linear systems are similar. In the hyperbolic case we have Hartman-Grobman theorem. We will be interested in the non-hyperbolic case.
This can be applied to problems in general relativity. To wit, consider a spatially homogeneous space-time (isometry group acts on space-like hypersurfaces so roughly speaking nothing depends on spatial positions ), so PDE becomes ODE. Concentrate on EVE or electrovac. The spat. homo. can be classified by Bianchi schemes. We like class A (type I, II, VI0, VII0, VIII, IX).
Wainwright-Hsu system describes the class A dynamics. (Unified in the sense that various Bianchi types in class A all embed into subregions of the phase space.
The system is 4 dimensional; variables are dimensionless. Fixed points are self-similar soltions and not stationary solutions of the EVE.
BKL hypothesis: a "general EVE Bianchi A" solution is approximated by a heteroclinic chain (not necessarily a cycle!) of Bianchi II solutions near big-bang singularity.
There exists recent partial results on this problem by various people. For example, constructions of asymptoticaly monotone solutions using Fuschian techniques. Here we treat the oscillatory case (proof of existence of the cycle case, but nowhere near generic).
Liebscher et al (2011) on Stability properties of "the triangle": it has a one-codimensional Lipschitz unstable (reverse time stable) manifold. (Recall, interested in big-bang! that is $t\to -\infty$. )
To consider the linearisation around the corners, we find that the eigenvalues of the linearisation have favourable property: there exists necessarily a 0 eigenmode (circle of Type I), But the key property is that there exists a negative eigenmode $-\mu$. And $\mu < \lambda_{1,2}$ for the two positive eigenmodes.
Bianchi VIII and IX lies on the unstable manifold. So the theorem is that there exists VIII and IX solutions which converge to triangle in backwards time.
VI0 with magnetic field was studied by Weaver. It is a model of IX. (It also has 4 degrees of freedom.) (Remark: pure magnetic field is not compatible with VIII and IX)
In VI0 with magnetic field there also exists "the triangle" with a DS similar to that of Wainwright and Hsu.
Joint work with Liebscher and Tchapada (spelling?) (2012)
One critical difference, in the VI0 + mag case it is possible that $\mu \geq \lambda_1$. But $\mu < \lambda_2$ still. ($\lambda_2$ is the "incoming direction".) This is not enough by itself (for general DS with heteroclinic cycles). We overcome using geometric nature of the underlying problem which gives "extra" invariant manifolds.
They come from other Bianchi types.
Method of proof: in the vacuum case, a generalised Poincare section, (6 steps, 3 for near corner, and 3 for excursions between corners), the discrete map turns out to be a contraction.
Excursions have growth, but with fixed factor. locally near corners, the passage can pick up arbitrarily small contraction factors. So one dominates the other.
In the magnetic case, the contraction breaks down, if we measure everything w.r.t. Euclidean metric. We can however introduce a new (singular along orbit) metric in which this becomes a contraction.
What happens in I and II with mag? V. Leblanc: you have one way in, but two ways out! How do you choose? No one knows.
Expectation: under stationarity, MOTS should equal event horizon.
Two natural approaches:
Use a combination of both methods.
Miao's Theorem: second approach is more likely to work in static case. Example
Theorem: Let $(\Sigma, g, N)$ ($N$ is lapse) be time-symmetric, static, vacuum, AF, initial data set. Assume $\partial\Sigma$ is the outermost minimal surface. Then it is isometric to the Schwarzschild half space.
Extended by Carrasco and Mars. (Does not need to be vacuu.) Note that ${N=0}$ is a embedded totally geodesic hypersurface in $\Sigma$. If $N > 0$ on the interior of $\Sigma$ and $=0$ on boundary, then Bunting + Massood-ul-Alam we are done. If $N = 0$ in the interior, then contradiction to outermost minimal surface. If $N > 0$ on the interior of $\Sigma$ and $>0$ on the boundary, can derive another contradiction to outermost minimal surface condition.
Killing initial data set. Defn. Additional compatibility conditions.
Can define a future Killing development.
Try to recover the hypotheses for proving uniqueness in black holes. For static case, need the topological boundary to be compact topological manifold without boundary.
Killing pre-horizons: bad boundaries...
Definition: a matter model is well-posed if the KID generates a unique, maximal, globally hyperbolic spacetime admitting a Killing vector field.
Mars & Reiris 2012: AF stationary initial data set with well-posed matter and if the boundary is future outer trapped, then the Killing development is isometrically embedded in the Cauchy development, and a neighborhood of the boundary is detached from the past of the future development.
In particular, such sets generate black hole space-times.
Mars & Reiris 2012: under the same hypothesis the boundary of the data must be disjoint from where the Killing field is time-like.
Furthermore, M&R can also rule out the problematic Killing pre-horizons.
With this, for a matter model for which an apropriate black hole uniqueness theorem, static AF KID can be dumped into a black hole in the uniqueness class (NEC, and $\partial \Sigma$ trapped assumed).
Extremal black holes: surface gravity is constant along Killing horizons; extremal is with surface gravity $\kappa \equiv 0$. This implies no bifurcate spheres.
Extremal Kerr(-Newman) and subsets. Majumdar-Papapetrou space times.
No redshift along event horizon. (Quantum mechanics: zero temperature and does not radiate.)
$\ell(\bar\chi) = 0$. On the extremal horizons the torsion $\eta$ solves an elliptic system. Chrusciel-Reall-Tod : no static vacuum extremal horizons with spherical topology.
Static electrovacuum spacetime with many black holes requires all holes are extremal (Chrusciel, Tod). Example Majumdar-Papapetrou.
For wave equation, the degeneracy of the redshift effect is the new aspect. Start by considering extremal RN backgrounds.
New features: Exists a conservation law along the event horizon for the spherical mean. (Transversal null derivative of $\phi$ + $\phi / M$ is conserved.)
This also gives a hierarchy of conservation laws for each angular frequency.
Theorem (S.A) If Lorentzian manifold contains an extremal axismmetric horizon $\mathcal{H}$. If $V$ is the killing field null and normal to the horizon, and $\Phi$ the axial killing tangential to the horizon, such that the two killing fields commute. If the distribution orthogonal to the two killing fields is integrable, then we have a conservation law on the horizon.
Ther conservation law holds for the spherical mean of an expression of the wave and first order derivatives of the wave.
It works for extremal Kerr. As well as MP muti blackholes.
The conservation laws are completely determined by the local properties of the extremal horizons, and hence do not depend on global aspects of the spacetime.
Generic solutions to the wave equation on extremal RN or Kerr, we have that the transversal derivative has non-decay, the higher transversal derivatives has pointwise blow-up toward $i^+$, and we have energy blow-up of higher incoming null derivatives.
However, for solutions on RN and axi-symmetric solutions to Kerr we have pointwise decay up to and including the event horizon.
Difficulties:
Unlike the subextremal cases where Irod and Mihalis can decouple trapping from superradiance, in the extremal Kerr for general solutions, the upper limit of superradiant frequencies is trapped. Energy may not even be bounded! We have essentially undamped quasinormal modes.
For the axisymmetric case, we can construct localised microlocal currents which decouple from the redshift effect.
Reall and Lucietti has also extended conservation laws to Teukolsky equation governing linearized gravitational perturbations of extremal Kerr.
Joint work with Sarp Akcay, Leor Barack, and Alessandro Nagar
Numerical results of Pretorius and Khurana 2007, Sperhake et al 2009.
Conjecture: with sufficient fine-tuning, almost all energy can be radiated.
This work: add radiation reaction in the self-force approximation.
Consider first dynamics of particle under a (self)force in Schwarzschild. To keep $g(u,u)$ constant at -1, need to assume the 4-force is orthogonal to $u$.
What is the critical solution?
The critical solution radiates and drop energy/momentum while sitting close to the circular orbit.
