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Dietrich Häfner

Publications and source records attributed to Dietrich Häfner.

14 recordsLinked to original sources
arXiv2026

The Unruh state for bosonic Teukolsky fields on subextreme Kerr spacetimes

We perform the quantization of Teukolsky scalars of spin $0$, $\pm 1$, and $\pm 2$ within the algebraic approach to quantum field theory. We first discuss the classical phase space, from which we subsequently construct the algebra. This sheds light on which fields are conjugates of each other. Further, we construct the Unruh state for this theory on Kerr and show that it is Hadamard on the black hole exterior and the interior up to the inner horizon. This shows not only that Hadamard states exist for this theory, but also extends the existence and Hadamard property of the Unruh state to (bosonic) Teukolsky fields on Kerr, where such a result was previously missing.

math-ph↗
arXiv2025

Linear stability of Kerr black holes in the full subextremal range

We prove, unconditionally, the linear stability of the Kerr family in the full subextremal range. On an analytic level, our proof is the same as that of our earlier paper in the slowly rotating case. The additional ingredients we use are, firstly, the mode stability result proved by Andersson, Whiting, and the first author and, secondly, computations related to the zero energy behavior of the linearized gauge-fixed Einstein equation in work by the second author.

gr-qc↗
arXiv2022

Mode analysis for the linearized Einstein equations on the Kerr metric : the large $\mathfrak{a}$ case

We give a complete analysis of mode solutions for the linearized Einstein equations and the $1-$form wave operator on the Kerr metric in the large $\mathfrak{a}$ case. By mode solutions we mean solutions of the form $e^{-it_*σ}\tilde{h}(r,θ,φ)$ where $t_*$ is a suitable time variable. The corresponding Fourier transformed $1-$form wave operator and linearized Einstein operator are shown to be Fredholm between suitable function spaces and $\tilde{h}$ has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy. No mode solutions exist for ${\Im}\, σ\ge 0,\, σ\neq 0$. For $σ=0$ mode solutions are Coulomb solutions for the $1-$form wave operator and linearized Kerr solutions plus pure gauge terms in the case of the linearized Einstein equations. If we fix a De Turck/wave map gauge, then the zero mode solutions for the linearized Einstein equations lie in a fixed $7-$dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by the third author and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime due to Aksteiner et al.

math.AP↗
arXiv2021

The Unruh state for massless fermions on Kerr spacetime and its Hadamard property

We give a rigorous definition of the Unruh state in the setting of massless Dirac fields on slowly rotating Kerr spacetimes. In the black hole exterior region, we show that it is asymptotically thermal at Hawking temperature on the past event horizon. Furthermore, we demonstrate that in the union of the exterior and interior regions, the Unruh state is pure and Hadamard. The main ingredients are the Häfner-Nicolas scattering theory, new microlocal estimates for characteristic Cauchy problems and criteria on the level of square-integrable solutions.

math-ph↗
arXiv2020

Existence of exponentially growing finite energy solutions for the charged Klein-Gordon equation on the De Sitter-Kerr-Newman metric

We show the existence of exponentially growing finite energy solutions for the charged Klein-Gordon equation on the De Sitter-Kerr-Newman metric for small charge and mass of the field and small angular momentum of the black hole. The mechanism behind is that the zero resonance that exists for zero charge, mass and angular momentum moves into the upper half plane.

math.AP↗
arXiv2019

Linear stability of slowly rotating Kerr black holes

We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equation: 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 equation have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications.

math.AP↗
arXiv2016

The decay of the $SU(2)$ Yang-Mills fields on the Schwarzschild black hole for spherically symmetric small energy initial data

We prove uniform decay estimates in the entire exterior of the Schwarzschild black hole for gauge invariant norms on the Yang-Mills fields valued in the Lie algebra associated to the Lie group $SU(2)$. We assume that the initial data are spherically symmetric satisfying a certain Ansatz, and have small energy, which eliminates the stationary solutions which do not decay. In particular, there don't exist any Coulomb type solutions satisfying this Ansatz. We first prove a Morawetz type estimate for the Yang-Mills fields within this setting, using the Yang-Mills equations directly. We then adapt the proof constructed in previous work by the first author to show local energy decay and uniform decay of the $L^{\infty}$ norm of the middle components in the entire exterior of the Schwarzschild black hole, including the event horizon.

math.AP↗
arXiv2013

Boundary values of resolvents of self-adjoint operators in Krein spaces

We prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space: if $H$ is a selfadjoint operator on a Krein space $\cH$, equipped with the Krein scalar product $\langle \cdot| \cdot \rangle$, $A$ is the generator of a $C_{0}-$group on $\cH$ and $I\subset \rr$ is an interval such that: \begin{itemize} \item[]1) $H$ admits a Borel functional calculus on $I$, \item[]2) the spectral projection $\one_{I}(H)$ is positive in the Krein sense, \item[]3) the following {\em positive commutator estimate} holds: \[ \Re \langle u| [H, ıA]u\rangle\geq c \langle u| u\rangle, \ u \in {\rm Ran}\one_{I}(H), \ c>0. \] \end{itemize} then assuming some smoothness of $H$ with respect to the group $\e^{ıt A}$, the following resolvent estimates hold: \[ \sup_{z\in I\pm ı]0, ν]}\| \langle A\rangle ^{-s}(H-z)^{-1}\langle A\rangle^{-s}\| <\infty, \ s>\12. \] As an application we consider abstract Klein-Gordon equations \[ \p_{t}^{2}ϕ(t)- 2 ık ϕ(t)+ hϕ(t)=0, \] and obtain resolvent estimates for their generators in {\em charge spaces} of Cauchy data.

math-ph↗
arXiv2013

Resolvent and propagation estimates for Klein-Gordon equations with non-positive energy

We study in this paper an abstract class of Klein-Gordon equations: \[ \p_{t}^{2}ϕ(t)- 2ık \p_{t}ϕ(t)+ h ϕ(t)=0, \] where $ϕ: \rr\to \cH$, $\cH$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $\cH$. We consider their generators $H$ (resp. $K$) in the two natural spaces of Cauchy data, the energy (resp. charge) spaces. We do not assume that the dynamics generated by $H$ or $K$ has any positive conserved quantity, in particular these operators may have complex spectrum. Assuming conditions on $h$ and $k$ which allow to use the theory of selfadjoint operators on Krein spaces, we prove weighted estimates on the boundary values of the resolvents of $H$, $K$ on the real axis. From these resolvent estimates we obtain corresponding propagation estimates on the behavior of the dynamics for large times. Examples include wave or Klein-Gordon equations on asymptotically euclidean or asymptotically hyperbolic manifolds, minimally coupled with an external electro-magnetic field decaying at infinity.

math-ph↗
arXiv2009

Creation of fermions by rotating charged black-holes

This work is devoted to the mathematical study of the Hawking effect for fermions in the setting of the collapse of a rotating charged star. We show that an observer who is located far away from the star and at rest with respect to the Boyer Lindquist coordinates observes the emergence of a thermal state when his proper time goes to infinity. We first introduce a model of the collapse of the star. We suppose that the space-time outside the star is given by the Kerr-Newman metric. The assumptions on the asymptotic behavior of the surface of the star are inspired by the asymptotic behavior of certain timelike geodesics in the Kerr-Newman metric. The Dirac equation is then written using coordinates and a Newman-Penrose tetrad which are adapted to the collapse. This coordinate system and tetrad are based on the so called simple null geodesics. The quantization of Dirac fields in a globally hyperbolic space-time is described. We formulate and prove a theorem about the Hawking effect in this setting. The proof of the theorem contains a minimal velocity estimate for Dirac fields that is slightly stronger than the usual ones and an existence and uniqueness result for solutions of a characteristic Cauchy problem for Dirac fields in the Kerr-Newman space-time. In an appendix we construct explicitly a Penrose compactification of block I of the Kerr-Newman space-time based on simple null geodesics.

math.AP↗