arXiv subjects

Peter Hintz

Publications and source records attributed to Peter Hintz.

At least 37 records · Page 2Linked to original sources

arXiv2024

Local theory of wave equations with timelike curves of conic singularities

We develop a general theory for the existence, uniqueness, and higher regularity of solutions to wave-type equations on Lorentzian manifolds with timelike curves of cone-type singularities. These singularities may be of geometric type (cone points with time-dependent cross sectional metric), of analytic type (such as asymptotically inverse square singularities or first order asymptotically scaling-critical singular terms), or any combination thereof. We can treat tensorial equations without any symmetry assumptions; we only require a condition of mode stability type for the stationary model operators defined at each point along the curve of cone points. In symmetric ultrastatic settings, we recover the solvability theory given by the functional calculus for the Friedrichs extension.

math.AP↗

arXiv2024

Universality of the quantum energy flux at the inner horizon of asymptotically de Sitter black holes

Recently, it was found that the energy flux of a free scalar quantum field on a Reissner-Nordström-de Sitter spacetime has a quadratic divergence towards the inner horizon of the black hole. Moreover, the leading divergence was found to be state independent as long as the spectral gap of the wave equation on the spacetime is sufficiently large. In this work, we show that the latter result can be extended to all subextremal Reissner-Nordström-de Sitter and subextremal Kerr-de Sitter spacetimes with a positive spectral gap.

gr-qc↗

arXiv2023

Asymptotically de Sitter metrics from scattering data in all dimensions

In spacetime dimensions $n+1\geq 4$, we show the existence of solutions of the Einstein vacuum equations which describe asymptotically de Sitter spacetimes with prescribed smooth data at the conformal boundary. This provides a short alternative proof of a special case of a result by Shlapentokh-Rothman and Rodnianski, and generalizes earlier results by Friedrich and Anderson to all dimensions.

gr-qc↗

arXiv2023

The linearized Einstein equations with sources

On vacuum spacetimes of general dimension, we study the linearized Einstein vacuum equations with a spatially compactly supported and (necessarily) divergence-free source. We prove that the vanishing of appropriate charges of the source, defined in terms of Killing vector fields on the spacetime, is necessary and sufficient for solvability within the class of spatially compactly supported metric perturbations. The proof combines classical results by Moncrief with the solvability theory of the linearized constraint equations with control on supports developed by Corvino and Chruściel-Delay.

gr-qc↗

arXiv2023

Microlocal analysis of operators with asymptotic translation- and dilation-invariances

On a suitable class of non-compact manifolds, we study (pseudo)differential operators which feature an asymptotic translation-invariance along one axis and an asymptotic dilation-invariance, or asymptotic homogeneity with respect to scaling, in all directions not parallel to that axis. Elliptic examples include generalized 3-body Hamiltonians at zero energy such as $Δ_x+V_0(x')+V(x)$ where $Δ_x$ is the Laplace operator on $\mathbb{R}^n_x=\mathbb{R}^{n-1}_{x'}\times\mathbb{R}_{x''}$, and $V_0$ and $V$ are potentials with at least inverse quadratic decay: this operator is approximately translation-invariant in $x''$ when $|x'|\lesssim 1$, and approximately homogeneous of degree $-2$ with respect to scaling in $(x',x'')$ when $|x'|\gtrsim|x''|$. Hyperbolic examples include wave operators on nonstationary perturbations of asymptotically flat spacetimes. We introduce a systematic framework for the (microlocal) analysis of such operators by working on a compactification $M$ of the underlying manifold. The analysis is based on a calculus of pseudodifferential operators which blends elements of Melrose's b-calculus and Vasy's 3-body scattering calculus. For fully elliptic operators in our 3b-calculus, we construct precise parametrices whose Schwartz kernels are polyhomogeneous conormal distributions on an appropriate resolution of $M\times M$. We prove the Fredholm property of such operators on a scale of weighted Sobolev spaces, and show that tempered elements of their kernels and cokernels have full asymptotic expansions on $M$.

math.AP↗

arXiv2023

Exterior stability of Minkowski space in generalized harmonic gauge

We give a short proof of the existence of a small piece of null infinity for $(3+1)$-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the standard wave coordinate gauge in which all non-physical metric degrees of freedom have strong decay at null infinity. Using a formulation of the gauge-fixed Einstein vacuum equations which implements constraint damping, we establish this strong decay regardless of the validity of the constraint equations. On a technical level, we use notions from geometric singular analysis to give a streamlined proof of semiglobal existence for the relevant quasilinear hyperbolic equation.

math.AP↗

arXiv2023

Linear waves on asymptotically flat spacetimes. I

We introduce a novel framework for the analysis of linear wave equations on nonstationary asymptotically flat spacetimes, under the assumptions of mode stability and absence of zero energy resonances for a stationary model operator. Our methods apply in all spacetime dimensions and to tensorial equations, and they do not require any symmetry or almost-symmetry assumptions on the spacetime metrics or on the wave type operators. Moreover, we allow for the presence of terms which are asymptotically scaling critical at infinity, such as inverse square potentials. For simplicity of presentation, we do not allow for normally hyperbolic trapping or horizons. In the first part of the paper, we study stationary wave type equations, i.e. equations with time-translation symmetry, and prove pointwise upper bounds for their solutions. We establish a relationship between pointwise decay rates and weights related to the mapping properties of the zero energy operator. Under a nondegeneracy assumption, we prove that this relationship is sharp by extracting leading order asymptotic profiles at late times. The main tool is the analysis of the resolvent at low energies. In the second part, we consider a class of wave operators without time-translation symmetry which settle down to stationary operators at a rate $t_*^{-δ}$ as an appropriate hyperboloidal time function $t_*$ tends to infinity. The main result is a sharp solvability theory for forward problems on a scale of polynomially weighted spacetime $L^2$-Sobolev spaces. The proof combines a regularity theory for the nonstationary operator with the invertibility of the stationary model established in the first part. The regularity theory is fully microlocal and utilizes edge-b-analysis near null infinity, as developed in joint work with Vasy, and 3b-analysis in the forward cone.

math.AP↗

arXiv2023

Microlocal analysis near null infinity in asymptotically flat spacetimes

We present a novel approach to the analysis of regularity and decay for solutions of wave equations in a neighborhood of null infinity in asymptotically flat spacetimes of any dimension. The classes of metrics and wave type operators we consider near null infinity include those arising in nonlinear stability problems for Einstein's field equations in $1+3$ dimensions. In a neighborhood of null infinity, in an appropriate compactification of the spacetime to a manifold with corners, the wave operators are of edge type at null infinity and totally characteristic at spacelike and future timelike infinity. On a corresponding scale of Sobolev spaces, we demonstrate how microlocal regularity propagates across or into null infinity via a sequence of radial sets. As an application, inspired by work of the second author with Baskin and Wunsch, we prove regularity and decay estimates for forward solutions of wave type equations on asymptotically flat spacetimes which are asymptotically homogeneous with respect to scaling in the forward timelike cone and have an appropriate structure at null infinity. These estimates are new even for the wave operator on Minkowski space. The results obtained here are also used as black boxes in a global theory of wave type equations on asymptotically flat and asymptotically stationary spacetimes developed by the first author.

math.DG↗

arXiv2022

Gluing small black holes into initial data sets

We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in $n\geq 3$ spatial dimensions. We glue an $ε$-rescaling of an asymptotically flat data set $(\hatγ,\hat k)$ into the neighborhood of a point $\mathfrak{p}\in X$ inside of another initial data set $(X,γ,k)$, under a local genericity condition (non-existence of KIDs) near $\mathfrak{p}$. As the scaling parameter $ε$ tends to $0$, the rescalings $\frac{x}ε$ of normal coordinates $x$ on $X$ around $\mathfrak{p}$ become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of $\mathfrak{p}$ on the other hand, the glued initial data converge back to $(γ,k)$. The initial data we construct enjoy polyhomogeneous regularity jointly in $ε$ and the (rescaled) spatial coordinates. Applying our construction to unit mass black hole data sets $(X,γ,k)$ and appropriate boosted Kerr initial data sets $(\hatγ,\hat k)$ produces initial data which conjecturally evolve into the extreme mass ratio inspiral of a unit mass and a mass $ε$ black hole. The proof combines a variant of the gluing method introduced by Corvino and Schoen with geometric singular analysis techniques originating in Melrose's work. On a technical level, we present a fully geometric microlocal treatment of the solvability theory for the linearized constraints map.

math.AP↗

arXiv2021

Mode stability and shallow quasinormal modes of Kerr-de Sitter black holes away from extremality

A Kerr-de Sitter black hole is a solution $(M,g_{Λ,\mathfrak{m},\mathfrak{a}})$ of the Einstein vacuum equations with cosmological constant $Λ>0$. It describes a black hole with mass $\mathfrak{m}>0$ and specific angular momentum $\mathfrak{a}\in\mathbb{R}$. We show that for any $ε>0$ there exists $δ>0$ so that mode stability holds for the linear scalar wave equation $\Box_{g_{Λ,\mathfrak{m},\mathfrak{a}}}ϕ=0$ when $|\mathfrak{a}/\mathfrak{m}|\in[0,1-ε]$ and $Λ\mathfrak{m}^2<δ$. In fact, we show that all quasinormal modes $σ$ in any fixed half space $\Imσ>-C\sqrtΛ$ are equal to $0$ or $-i\sqrt{Λ/3}(n+o(1))$, $n\in\mathbb{N}$, as $Λ\mathfrak{m}^2\searrow 0$. We give an analogous description of quasinormal modes for the Klein-Gordon equation. We regard a Kerr-de Sitter black hole with small $Λ\mathfrak{m}^2$ as a singular perturbation either of a Kerr black hole with the same angular momentum-to-mass ratio, or of de Sitter spacetime without any black hole present. We use the mode stability of subextremal Kerr black holes, proved by Whiting and Shlapentokh-Rothman, as a black box; the quasinormal modes described by our main result are perturbations of those of de Sitter space. Our proof is based on careful uniform a priori estimates, in a variety of asymptotic regimes, for the spectral family and its de Sitter and Kerr model problems in the singular limit $Λ\mathfrak{m}^2\searrow 0$.

gr-qc↗

arXiv2021

Quasinormal modes and dual resonant states on de Sitter space

The existence of quasinormal modes (QNMs) for waves propagating on pure de Sitter space has been called into question in several works. We definitively prove the existence of quasinormal modes for massless and massive scalar fields in all dimensions and for all scalar field masses, and present a simple method for the explicit calculation of QNMs and the corresponding mode solutions. By passing to coordinates which are regular at the cosmological horizon, we demonstrate that certain QNMs only appear in the QNM expansion of the field when the initial data do not vanish near the cosmological horizon. The key objects in the argument are dual resonant states. These are distributional mode solutions of the adjoint field equation satisfying a generalized incoming condition at the horizon, and they characterize the amplitudes with which QNMs contribute to the QNM expansion of the field.

gr-qc↗

arXiv2021

Elliptic parametrices in the 0-calculus of Mazzeo and Melrose

The purpose of this note is to spell out the details of the construction of parametrices for fully elliptic uniformly degenerate pseudodifferential operators on manifolds $X$ with boundary. Following the original work by Mazzeo-Melrose on the 0-calculus, the parametrices are shown to have (polyhomogeneous) conormal Schwartz kernels on the 0-double space, a resolution of $X^2$. The extended 0-double space introduced by Lauter plays a useful role in the construction.

math.AP↗

arXiv2021

Quasinormal modes of small Schwarzschild-de Sitter black holes

We study the behavior of the quasinormal modes (QNMs) of massless and massive linear waves on Schwarzschild-de Sitter black holes as the black hole mass tends to 0. Via uniform estimates for a degenerating family of ODEs, we show that in bounded subsets of the complex plane and for fixed angular momenta, the QNMs converge to those of the static model of de Sitter space. Detailed numerics illustrate our results and suggest a number of open problems.

gr-qc↗

arXiv2021

The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

We consider the semilinear wave equation $\Box_g u+a u^4=0$, $a\neq 0$, on a Lorentzian manifold $(M,g)$ with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric $g$ and the coefficient $a$ up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.

math.AP↗

arXiv2021

An inverse boundary value problem for a semilinear wave equation on Lorentzian manifolds

We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness.

math.AP↗

arXiv2021

Semiclassical propagation through cone points

We introduce a general framework for the study of the diffraction of waves by cone points at high frequencies. We prove that semiclassical regularity propagates through cone points with an almost sharp loss even when the underlying operator has leading order terms at the conic singularity which fail to be symmetric. We moreover show improved regularity along strictly diffractive geodesics. Applications include high energy resolvent estimates for complex- or matrix-valued inverse square potentials and for the Dirac-Coulomb equation. We also prove a sharp propagation estimate for the semiclassical conic Laplacian. The proofs use the semiclassical cone calculus, introduced recently by the author, and combine radial point estimates with estimates for a scattering problem on an exact cone. A second microlocal refinement of the calculus captures semiclassical conormal regularity at the cone point and thus facilitates a unified treatment of semiclassical cone and b-regularity.

math.AP↗

arXiv2020

Resolvents and complex powers of semiclassical cone operators

We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter $h$ tends to $0$. An example of such an operator is the shifted semiclassical Laplacian $h^2Δ_g+1$ on a manifold $(X, g)$ of dimension $n\geq 3$ with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space $[0,1)_h\times X\times X$ of $h$-dependent integral kernels; the construction of complex powers relies on a calculus with a second semiclassical parameter. As an application, we characterize the domains of $(h^2Δ_g+1)^{w/2}$ for $\mathrm{Re}\,w\in(-\frac{n}{2},\frac{n}{2})$ and use this to prove the propagation of semiclassical regularity through a cone point on a range of weighted semiclassical function spaces.

math.AP↗

arXiv2020

Uniqueness of Kerr-Newman-de Sitter black holes with small angular momenta

We show that a stationary solution of the Einstein-Maxwell equations which is close to a non-degenerate Reissner-Nordström-de Sitter solution is in fact equal to a slowly rotating Kerr-Newman-de Sitter solution. The proof uses the non-linear stability of the Kerr-Newman-de Sitter family of black holes for small angular momenta, recently established by the author, together with an extension argument for Killing vector fields. Our black hole uniqueness result only requires the solution to have high but finite regularity; in particular, we do not make any analyticity assumptions.

gr-qc↗