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Stefan Hollands

Publications and source records attributed to Stefan Hollands.

At least 19 recordsLinked to original sources
arXiv2026

A Quantum Dominant Energy Condition

We propose a quantum dominant energy condition (QDEC) for the stress tensor in the context of quantum field theory in curved spacetimes. A rigorous proof is given for the case of Rindler wedges, and a heuristic discussion about possible generalizations to more general geometric setups, including curved spacetime, is provided. In Minkowski spacetime, we establish a connection with state recovery bounds. We illustrate the QDEC for coherent states of the free scalar field, where it turns out to be related to the ordinary DEC for the stress tensor of the classical solution that defines the coherent state.

gr-qc↗
arXiv2026

Hidden Symmetry of Kerr-deSitter from Manifest Symmetry of Painlevé VI

We show that the recently discovered ``mass symmetries" of the radial Teukolsky equation for Kerr-deSitter black holes can be understood from a corresponding symmetry of the Painlevé VI equation via the classical theory of ``isomonodromic deformations". It is known that a subset of the mass symmetries transforms the Teukolsky equation to a wave equation on a ``dual" Lorentzian spacetime. We find that this dual spacetime again represents a black hole, and use this insight to re-obtain a previously known partial mode stability result in a geometric manner. Further special geometric features of the dual spacetime are pointed out.

gr-qc↗
arXiv2026

New Horizons in Effective Field Theory?

We consider the most general parity symmetric effective scalar tensor theory in four dimensions containing terms up to fourth derivative order in the Lagrangian. It has been shown [H.S. Reall, Phys. Rev. D 103 (2021), 084027] that this theory has three polarizations generically goverened by different (nested) propagation cones, neither of which in general coincides with the lightcone as defined by the metric. Consequently, the notion of black hole horizon must be defined relative to the widest propagation cone, and not with respect to the metric. We provide two theorems stating that, nevertheless, the horizon of a \emph{stationary} black hole is null with respect to the metric, and that, in fact, all three propagation cones touch on the horizon. The conditions in these theorems allow for rotating black holes. Our theorems thereby suggest that the notion of Killing horizon, central in most discussions of black hole thermodynamics, retains its fundamental status, and that certain thermodynamic paradoxes associated with multiple propagation cones are evaded.

gr-qc↗
arXiv2026

Bekenstein's bound for wave packets

Let $B$ be a spatial region of width $2R$ and $Φ$ a Klein-Gordon wave packet localized in $B$ at time zero. We show the inequality $S \leq 2πR E$; here, $S$ is the entropy of $Φ$ contained in a region $B$, and $E$ is the energy content of $Φ$ within $B$. We consider a wider setting and formulate a variational problem aimed at minimizing our bound when $Φ$ is not localized in $B$. Our inequality holds in more generality in the framework of local, Poincaré covariant nets of standard subspaces and is related to the Bekenstein inequality. We point out a general bound that is compatible with the recent numerical computations by Bostelmann, Cadamuro, and Minz concerning the one-particle modular Hamiltonian of a scalar massive quantum Klein-Gordon field. We also provide a version of the entropy balance and ant formulas for wave packets.

math-ph↗
arXiv2025

A New Proof of the QNEC

We give a simplified proof of the quantum null energy condition (QNEC). Our proof is based on an explicit formula for the shape derivative of the relative entropy, with respect to an entangling cut. It allows bypassing the analytic continuation arguments of a previous proof by Ceyhan and Faulkner and can be used e.g., for defining entropy current fluctuations.

hep-th↗
arXiv2025

Spin-2 Green's Functions on Kerr in Radiation Gauge

We construct retarded and advanced Green's functions for gravitational perturbations in Kerr in an ingoing radiation gauge. Our Green's functions have a frequency domain piece that has previously been obtained by Ori [Phys. Rev. D 67 (2003)] based on the Chrzanowski-Cohen-Kegeles metric reconstruction method. As is well known, this piece by itself is not sufficient to obtain an actual Green's function. We show how to complete it with a piece based on a method by Green et al. [Class. Quant. Grav. 37 (2020)]. The completion piece has a completely explicit form in the time-domain and is supported on pairs of points on the same outgoing principal null geodesic which are in the appropriate causal order. We expect our Green's functions to be useful for gravitational self-force calculations and other perturbation problems on Kerr spacetime.

gr-qc↗
arXiv2025

Modular Time Evolution and the QNEC

We establish an inequality restricting the evolution of states in quantum field theory with respect to the modular flow of a wedge, $Δ^{is}$, for large $|s|$. Our bound is related to the quantum null energy condition, QNEC. In one interpretation, it can be seen as providing a ``chaos-bound'' $\le 2π$ on the Lyapunov exponent with respect to Rindler time, $s$. Mathematically, our inequality is a statement about half-sided modular inclusions of von Neumann algebras.

hep-th↗
arXiv2025

Bekenstein Bound for Approximately Local Charged States

We generalize the energy-entropy ratio inequality in quantum field theory (QFT) established by one of us from localized states to a larger class of states. The states considered in this paper can be in a charged (non-vacuum) representation of the QFT or may be only approximately localized in the region under consideration. Our inequality is $S(Ψ|\!| Ω) \le 2πR \, ( Ψ, H_ρΨ) + \log d(ρ) + \varepsilon$, where $S$ is the relative entropy, where $R$ is a "radius" (width) characterizing the size of the region, $d(ρ)$ is the statistical (quantum) dimension of the given charged sector $ρ$ hosting the quantum state $Ψ$, $Ω$ is the vacuum state, $H_ρ$ is the Hamiltonian in the charged sector, and $\varepsilon$ is a tolerance measuring the deviation of $Ψ$ from the vacuum according to observers in the causal complement of the region.

hep-th↗
arXiv2024

Extremal Black Hole Weather

We consider weakly non-linear gravitational perturbations of a near-extremal Kerr black hole governed by the second order vacuum Einstein equation. Using the GHZ formalism [Green et al., Class. Quant. Grav. 7(7):075001, 2020], these are parameterized by a Hertz potential. We make an ansatz for the Hertz potential as a series of zero-damped quasinormal modes with time-dependent amplitudes, and derive a non-linear dynamical system for them. We find that our dynamical system has a time-independent solution within the near horizon scaling limit. This equilibrium solution is supported on axisymmetric modes, with amplitudes scaling as $c_\ell \sim C^{\rm low} 2^{-\ell/2} \ell^{-\frac{7}{2}}$ for large polar angular momentum mode number $\ell$, where $C^{\rm low}$ is a cumulative amplitude of the low $\ell$ modes. We interpret our result as evidence that the dynamical evolution will approach, for a parametrically long time as extremality is approached, a distribution of mode amplitudes dyadically exponentially suppressed in $\ell$, hence as the endpoint of an inverse cascade. It is reminiscent of weather-like phenomena in certain models of atmospheric dynamics of rotating bodies. During the timescale considered, the decay of the QNMs themselves plays no role given their parametrically long half-life. Hence, our result is due entirely to weakly non-linear effects.

gr-qc↗
arXiv2024

Metric Reconstruction in Kerr Spacetime

Metric reconstruction is the general problem of parameterizing GR in terms of its two ``true degrees of freedom'', e.g., by a complex scalar ``potential'' -- in practice mostly with the aim of simplifying the Einstein equation (EE) within perturbative approaches. In this paper, we re-analyze the metric reconstruction procedure by Green, Hollands, and Zimmerman (GHZ) [Class. Quant. Grav. \textbf{37}, 075001 (2020)], which is a generalization of the Chrzanowski-Cohen-Kegeles (CCK) approach. Contrary to the CCK method, that by GHZ is applicable not only to the vacuum, but also to the sourced linearized Einstein equation (EE). Our main innovation is a version of the GHZ method giving an efficient integration scheme for the initial value problem of the sourced linear EE. By iteration, our scheme gives the metric to as high an order in perturbation theory around Kerr as one might wish, in principle. At each order, the metric perturbation is a sum of a corrector, obtained by solving a triangular system of transport equations, a reconstructed piece, obtained from a Hertz potential as in the CCK approach, and an algebraically special perturbation, determined by the ADM quantities. As a by-product, we determine the precise relations between the asymptotic tail of the Hertz potential in the GHZ and CCK schemes, and the quantities relevant for gravitational radiation, namely, the energy flux, news- and memory tensors, and their associated BMS-supertranslations. We also discuss ways of transforming the metric perturbation to Lorenz gauge.

gr-qc↗
arXiv2024

The Entropy of Dynamical Black Holes

We propose a new formula for the entropy of a dynamical black hole$-$valid to leading order for perturbations off of a stationary black hole background$-$in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in $n$ dimensions. In stationary eras, this formula agrees with the usual Noether charge formula, but in nonstationary eras, we obtain a nontrivial correction term. In general relativity, our formula for the entropy of a dynamical black hole is $1/4$ of the horizon area plus a term involving the integral of the expansion of the null generators of the horizon, which we show is $1/4$ of the area of the apparent horizon to leading order. Our formula for entropy in a general theory of gravity obeys a "local physical process version" of the first law of black hole thermodynamics. For first order perturbations sourced by external matter that satisfies the null energy condition, our entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the second law is obeyed at leading order if and only if the "modified canonical energy flux" is positive (as is the case in general relativity but presumably would not hold in general theories). We obtain a general relationship between our formula for the entropy of a dynamical black hole and a formula proposed independently by Dong and by Wall. We then consider the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole. We show that the validity of the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using our notion of black hole entropy but using a modified notion of von Neumann entropy for matter. On the other hand, the generalized second law for the Dong-Wall entropy is equivalent to an integrated version of QNEC, using the unmodified von Neumann entropy for the entropy of matter.

hep-th↗
arXiv2024

Infinite quantum twisting at the Cauchy horizon of rotating black holes

We present a numerical calculation of the expectation value of the quantum angular-momentum current flux density for a scalar field in the Unruh state near the inner horizon of a Kerr-de Sitter black hole. Our results indicate that this flux diverges as $V_-^{-1}$ in a suitable Kruskal coordinate such that $V_-=0$ at the inner horizon. Depending on the parameter values of the scalar field and black hole that we consider, and depending on the polar angle (latitude), this flux can have different signs. In the near extremal cases considered, the angle average of the expectation value of the quantum angular momentum current flux is of the opposite sign as the angular momentum of the background itself, suggesting that, in the cases considered, quantum effects tend to decrease the total angular momentum of the spheres away from the extremal value. We also numerically calculate the energy flux component, which provides the leading order divergence of the quantum stress energy tensor, dominant over the classical stress energy tensor, at the inner horizon. Taking our expectation value of the quantum stress tensor as the source in the semiclassical Einstein equation, our analysis suggests that the spheres approaching the inner horizon can undergo an infinite twisting due to quantum effects along latitudes separating regions of infinite expansion and contraction.

gr-qc↗
arXiv2024

Conserved currents for Kerr and orthogonality of quasinormal modes

We introduce a bilinear form for Weyl scalar perturbations of Kerr. The form is symmetric and conserved, and we show that, when combined with a suitable renormalization prescription involving complex r integration contours, quasinormal modes are orthogonal in the bilinear form for different (l, m, n). These properties are apparently not evident consequences of standard properties for the radial and angular solutions to the decoupled Teukolsky relations and rely on the Petrov type D character of Kerr and its t-$ϕ$ reflection isometry. We show that quasinormal mode excitation coefficients are given precisely by the projection with respect to our bilinear form. These properties can make our bilinear form useful to set up a framework for nonlinear quasinormal mode coupling in Kerr. We also provide a general discussion on conserved local currents and their associated local symmetry operators for metric and Weyl perturbations, identifying a collection containing an increasing number of derivatives.

gr-qc↗
arXiv2024

Relativistic perturbation theory for black-hole boson clouds

We develop a relativistic perturbation theory for scalar clouds around rotating black holes. We first introduce a relativistic product and corresponding orthogonality relation between modes, extending a recent result for gravitational perturbations. We then derive the analog of time-dependent perturbation theory in quantum mechanics, and apply it to calculate self-gravitational frequency shifts. This approach supersedes the non-relativistic "gravitational atom" approximation, brings close agreement with numerical relativity, and has practical applications for gravitational-wave astronomy.

gr-qc↗
arXiv2023

The Operator Product Expansion in Quantum Field Theory

Operator product expansions (OPEs) in quantum field theory (QFT) provide an asymptotic relation between products of local fields defined at points $x_1, \dots, x_n$ and local fields at point $y$ in the limit $x_1, \dots, x_n \to y$. They thereby capture in a precise way the singular behavior of products of quantum fields at a point as well as their ``finite trends.'' In this article, we shall review the fundamental properties of OPEs and their role in the formulation of interacting QFT in curved spacetime, the ``flow relations'' in coupling parameters satisfied by the OPE coefficients, the role of OPEs in conformal field theories, and the manner in which general theorems -- specifically, the PCT theorem -- can be formulated using OPEs in a curved spacetime setting.

hep-th↗
arXiv2023

Complexity in algebraic QFT

We consider a notion of complexity of quantum channels in relativistic continuum quantum field theory (QFT) defined by the distance to the trivial (identity) channel. Our distance measure is based on a specific divergence between quantum channels derived from the Belavkin-Staszewski (BS) divergence. We prove in the prerequisite generality necessary for the algebras in QFT that the corresponding complexity has several reasonable properties: (i) the complexity of a composite channel is not larger than the sum of its parts, (ii) it is additive for channels localized in spacelike separated regions, (iii) it is convex, (iv) for an $N$-ary measurement channel it is $\log N$, (v) for a conditional expectation associated with an inclusion of QFTs with finite Jones index it is given by $\log (\text{Jones Index})$. The main technical tool in our work is a new variational principle for the BS divergence.

quant-ph↗
arXiv2022

A stationary black hole must be axisymmetric in effective field theory

The black hole rigidity theorem asserts that a rotating stationary black hole must be axisymmetric. This theorem holds for General Relativity with suitable matter fields, in four or more dimensions. We show that the theorem can be extended to any diffeomorphism invariant theory of vacuum gravity, assuming that this is interpreted in the sense of effective field theory, with coupling constants determined in terms of a ``UV scale'', and that the black hole solution can locally be expanded as a power series in this scale.

gr-qc↗