Igor Rodnianski’s research while affiliated with Princeton University and other places

We prove global existence, boundedness and decay for small data solutions $\psi$ to a general class of quasilinear wave equations on Kerr black hole backgrounds in the full sub-extremal range $|a|<M$. The method extends our previous [DHRT22], which considered such equations on a wide class of background spacetimes, including Kerr, but restricted in that case to the very slowly rotating regime $|a|\ll M$ (which may be treated simply as a perturbation of Schwarzschild a=0). To handle the general $|a|<M$ case, our present proof is based on two ingredients: (i) the linear inhomogeneous estimates on Kerr backgrounds proven in [DRSR16], further refined however in order to gain a derivative in elliptic frequency regimes, and (ii) the existence of appropriate physical space currents satisfying degenerate coercivity properties, but which now must be tailored to a finite number of wave packets defined by suitable frequency projection. The above ingredients can be thought of as adaptations of the two basic ingredients of [DHRT22], exploiting however the peculiarities of the Kerr geometry. The novel frequency decomposition in (ii), inspired by the boundedness arguments of [DR11, DRSR16], is defined using only azimuthal and stationary frequencies, and serves both to separate the superradiant and non-superradiant parts of the solution and to localise trapping to small regions of spacetime. The strengthened (i), on the other hand, allows us to relax the required coercivity properties of our wave-packet dependent currents, so as in particular to accept top order errors provided that they are localised to the elliptic frequency regime. These error terms are analysed with the help of the full Carter separation.

This is the second paper in a series of papers addressing the characteristic gluing problem for the Einstein vacuum equations. We solve the codimension-10 characteristic gluing problem for characteristic data which are close to the Minkowski data. We derive an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges that are conserved by the linearized null constraint equations and act as obstructions to the gluing problem. The gauge-dependent charges can be matched by applying angular and transversal gauge transformations of the characteristic data. By making use of a special hierarchy of radial weights of the null constraint equations, we construct the null lapse function and the conformal geometry of the characteristic hypersurface, and we show that the aforementioned charges are in fact the only obstructions to the gluing problem. Modulo the gauge-invariant charges, the resulting solution of the null constraint equations is $C^{m+2}$ for any specified integer $m\ge 0$ in the tangential directions and $C^2$ in the transversal directions to the characteristic hypersurface. We also show that higher-order (in all directions) gluing is possible along bifurcated characteristic hypersurfaces (modulo the gauge-invariant charges).

This is the third paper in a series of papers adressing the characteristic gluing problem for the Einstein vacuum equations. We provide full details of our characteristic gluing (including the 10 charges) of strongly asymptotically flat data to the data of a suitably chosen Kerr spacetime. The choice of the Kerr spacetime crucially relies on relating the 10 charges to the ADM energy, linear momentum, angular momentum and the center-of-mass. As a corollary, we obtain an alternative proof of the Corvino-Schoen spacelike gluing construction for strongly asymptotically flat spacelike initial data.

For ( t , x ) ∈ ( 0 , ∞ ) × T D (t,x) \in (0,\infty )\times \mathbb {T}^{\mathfrak {D}} , the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as t ↓ 0 t \downarrow 0 , i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents q ~ 1 , ⋯ , q ~ D ∈ R \widetilde {q}_1,\cdots ,\widetilde {q}_{\mathfrak {D}} \in \mathbb {R} , which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at { t = 1 } \lbrace t = 1 \rbrace , as long as the exponents are “sub-critical” in the following sense: max I , J , B = 1 , ⋯ , D I > J { q ~ I + q ~ J − q ~ B } > 1 \underset {\substack {I,J,B=1,\cdots , \mathfrak {D}\\ I > J}}{\max } \{\widetilde {q}_I+\widetilde {q}_J-\widetilde {q}_B\}>1 . Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with D = 3 \mathfrak {D}= 3 and q ~ 1 ≈ q ~ 2 ≈ q ~ 3 ≈ 1 / 3 \widetilde {q}_1 \approx \widetilde {q}_2 \approx \widetilde {q}_3 \approx 1/3 , which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einstein-vacuum equations for D ≥ 38 \mathfrak {D}\geq 38 with max I = 1 , ⋯ , D | q ~ I | > 1 / 6 \underset {I=1,\cdots ,\mathfrak {D}}{\max } |\widetilde {q}_I| > 1/6 . In this paper, we prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the 1 + D 1+\mathfrak {D} -dimensional Einstein-scalar field system for all D ≥ 3 \mathfrak {D}\geq 3 and the 1 + D 1+\mathfrak {D} -dimensional Einstein-vacuum equations for D ≥ 10 \mathfrak {D}\geq 10 ; both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in 1 + 3 1+3 dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized U ( 1 ) U(1) -symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized U ( 1 ) U(1) -symmetric solutions. Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to t t , and to handle this difficulty, we use t t -weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the t t -weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime.

We first introduce a new model for a two-dimensional gauge-covariant wave equation with space-time white noise. In our main theorem, we obtain the probabilistic global well-posedness of this model in the Lorenz gauge. Furthermore, we prove the failure of a probabilistic null-form estimate, which exposes a potential obstruction towards the probabilistic well-posedness of a stochastic Maxwell-Klein-Gordon equation.

We prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds, allowing in particular for the presence of horizons, ergoregions and trapped null geodesics, and including as a special case the Schwarzschild and very slowly rotating $\vert a \vert \ll M$ Kerr family of black holes in general relativity. There are two distinguishing aspects of our approach. The first aspect is its dyadically localised nature: The nontrivial part of the analysis is reduced entirely to time-translation invariant $r^p$-weighted estimates, in the spirit of [DR09], to be applied on dyadic time-slabs which for large r are outgoing. Global existence and decay then both immediately follow by elementary iteration on consecutive such time-slabs, without further global bootstrap. The second, and more fundamental, aspect is our direct use of a "blackbox" linear inhomogeneous energy estimate on exactly stationary metrics, together with a novel but elementary physical space top order identity that need not capture the structure of trapping and is robust to perturbation. In the specific example of Kerr black holes, the required linear inhomogeneous estimate can then be quoted directly from the literature [DRSR16], while the additional top order physical space identity can be shown easily in many cases (we include in the Appendix a proof for the Kerr case $\vert a \vert \ll M$, which can in fact be understood in this context simply as a perturbation of Schwarzschild). In particular, the approach circumvents the need either for producing a purely physical space identity capturing trapping or for a careful analysis of the commutation properties of frequency projections with the wave operator of time-dependent metrics.

In this paper we develop a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to the previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the 10-dimensional spaces of obstructions to the null and spacelike (asymptotically flat) gluing problems, previously known in the literature. In particular, we show that any asymptotically flat spacelike initial data can be glued to the Schwarzschild initial data of mass M for any $M>0$ sufficiently large. More generally, compared to the celebrated result of Corvino-Schoen, our methods allow us to choose ourselves the Kerr spacelike initial data that is being glued onto. As in our earlier work, our primary focus is the analysis of the null problem, where we develop a new technique of combining low-frequency linear analysis with high-frequency nonlinear control. The corresponding spacelike results are derived a posteriori by solving a characteristic initial value problem.