Peter Hintz

Last Name

Hintz

First Name

Peter

ORCID

0000-0002-7346-9016

Organisational unit

09749 - Hintz, Peter (ehemalig) / Hintz, Peter (former)

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Publications 1 - 10 of 19

Resolvents and complex powers of semiclassical cone operators
Hintz, Peter (2022)
Mathematische Nachrichten
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 h2 Δg + 1 on a manifold (X,g) of dimension n≥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 ×X×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 (h2 Δg + 1)w/2 for Re w ∈ (−n/2,n/2) and use this to prove the propagation of semiclassical regularity through a cone point on a range of weighted semiclassical function spaces.
The linearized Einstein equations with sources
Hintz, Peter (2024)
Letters in Mathematical Physics
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–Schoen and Chruściel–Delay.
An Inverse Boundary Value Problem for a Semilinear Wave Equation on Lorentzian Manifolds
Hintz, Peter; Uhlmann, Gunther; Zhai, Jian (2022)
International Mathematics Research Notices
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.
The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds
Hintz, Peter; Uhlmann, Gunther; Zhai, Jian (2022)
Communications in Partial Differential Equations
We consider the semilinear wave equation □gu + au4 = 0, a ≠ 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.
Asymptotically de Sitter metrics from scattering data in all dimensions
Hintz, Peter (2024)
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
In space-time dimensions n+1≥4, we show the existence of solutions of the Einstein vacuum equations which describe asymptotically de Sitter space-times 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 of Friedrich and Anderson to all dimensions. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.
Linear stability of slowly rotating Kerr black holes
Häfner, Dietrich; Hintz, Peter; Vasy, András (2021)
Inventiones mathematicae
We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory, for the analysis of resolvents of operators on asymptotically flat spaces. With the mode stability of the Schwarzschild metric as well as of certain scalar and 1-form wave operators on the Schwarzschild spacetime as an input, we establish the linear stability of slowly rotating Kerr black holes using perturbative arguments; in particular, our proof does not make any use of special algebraic properties of the Kerr metric. The heart of the paper is a detailed description of the resolvent of the linearization of a suitable hyperbolic gauge-fixed Einstein operator at low energies. As in previous work by the second and third authors on the nonlinear stability of cosmological black holes, constraint damping plays an important role. Here, it eliminates certain pathological generalized zero energy states; it also ensures that solutions of our hyperbolic formulation of the linearized Einstein equations have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications.
Gluing Small Black Holes into Initial Data Sets
Hintz, Peter (2024)
Communications in Mathematical Physics
We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in n≥3 spatial dimensions. We glue an ϵ-rescaling of an asymptotically flat data set (γ^,k^) into the neighborhood of a point p∈X inside of another initial data set (X,γ,k), under a local genericity condition (non-existence of KIDs) near p. As the scaling parameter ϵ tends to 0, the rescalings xϵ of normal coordinates x on X around p become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of 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 (γ^,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.
Quasinormal modes of small Schwarzschild–de Sitter black holes
Hintz, Peter; Xie, YuQing (2022)
Journal of Mathematical Physics
We study the behavior of 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 ordinary differential equations, 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.
Correction to: Linear stability of slowly rotating Kerr black holes
Häfner, Dietrich; Hintz, Peter; Vasy, András (2024)
Inventiones mathematicae
Correction to: Invent. Math. (2021) 223:1227–1406 https://doi.org/10.1007/s00222-020-01002-4
Black hole gluing in the Sitter space
Hintz, Peter (2021)
Communications in Partial Differential Equations
We construct dynamical many-black-hole spacetimes with well-controlled asymptotic behavior as solutions of the Einstein vacuum equation with positive cosmological constant. We accomplish this by gluing Schwarzschild–de Sitter or Kerr–de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions on the black hole parameters. We give a self-contained treatment of solving the Einstein equation directly for the metric, given the scattering data we encounter at the future conformal boundary. The main step in the construction is the solution of a linear divergence equation for trace-free symmetric 2-tensors; this is closely related to Friedrich’s analysis of scattering problems for the Einstein equation on asymptotically simple spacetimes.

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Peter Hintz

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