arXiv subjects

Igor Rodnianski

Publications and source records attributed to Igor Rodnianski.

At least 37 records · Page 2Linked to original sources

arXiv2018

On the nature of Hawking's incompleteness for the Einstein-vacuum equations: The regime of moderately spatially anisotropic initial data

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity. Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking's celebrated "singularity" theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness. Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior. Of independent interest is our method of proof, which is more robust than earlier approaches in that i) it does not rely on approximate monotonicity identities and ii) it accommodates the possibility that the solution develops very singular high-order spatial derivatives, whose blowup rates are allowed to be, within the scope of our bootstrap argument, much worse than those of the base-level quantities driving the fundamental blowup. For these reasons, our approach could be used to obtain similar blowup results for various Einstein-matter systems in any number of spatial dimensions for solutions corresponding to an open set of moderately spatially anisotropic initial data, thus going beyond the nearly spatially isotropic regime treated in earlier works.

math.AP↗

arXiv2018

A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation

We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on $(0,\infty) \times \mathbb{T}^3$. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a "Big Bang" singularity at $\lbrace t = 0 \rbrace$. We place initial data for the linearized system along $\lbrace t = 1 \rbrace \simeq \mathbb{T}^3$ and study the linear solution's behavior in the collapsing direction $t \downarrow 0$. Our first main result is the proof of an approximate $L^2$ monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) solution. In particular, we show that as $t \downarrow 0$, various time-rescaled components of the linear solution converge to regular functions defined along $\lbrace t = 0 \rbrace$. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well-posed only in the direction $t \downarrow 0$. Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction $t \downarrow 0$ under small perturbations of its data at $\lbrace t = 1 \rbrace$.

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arXiv2018

The Asymptotically Self-Similar Regime for the Einstein Vacuum Equations

We develop a local theory for the construction of singular spacetimes in all spacetime dimensions which become asymptotically self-similar as the singularity is approached. The techniques developed also allow us to construct and classify exact self-similar solutions which correspond to the formal asymptotic expansions of Fefferman and Graham's ambient metric.

gr-qc↗

arXiv2017

Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case $|a|\ll M$

We prove boundedness and polynomial decay statements for solutions of the spin $\pm2$ Teukolsky equation on a Kerr exterior background with parameters satisfying $|a|\ll M$. The bounds are obtained by introducing generalisations of the higher order quantities $P$ and $\underline{P}$ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters $|a|<M$. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.

gr-qc↗

arXiv2017

Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations

In this paper, we study the problem of the nonlinear interaction of impulsive gravitational waves for the Einstein vacuum equations. The problem is studied in the context of a characteristic initial value problem with data given on two null hypersurfaces and containing curvature delta singularities. We establish an existence and uniqueness result for the spacetime arising from such data and show that the resulting spacetime represents the interaction of two impulsive gravitational waves germinating from the initial singularities. In the spacetime, the curvature delta singularities propagate along 3-dimensional null hypersurfaces intersecting to the future of the data. To the past of the intersection, the spacetime can be thought of as containing two independent, non-interacting impulsive gravitational waves and the intersection represents the first instance of their nonlinear interaction. Our analysis extends to the region past their first interaction and shows that the spacetime still remains smooth away from the continuing propagating individual waves. The construction of these spacetimes are motivated in part by the celebrated explicit solutions of Khan-Penrose and Szekeres. The approach of this paper can be applied to an even larger class of characteristic data and in particular implies an extension of the theorem on formation of trapped surfaces by Christodoulou and Klainerman-Rodnianski, allowing non-trivial data on the initial incoming hypersurface.

gr-qc↗

arXiv2016

The linear stability of the Schwarzschild solution to gravitational perturbations

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge--Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of nonlinear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.

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arXiv2014

Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M

This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal |a| < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.), World Scientific, Singapore, 2011, pp. 132-189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincare], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notations so that it can be read independently of previous work.

gr-qc↗

arXiv2014

A scattering theory for the wave equation on Kerr black hole exteriors

We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards.

gr-qc↗

arXiv2014

The Bounded L2 Curvature Conjecture

This is the main paper in a sequence in which we give a complete proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, $L^2$ bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1+1 (based on Strichartz estimates) were obtained in [Ba-Ch1] [Ba-Ch2] [Ta1] [Ta2] [Kl-R1] and optimized in [Kl-R2] [Sm-Ta], the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. To achieve our goals we recast the Einstein vacuum equations as a quasilinear $so(3,1)$-valued Yang-Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of \textit{null structure} compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including $L^2$ error bounds which is carried out in [Sz1]-[Sz4], as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in \cite{Sz5}.

math.AP↗

arXiv2014

Stable Big Bang Formation in Near-FLRW Solutions to the Einstein-Scalar Field and Einstein-Stiff Fluid Systems

We prove a stable singularity formation result for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our results apply to small perturbations of the spatially flat FLRW solution with topology $(0,\infty) \times \mathbb{T}^3.$ The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards the future and that has a "Big Bang" singularity at $\lbrace 0 \rbrace \times \mathbb{T}^3,$ where its curvature blows up. We place data on a Cauchy hypersurface $Σ_1'$ that are close to the FLRW data induced on $\lbrace 1 \rbrace \times \mathbb{T}^3.$ We study the perturbed solution in the collapsing direction and prove that its basic features closely resemble those of the FLRW solution. In particular, we construct constant mean curvature-transported spatial coordinates for the perturbed solution covering $(t,x) \in (0,1] \times \mathbb{T}^3$ and show that it also has a Big Bang at $\lbrace 0 \rbrace \times \mathbb{T}^3,$ where its curvature blows up. The blow-up confirms Penrose's Strong Cosmic Censorship hypothesis for the "past-half" of near-FLRW solutions. The most difficult aspect of the proof is showing that the solution exists for $(t,x) \in (0,1] \times \mathbb{T}^3,$ and to this end, we derive energy estimates that are allowed to mildly blow-up as $t \downarrow 0.$ To close these estimates, we use the most important ingredient in our analysis: an $L^2-$type energy approximate monotonicity inequality that holds for near-FLRW solutions. In the companion article "A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation," we used the approximate monotonicity to prove a stability result for solutions to linearized versions of the equations. The present article shows that the linear stability result can be upgraded to control the nonlinear terms.

math.AP↗

arXiv2014

Local Propagation of Impulsive Gravitational Waves

In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of non-regular characteristic data.

gr-qc↗

arXiv2014

Type II blow up for the energy supercritical NLS

We consider the energy super critical nonlinear Schrödinger equation $$i\pa_tu+Δu+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical regime, we construct a family of smooth finite time blow up solutions which become singular via concentration of a universal profile with the so called type II quantized blow up rates. The essential feature of these solutions is that all norms below scaling remain bounded. Our analysis fully revisits the construction of type II blow up solutions for the corresponding heat equation, which was done using maximum principle techniques following. Instead we develop a robust energy method, in continuation of the works in the energy and mass critical cases. This shades a new light on the essential role played by the solitary wave and its tail in the type II blow up mechanism, and the universality of the corresponding singularity formation in both energy critical and super critical regimes.

math.AP↗

arXiv2013

A fully anisotropic mechanism for formation of trapped surfaces in vacuum

We present a new, fully anisotropic, criterion for formation of trapped surfaces in vacuum. More precisely we provide conditions on null data, concentrated in a neighborhood of a short null geodesic segment (possibly flat everywhere else) whose future development contains a trapped surface. This extends considerably the previous result of Christodoulou \cite{Chr:book} which required instead a uniform condition along all null geodesic generators. To obtain our result we combine Christodoulou's mechanism for the formation of a trapped surface with a new deformation process which takes place along incoming null hypersurfaces.

gr-qc↗

arXiv2013

A scattering theory construction of dynamical vacuum black holes

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full "functional" degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift.

gr-qc↗

arXiv2013

Overview of the proof of the Bounded $L^2$ Curvature Conjecture

This memoir contains an overview of the proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to another, more subtle, scaling tied to its causal geometry. Indeed, $L^2$ bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1+1 (based on Strichartz estimates) were obtained in \cite{Ba-Ch1}, \cite{Ba-Ch2}, \cite{Ta1}, \cite{Ta2}, \cite{Kl-R1} and optimized in \cite{Kl-R2}, \cite{Sm-Ta}, the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. The entire proof is obtained in a sequence of 6 papers.

math.AP↗

arXiv2011

Effective limiting absorption principles, and applications

We investigate quantitative (or effective) versions of the limiting absorption principle, for the Schrödinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form $$ \| R(λ+i\eps) f \|_{H^{0,-1/2-σ}} \leq C(λ, H) \| f \|_{H^{0,1/2+σ}}.$$ We are particularly interested in the exact nature of the dependence of the constants $C(λ,H)$ on both $λ$ and $H$. It turns out that the answer to this question is quite subtle, with distinctions being made between low energies $λ\ll 1$, medium energies $λ\sim 1$, and large energies $λ\gg 1$, and there is also a non-trivial distinction between "qualitative" estimates on a single operator $H$ (possibly obeying some spectral condition such as non-resonance, or a geometric condition such as non-trapping), and "quantitative" estimates (which hold uniformly for all operators $H$ in a certain class). Using elementary methods (integration by parts and ODE techniques), we give some sharp answers to these questions. As applications of these estimates, we present a global-in-time local smoothing estimate and pointwise decay estimates for the associated time-dependent Schrödinger equation, as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation, under some additional assumptions on the operator $H$.

math.AP↗

arXiv2011

Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map

We consider the energy critical Schrödinger map problem with the 2-sphere target for equivariant initial data of homotopy index $k=1$. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

math.AP↗

arXiv2011

Blow up dynamics for smooth data equivariant solutions to the energy critical Schrodinger map problem

We consider the energy critical Schrodinger map to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map in the scale invariant norm which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. The concentration rate is given by $$λ(t)=κ(u)\frac{T-t}{|\log (T-t)|^2}(1+o(1))$$ for some $κ(u)>0$. The detailed proofs of the results will appear in a companion paper.

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