arXiv subjects

Benjamin Withers

Publications and source records attributed to Benjamin Withers.

At least 19 recordsLinked to original sources

arXiv2026

OPE = QNM

At microscopic scales the linear response of thermal states of large-$N$ CFTs is governed by a thermal operator product expansion (OPE), while at large scales response is governed by collective excitations known as quasinormal modes (QNM). We show that the OPE and QNM representations of the retarded correlator in mixed time and spatial momentum coordinates have an overlapping region of convergence in the complex time plane, giving a map between OPE and QNM data. We show that large-overtone QNM asymptotics are related to OPE singularities, while low-overtone QNM data appear in analytic continuation from short to large times. Using this approach we obtain new analytic results for QNM asymptotics, and numerically obtain low-overtone QNMs from OPE data for the Schwarzschild-AdS$_5$ black brane. We further show that the OPE spectrum is intimately related to QNM data through a set of sum rules which we derive in Mellin space. Finally, using the lightcone OPE, we argue that stress tensor correlators at large spatial momentum thermalise slower as the conformal collider bounds approach saturation. This work points to a new thermal bootstrap programme where OPE and QNM data constrain each other.

hep-th↗

arXiv2026

Bouncing singularities in Schwarzschild: a geometric origin of the QNM convergence region

We show analytically that the convergence of the QNM expansion of the retarded Green's function of the Schwarzschild spacetime is set by a singularity in the complex time plane. The singularity has a simple geometric origin: it is an example of a `bouncing singularity' in the language of AdS/CFT literature, caused by a null geodesic which bounces from the black hole singularity. Our work explains why the QNM convergence region at real times is bounded by null ray which scatters from the gravitational potential at a seemingly unremarkable point ($r_* = 0$ in the conventions of previous work) -- this ray is the same distance from the origin as the bouncing singularity in the relevant complex plane. The same set of singularities are responsible for an annular region of convergence for the Matsubara mode sum which describes the early time behaviour of the Schwarzschild Green's function for perturbations close to the horizon.

gr-qc↗

arXiv2026

Analytic structure of holographic thermal correlators from Fourier series

We compute the holographic Euclidean two-point function of scalar operators in a thermal state. We work directly using the Fourier series on the thermal circle. The Fourier series does not converge as a function, but instead converges as a distribution, consistent with QFT expectations. The result is manifestly periodic and consistent with analyticity in the strip $0<\mathfrak{Re}(\tau)<\beta$. Expanding in $\tau$ we obtain all OPE coefficients, including the double-trace sector. Thus our approach has an advantage compared to recent work where double-traces were bootstrapped from stress-tensor data. Bouncing singularities appear as non-perturbative sectors in the transseries for Fourier coefficients, but their transseries parameters are all zero in the case of the Euclidean correlator.

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arXiv2025

Price's law from quasinormal modes

We show that Price's power-law tail for perturbations of Schwarzschild, $t^{-2\ell-3}$ as $t\to \infty$, can be obtained from a sum of Schwarzschild-de Sitter quasinormal modes in the limit $\Lambda \to 0^+$.

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arXiv2025

Stable evolution of relativistic hydrodynamics order-by-order in gradients

We provide a systematic framework for solving the initial value problem for relativistic hydrodynamics formulated as a gradient expansion. Secular growth is handled by a suitable covariant resummation scheme, which reorganises the degrees of freedom at each order in the expansion while preserving the sum. Our scheme can be applied to any order in the gradient expansion; we provide the explicit formulation at first and second orders. When working to first order, we find that the BDNK equations of motion emerge as an intermediate step in a calculation performed in the Landau frame. We show that non-hydrodynamic modes appear only in such intermediate calculations and cancel when evaluating solutions to the required order. Our procedure does not introduce any other fields or require any additional initial data beyond those appearing in the theory of ideal fluids.

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arXiv2025

Beyond quasinormal modes: a complete mode decomposition of black hole perturbations

We show that retarded Green's functions of black hole spacetimes can be expressed as a convergent mode sum everywhere in spacetime. At late times a quasinormal mode sum converges, while at early times a Matsubara (or, Euclidean) mode sum converges. The two regions are separated by a lightcone which scatters from the black hole potential. The Matsubara sum is a Fourier series on the Euclidean thermal circle associated to the early time region. We illustrate our results for P\"oschl-Teller, BTZ, and Schwarzschild. In the case of Schwarzschild, we express the branch cut contribution as a convergent sum of de Sitter quasinormal modes as $\Lambda\to 0^+$, and exploit recent exact solutions to the Heun connection problem. In each case we analytically show convergence by studying the asymptotics of residue sums and also provide numerical demonstrations.

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arXiv2025

Transients in black hole perturbation theory

Black hole quasinormal modes arise as eigenmodes of a non-normal Hamiltonian and consequently they do not obey orthogonality relations with respect to commonly used inner products, for example, the energy inner product. A direct consequence of this is the appearance of transient phenomena. This review summarises current developments on the topic, both in frequency- and time-domain. In particular, we discuss the appearance of i) transient plateaus: arbitrarily long-lived sums of quasinormal modes, corresponding to localised energy packets near the future horizon; ii) transient growth, with the latter either appearing in the vicinity of black hole phase transitions or in the context of higher-derivative Sobolev norms.

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arXiv2025

QNM orthogonality relations for AdS black holes

We present orthogonality relations for quasinormal modes of a wide class of asymptotically AdS black holes. The definition is obtained from a standard product, modified by a CPT operator and placed on a complex radial contour which avoids branch points of the modes. They are inspired by existing constructions for de Sitter and Kerr spacetimes. The CPT operator is needed to map right eigenfunctions of the Hamiltonian into left eigenfunctions. The radial contour connects two copies of the dual QFT on a thermal Schwinger-Keldysh contour, making contact with real-time holography and the double cone wormhole.

hep-th↗

arXiv2025

Non-modal effects in black hole perturbation theory: Transient Superradiance

We study the non-modal stability of black hole spacetimes under linear perturbations. We show that large-amplitude growth can occur at finite time, despite asymptotic decay of linear perturbations. In the example presented, the physical mechanism is a transient form of superradiance, and is qualitatively similar to the transition to turbulence in Navier-Stokes shear flows. As part of the construction we provide a theorem for the positivity of QNM energies, and introduce a truncated-Hamiltonian approach to black hole pseudospectra which does not suffer from convergence issues.

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arXiv2024

Exact low-temperature Green's functions in AdS/CFT: From Heun to confluent Heun

We obtain exact expressions for correlation functions of charged scalar operators at finite density and low temperature in CFT$_4$ dual to the RN-AdS$_5$ black brane. We use recent developments in the Heun connection problem in black hole perturbation theory arising from Liouville CFT and the AGT correspondence. The connection problem is solved perturbatively in an instanton counting parameter, which is controlled in a double-scaling limit where $\omega, T \to 0$ holding $\omega/T$ fixed. This provides analytic control over the emergence of the zero temperature branch cut as a confluent limit of the Heun equation. From the Green's function we extract analytic results for the critical temperature of the holographic superconductor, as well as dispersion relations for both gapped and gapless low temperature quasinormal modes. We demonstrate precise agreement with numerics.

hep-th↗

arXiv2024

Transient dynamics of quasinormal mode sums

Quasinormal modes of spacetimes with event horizons are typically governed by a non-normal operator. This gives rise to spectral instabilities, a topic of recent interest in the black hole pseudospectrum programme. In this work we show that non-normality leads to the existence of arbitrarily long-lived sums of short-lived quasinormal modes, corresponding to localising packets of energy near the future horizon. There exist sums of $M$ quasinormal modes whose lifetimes scale as $\log{M}$. This transient behaviour results from large cancellations between non-orthogonal quasinormal modes. We provide simple closed-form examples for a massive scalar field in the static patch of dS$_{d+1}$ and the BTZ black hole. We also provide numerical examples for scalar perturbations of Schwarzschild-AdS$_{d+1}$, and gravitational perturbations of Schwarzschild in asymptotically flat spacetime, using hyperboloidal foliations. The existence of these perturbations is linked to certain properties of black hole pseudospectra. We comment on implications for thermalisation times in holographic plasmas.

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arXiv2023

The Ambient Space Formalism

We present a new formalism to solve the kinematical constraints due to Weyl invariance for CFTs in curved backgrounds and/or non-trivial states, and we apply it to thermal CFTs and to CFTs on squashed spheres. The ambient space formalism is based on constructing a class of geometric objects that are Weyl covariant and identifying them as natural building blocks of correlation functions. We construct (scalar) $n$-point functions and we illustrate the formalism with a detailed computation of 2-point functions. We compare our results for thermal 2-point functions with results that follow from thermal OPEs and holographic computations, finding exact agreement. In our holographic computation we also obtain the OPE coefficient of the leading double-twist contribution, and we discuss how the double-twist coefficients may be computed from the multi-energy-momentum contributions, given knowledge of the analytic structure of the correlator. The 2-point function for the CFT on squashed spheres is a new result. We also discuss the relation of our work to flat holography.

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arXiv2023

Black hole excited states from broken translations in Euclidean time

We prepare an excited finite temperature state in ${\cal N}=4$ SYM by means of a Euclidean path integral with a relevant deformation. The deformation explicitly breaks imaginary-time translations along the thermal circle whilst preserving its periodicity. We then study how the state relaxes to thermal equilibrium in real time. Computations are performed using real-time AdS/CFT, by constructing novel mixed-signature black holes in numerical relativity corresponding to Schwinger-Keldysh boundary conditions. These correspond to deformed cigar geometries in the Euclidean, glued to a pair of dynamical spacetimes in the Lorentzian. The maximal extension of the Lorentzian black hole exhibits a `causal shadow', a bulk region which is spacelike separated from both boundaries. We show that causal shadows are generic in path-integral prepared states where imaginary-time translations along the thermal circle are broken.

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arXiv2023

The Hydrohedron: Bootstrapping Relativistic Hydrodynamics

As an effective theory, relativistic hydrodynamics is fixed by symmetries up to a set of transport coefficients. A lot of effort has been devoted to explicit calculations of these coefficients. Here we propose a shift in perspective: we deploy bootstrap techniques to rule out theories that are inconsistent with microscopic causality. What remains is a universal convex geometry in the space of transport coefficients, which we call the hydrohedron. The landscape of all consistent theories necessarily lie inside or on the edges of the hydrohedron. We analytically construct cross-sections of the hydrohedron corresponding to bounds on transport coefficients that appear in sound and diffusion modes for theories without stochastic fluctuations.

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arXiv2022

Rigorous bounds on transport from causality

We use causality to derive a number of simple and universal constraints on dispersion relations, which describe the location of singularities of retarded two-point functions in relativistic quantum field theories. We prove that all causal dissipative dispersion relations have a finite radius of convergence. We then give two-sided bounds on all transport coefficients in units of this radius, including an upper bound on diffusivity.

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arXiv2022

Thermal three-point functions from holographic Schwinger-Keldysh contours

We compute fully retarded scalar three-point functions of holographic CFTs at finite temperature using real-time holography. They describe the nonlinear response of a holographic medium under scalar forcing, and display single and higher-order poles associated to resonant QNM excitations. This involves computing the bulk-to-bulk propagator on a piecewise mixed-signature spacetime, the dual of the Schwinger-Keldysh contour. We show this construction is equivalent to imposing ingoing boundary conditions on a single copy of a black hole spacetime, similar to the case of the two-point function. We also compute retarded scalar correlators with stress-tensor insertions in general CFTs by solving Ward identities on the Schwinger-Keldysh contour.

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arXiv2022

An embedding formalism for CFTs in general states on curved backgrounds

We present a generalisation of the embedding space formalism to conformal field theories (CFTs) on non-trivial states and curved backgrounds, based on the ambient metric of Fefferman and Graham. The ambient metric is a Lorentzian Ricci-flat metric in $d+2$ dimensions and replaces the Minkowski metric of the embedding space. It is canonically associated with a $d$-dimensional conformal manifold, which is the physical spacetime where the CFT${}_d$ lives. We propose a construction of CFT${}_d$ $n$-point functions in non-trivial states and on curved backgrounds using appropriate geometric invariants of the ambient space as building blocks. This captures the contributions of non-vanishing 1-point functions of multi-stress-energy tensors, at least in holographic CFTs. We apply the formalism to 2-point functions of thermal CFT, finding exact agreement with a holographic computation and expectations based on thermal operator product expansions (OPEs), and to CFTs on squashed spheres where no prior results are known and existing methods are difficult to apply, demonstrating the utility of the method.

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arXiv2022

Convergence of the Fefferman-Graham expansion and complex black hole anatomy

Given a set of sources and one-point function data for a Lorentzian holographic QFT, does the Fefferman-Graham expansion converge? If it does, what sets the radius of convergence, and how much of the interior of the spacetime can be reconstructed using this expansion? As a step towards answering these questions we consider real analytic CFT data, where in the absence of logarithms, the radius is set by singularities of the complex metric reached by analytically continuing the Fefferman-Graham radial coordinate. With the conformal boundary at the origin of the complex radial plane, real Lorentzian submanifolds appear as piecewise paths built from radial rays and arcs of circles centred on the origin. This allows singularities of Fefferman-Graham metric functions to be identified with gauge-invariant singularities of maximally extended black hole spacetimes, thereby clarifying the physical cause of the limited radius of convergence in such cases. We find black holes with spacelike singularities can give a radius of convergence equal to the horizon radius, however for black holes with timelike singularities the radius is smaller. We prove that a finite radius of convergence does not necessarily follow from the existence of an event horizon, a spacetime singularity, nor from caustics of the Fefferman-Graham gauge, by providing explicit examples of spacetimes with an infinite radius of convergence which contain such features.

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