Publications (229)
Dirichlet walls and the end of time
Hanako Helton, Gary T. Horowitz, Donald Marolf
The paper investigates how imposing Dirichlet boundary conditions on a finite surface in Einstein-Hilbert gravity can cause evolutions to end in finite time due to singularities re…
Eternal Black Holes and Superselection in AdS/CFT
Donald Marolf, Aron C. Wall
It has been argued i) that Lorentz-signature solutions with wormholes connecting n asymptotically AdS regions describe bulk quantum states dual to n entangled but non-interacting C…
Loop Representations for 2+1 Gravity on a Torus
Donald Marolf
We study the loop representation of the quantum theory for 2+1 dimensional general relativity on a manifold, , where is the torus, and…
Spacetime Embedding Diagrams for Black Holes
Donald Marolf
We show that the 1+1 dimensional reduction (i.e., the radial plane) of the Kruskal black hole can be embedded in 2+1 Minkowski spacetime and discuss how features of this spacetime…
Holography and Eternal Inflation
David A. Lowe, Donald Marolf
We show that eternal inflation is compatible with holography. In particular, we emphasize that if a region is asymptotically de Sitter in the future, holographic arguments by thems…
de Sitter invariance of the dS graviton vacuum
Atsushi Higuchi, Donald Marolf, Ian A. Morrison
The two-point function of linearized gravitons on de Sitter space is infrared divergent in the standard transverse traceless synchronous gauge defined by cosmological coordin…
Counting Supertubes
Belkis Cabrera Palmer, Donald Marolf
The quantum states of the supertube are counted by directly quantizing the linearized Born-Infeld action near the round tube. The result is an entropy $S = 2Ï\sqrt{2 (Q_{D0}Q_{F1}…
Group Averaging of massless scalar fields in 1+1 de Sitter
Donald Marolf, Ian A. Morrison
Perturbative gravity in global de Sitter space is subject to so-called linearization stability constraints: If they are to couple consistently to the gravitational field, quantum s…
Non-Thermal Behavior in Conformal Boundary States
Kevin Kuns, Donald Marolf
Cardy has recently observed that certain carefully tuned states of 1+1 CFTs on a timelike strip are periodic with period set by the light-crossing time. The states in question are…
Observations of Hawking radiation: the Page curve and baby universes
Donald Marolf, Henry Maxfield
We reformulate recent insights into black hole information in a manner emphasizing operationally-defined notions of entropy, Lorentz-signature descriptions, and asymptotically flat…
On the stability of gravity with Dirichlet walls
Tomas Andrade, William R. Kelly, Donald Marolf +1
Dirichlet walls -- timelike boundaries at finite distance from the bulk on which the induced metric is held fixed -- have been used to model AdS spacetimes with a finite cutoff. In…
Pull Back Invariant Matter Couplings
Donald Marolf
An algorithm is described for the construction of actions for scalar, spinor, and vector gauge fields that remains well-defined when the metric is degenerate and that involve no co…
Stringy negative-tension branes and the second law of thermodynamics
Donald Marolf, Simon F. Ross
Negative energy objects generally lead to instabilities and a number of other disturbing behaviors. In particular, negative energy fluxes lead to a breakdown of the classical area…
Marginally Trapped Surfaces and AdS/CFT
Brianna Grado-White, Donald Marolf
It has been proposed that the areas of marginally trapped or anti-trapped surfaces (also known as leaves of holographic screens) may encode some notion of entropy. To connect this…
On the Stress Tensor for Asymptotically Flat Gravity
Robert B. Mann, Donald Marolf, Robert McNees +1
The recent introduction of a boundary stress tensor for asymptotically flat spacetimes enabled a new construction of energy, momentum, and Lorentz charges. These charges are known…
The action of HRT-areas as operators in semiclassical gravity
Molly Kaplan, Donald Marolf
We study the action of Hubeny-Rangamani-Takayanagi (HRT) area operators on the covariant phase space of classical solutions. It has been previously proposed that this action genera…
AdS Euclidean wormholes
Donald Marolf, Jorge E. Santos
We explore the construction and stability of asymptotically anti-de Sitter Euclidean wormholes in a variety of models. In simple ad hoc low-energy models, it is not hard to constru…
Resource Letter NSST-1: The Nature and Status of String Theory
Donald Marolf
This Resource Letter provides a guide to some of the the introductory and review literature in string theory. It is in no way complete, though it is intended to be of use to studen…
Causal holographic information does not satisfy the linearized quantum focusing condition
Zicao Fu, Donald Marolf, Marvin Qi
The Hubeny-Rangamani causal holographic information (CHI) defined by a region of a holographic quantum field theory (QFT) is a modern version of the idea that the area of event…
A rough end for smooth microstate geometries
Donald Marolf, Ben Michel, Andrea Puhm
Supersymmetric microstate geometries with five non-compact dimensions have recently been shown by Eperon, Reall, and Santos (ERS) to exhibit a non-linear instability featuring the…
Adiabatic corrections to holographic entanglement in thermofield doubles and confining ground states
Donald Marolf, Jason Wien
We study entanglement in states of holographic CFTs defined by Euclidean path integrals over geometries with slowly varying metrics. In particular, our CFT spacetimes have fi…
Group Averaging for de Sitter free fields
Donald Marolf, Ian A. Morrison
Perturbative gravity about global de Sitter space is subject to linearization-stability constraints. Such constraints imply that quantum states of matter fields couple consistently…
A paucity of bulk entangling surfaces: AdS wormholes with de Sitter interiors
Sebastian Fischetti, Donald Marolf, Aron C. Wall
We study and construct spacetimes, dubbed planar AdS-dS-wormholes, satisfying the null energy condition and having two asymptotically AdS boundaries connected through a (non-traver…
Energy Conditions and Junction Conditions
Donald Marolf, Sho Yaida
We consider the familiar junction conditions described by Israel for thin timelike walls in Einstein-Hilbert gravity. One such condition requires the induced metric to be continuou…
Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry
Stefan Hollands, Donald Marolf
We use a covariant phase space formalism to give a general prescription for defining Hamiltonian generators of bosonic and fermionic symmetries in diffeomorphism invariant theories…
Violating the Quantum Focusing Conjecture and Quantum Covariant Entropy Bound in dimensions
Zicao Fu, Jason Koeller, Donald Marolf
We study the Quantum Focussing Conjecture (QFC) in curved spacetime. Noting that quantum corrections from integrating out massive fields generally induce a Gauss-Bonnet term, we st…
On the Stress Tensor of Kerr/CFT
Aaron J. Amsel, Donald Marolf, Matthew M. Roberts
The recently-conjectured Kerr/CFT correspondence posits a field theory dual to dynamics in the near-horizon region of an extreme Kerr black hole with certain boundary conditions. W…
An Apologia for Firewalls
Ahmed Almheiri, Donald Marolf, Joseph Polchinski +2
We address proposed alternatives to the black hole firewall. We show that embedding the interior Hilbert space of an old black hole into the Hilbert space of the early radiation is…
Black hole entropy: inside or out?
Ted Jacobson, Donald Marolf, Carlo Rovelli
A trialogue. Ted, Don, and Carlo consider the nature of black hole entropy. Ted and Carlo support the idea that this entropy measures in some sense ``the number of black hole micro…
The non-zero energy of 2+1 Minkowski space
Donald Marolf, Leonardo Patiño
We compute the energy of 2+1 Minkowski space from a covariant action principle. Using Ashtekar and Varadarajan's characterization of 2+1 asymptotic flatness, we first show that the…
Perfect mirrors and the self-accelerating box paradox
Donald Marolf, Rafael Sorkin
We consider the question raised by Unruh and Wald of whether mirrored boxes can self-accelerate in flat spacetime (the ``self-accelerating box paradox''). From the point of view of…
Inextendible Schwarzschild black hole with a single exterior: How thermal is the Hawking radiation?
Jorma Louko, Donald Marolf
Several approaches to Hawking radiation on Schwarzschild spacetime rely in some way or another on the fact that the Kruskal manifold has two causally disconnected exterior regions.…
One-loop universality of holographic codes
Xi Dong, Donald Marolf
Recent work showed holographic error correcting codes to have simple universal features at . In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are assoc…
Poisson Bracket on the Space of Histories
Donald Marolf
We extend the Poisson bracket from a Lie bracket of phase space functions to a Lie bracket of functions on the space of canonical histories and investigate the resulting algebras.…
Conformal field theories in anti-de Sitter space
Ofer Aharony, Donald Marolf, Mukund Rangamani
In this paper we discuss the dynamics of conformal field theories on anti-de Sitter space, focussing on the special case of the N=4 supersymmetric Yang-Mills theory on AdS_4. We ar…
A coarse-grained generalized second law for holographic conformal field theories
William Bunting, Zicao Fu, Donald Marolf
We consider the universal sector of a -dimensional large- strongly-interacting holographic CFT on a black hole spacetime background . When our CFT is coupled to dynami…
Statistical Effects and the Black Hole/D-brane Correspondence
Donald Marolf
The horizon area and curvature of three-charge BPS black strings are studied in the D-brane ensemble for the stationary black string. The charge distributions along the string are…
Comparing Formulations of Generalized Quantum Mechanics for Reparametrization-Invariant Systems
James B. Hartle, Donald Marolf
A class of decoherence schemes is described for implementing the principles of generalized quantum theory in reparametrization-invariant `hyperbolic' models such as minisuperspace…
Holographic confinement in inhomogenous backgrounds
Donald Marolf, Jason Wien
As noted by Witten, compactifying a -dimensional holographic CFT on an gives a class of -dimensional confining theories with gravity duals. The prototypical bulk so…
Loop constraints: A habitat and their algebra
Jerzy Lewandowski, Donald Marolf
This work introduces a new space $\T'_*$ of `vertex-smooth' states for use in the loop approach to quantum gravity. Such states provide a natural domain for Euclidean Hamiltonian c…
Plane waves and spacelike infinity
Donald Marolf, Simon F. Ross
In an earlier paper, we showed that the causal boundary of any homogeneous plane wave satisfying the null convergence condition consists of a single null curve. In Einstein-Hilbert…
Real-time gravitational replicas: Low dimensional examples
Sean Colin-Ellerin, Xi Dong, Donald Marolf +2
We continue the study of real-time replica wormholes initiated in arXiv:2012.00828. Previously, we had discussed the general principles and had outlined a variational principle for…
Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf +2
Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inne…
Outgoing gravitational shock-wave at the inner horizon: The late-time limit of black hole interiors
Donald Marolf, Amos Ori
We investigate the interiors of 3+1 dimensional asymptotically flat charged and rotating black holes as described by observers who fall into the black holes at late times, long aft…
Quantum Observables and Recollapsing Dynamics
Donald Marolf
Within a simple quantization scheme, observables for a large class of finite dimensional time reparametrization invariant systems may be constructed by integration over the manifol…
Asymptotic Symmetries from finite boxes
Tomas Andrade, Donald Marolf
It is natural to regulate an infinite-sized system by imposing a boundary condition at finite distance, placing the system in a "box." This breaks symmetries, though the breaking i…
On the nature of ensembles from gravitational path integrals
Donald Marolf
Spacetime wormholes in gravitational path integrals have long been interpreted in terms of ensembles of theories. Here we probe what sort of theories such ensembles might contain.…
Bulk and Transhorizon Measurements in AdS/CFT
Idse Heemskerk, Donald Marolf, Joseph Polchinski +1
We discuss the construction of bulk operators in asymptotically AdS spacetimes, including the interiors of AdS black holes. We use this to address the question "If Schrodinger's ca…
Entanglement and Correlations near Extremality: CFTs dual to Reissner-Nordström AdS
Tomas Andrade, Sebastian Fischetti, Donald Marolf +2
We use the AdS/CFT correspondence to study models of entanglement and correlations between two CFTs in thermofield double states at finite chemical potential. Our bulk spacet…
Constrained HRT Surfaces and their Entropic Interpretation
Xi Dong, Donald Marolf, Pratik Rath
Consider two boundary subregions and that lie in a common boundary Cauchy surface, and consider also the associated HRT surface for . In that context, the constra…
Half-Branes, Singular Brane Intersections, and Kaluza-Klein reduction
Donald Marolf
Two subtle aspects of brane intersections are investigated. The first concerns the `half-branes' that arise in discussions of the Hanany-Witten effect, often in the D0/D8 setting.…
An illustration of 2+1 gravity loop transform troubles
Donald Marolf
A nonperturbative approach to quantum gravity that has generated much discussion is the attempt to construct a ``loop representation." Despite it's success in linear quantum theori…
Simple sufficient conditions for the generalized covariant entropy bound
Raphael Bousso, Eanna E. Flanagan, Donald Marolf
The generalized covariant entropy bound is the conjecture that the entropy of the matter present on any non-expanding null hypersurface L will not exceed the difference between the…
Conservation Laws in the Quantum Mechanics of Closed Systems
James B. Hartle, Raymond Laflamme, Donald Marolf
We investigate conservation laws in the quantum mechanics of closed systems. We review an argument showing that exact decoherence implies the exact conservation of quantities that…
Notes on Spacetime Thermodynamics and the Observer-dependence of Entropy
Donald Marolf, Djordje Minic, Simon Ross
Due to the Unruh effect, accelerated and inertial observers differ in their description of a given quantum state. The implications of this effect are explored for the entropy assig…
Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers
William Donnelly, Ben Michel, Donald Marolf +1
We generalize the Pastawski-Yoshida-Harlow-Preskill (HaPPY) holographic quantum error-correcting code to provide a toy model for bulk gauge fields or linearized gravitons. The key…
Interpolating Between Topologies: Casimir Energies
Donald Marolf
A set of models is considered which, in a certain sense, interpolates between 1+1 free quantum field theories on topologically distinct backgrounds. The intermediate models may be…
The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole
Ahmed Almheiri, Netta Engelhardt, Donald Marolf +1
Bulk quantum fields are often said to contribute to the generalized entropy only at . Nonetheless, in the context of evaporating black hol…
Bag-of-gold spacetimes, Euclidean wormholes, and inflation from domain walls in AdS/CFT
Zicao Fu, Donald Marolf
We use Euclidean path integrals to explore the set of bulk asymptotically AdS spacetimes with good CFT duals. We consider simple bottom-up models of bulk physics defined by Einstei…
Banishing AdS ghosts with a UV cutoff
Tomas Andrade, Thomas Faulkner, Donald Marolf
A recent attempt to make sense of scalars in AdS with "Neumann boundary conditions" outside of the usual BF-window led to pathologies including…
No Dynamics in the Extremal Kerr Throat
Aaron J. Amsel, Gary T. Horowitz, Donald Marolf +1
Motivated by the Kerr/CFT conjecture, we explore solutions of vacuum general relativity whose asymptotic behavior agrees with that of the extremal Kerr throat, sometimes called the…
Time-independence of gravitational Rényi entropies and unitarity in quantum gravity
Donald Marolf, Zhencheng Wang
The Hubeny-Rangamani-Takayanagi surface computing the entropy of a domain of dependence on an asymptotically AdS boundary is known to be causally inaccessible…
Boundary Conditions and Dualities: Vector Fields in AdS/CFT
Donald Marolf, Simon Ross
In AdS, scalar fields with masses slightly above the Breitenlohner-Freedman bound admit a variety of possible boundary conditions which are reflected in the Lagrangian of the dual…
Proof of Classical Versions of the Bousso Entropy Bound and of the Generalized Second Law
Eanna E. Flanagan, Donald Marolf, Robert M. Wald
Bousso has conjectured that in any spacetime satisfying Einstein's equation and satisfying the dominant energy condition, the "entropy flux" S through any null hypersurface L gener…
The Generalized Peierls Bracket
Donald Marolf
We first extend the Peierls algebra of gauge invariant functions from the space of classical solutions to the space of histories used in path integration and…
AdS flowing black funnels: Stationary AdS black holes with non-Killing horizons and heat transport in the dual CFT
Sebastian Fischetti, Donald Marolf, Jorge E. Santos
We construct stationary non-equilibrium black funnels locally asymptotic to global AdS4 in vacuum Einstein-Hilbert gravity with negative cosmological constant. These are non-compac…
Branes, Charge and Intersections
Donald Marolf
This is a brief summary of lectures given at the Fourth Mexican School on Gravitation and Mathematical Physics. The lectures gave an introduction to branes in eleven-dimensional su…
A perturbative perspective on self-supporting wormholes
Zicao Fu, Brianna Grado-White, Donald Marolf
We describe a class of wormholes that generically become traversable after incorporating gravitational back-reaction from linear quantum fields satisfying appropriate (periodic or…
The dangers of extremes
Donald Marolf
While extreme black hole spacetimes with smooth horizons are known at the level of mathematics, we argue that the horizons of physical extreme black holes are effectively singular.…
Supersymmetric Multi-trace Boundary Conditions in AdS
Aaron J. Amsel, Donald Marolf
Boundary conditions for massive fermions are investigated in AdS_d for . For fermion masses in the range with the AdS length, the standard not…
A tale of two superpotentials: Stability and Instability in Designer Gravity
Aaron J. Amsel, Thomas Hertog, Stefan Hollands +1
We investigate the stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound. The bounda…
The Spectral Analysis Inner Product for Quantum Gravity
Donald Marolf
This submission to the Proceedings of the Seventh Marcel-Grossman Conference is an advertisement for the use of the ``spectral analysis inner product" for minisuperspace models in…
Geons with spin and charge
Jorma Louko, Robert B. Mann, Donald Marolf
We construct new geon-type black holes in D>3 dimensions for Einstein's theory coupled to gauge fields. A static nondegenerate vacuum black hole has a geon quotient provided the sp…
Phases of Holographic Hawking Radiation on spatially compact spacetimes
Donald Marolf, Jorge E. Santos
We study phases of equilibrium Hawking radiation in -dimensional holographic CFTs on spatially compact spacetimes with two black holes. In the particular phases chosen the dual…
State-Dependent Divergences in the Entanglement Entropy
Donald Marolf, Aron C. Wall
We show the entanglement entropy in certain quantum field theories to contain state-dependent divergences. Both perturbative and holographic examples are exhibited. However, quanti…
Almost Ideal Clocks in Quantum Cosmology: A Brief Derivation of Time
Donald Marolf
A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock.' Previously, this formalism was succ…
T-duality and the case of the disappearing brane
Donald Marolf
When an NS5-brane crosses a D6-brane, a D4-brane is created stretching between them by the Hanany-Witten effect. However, the T-dual situation involves a Kaluza-Klein monopole cros…
Some brane theoretic no-hair results (and their field theory duals)
Donald Marolf
This contribution to the proceedings of the 1999 Canadian Conference on General Relativity and Relativistic Astrophysics is a brief exposition of earlier work, with Sumati Surya (h…
Null states and time evolution in a toy model of black hole dynamics
Xi Dong, Maciej Kolanowski, Xiaoyi Liu +2
Spacetime wormholes can provide non-perturbative contributions to the gravitational path integral that make the actual number of states in a gravitational system much smaller…
A new recipe for causal completions
Donald Marolf, Simon F. Ross
We discuss the asymptotic structure of spacetimes, presenting a new construction of ideal points at infinity and introducing useful topologies on the completed space. Our construct…
't Hooft suppression and holographic entropy
William R. Kelly, Kevin Kuns, Donald Marolf
Recent works have related the bulk first law of black hole mechanics to the first law of entanglement in a dual CFT. These are first order relations, and receive corrections for fi…
Flat entanglement spectra in fixed-area states of quantum gravity
Xi Dong, Daniel Harlow, Donald Marolf
We use the Einstein-Hilbert gravitational path integral to investigate gravitational entanglement at leading order . We argue that semiclassical states prepared by a Euclid…
CFT sewing as the dual of AdS cut-and-paste
Donald Marolf
The CPT map allows two states of a quantum field theory to be sewn together over CPT-conjugate partial Cauchy surfaces to make a state on a new spacetime. We study the ho…
Energy Bounds in Designer Gravity
Aaron J. Amsel, Donald Marolf
We consider asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound in d greater than or equal to 4…
A Uniqueness Theorem for Constraint Quantization
Domenico Giulini, Donald Marolf
This work addresses certain ambiguities in the Dirac approach to constrained systems. Specifically, we investigate the space of so-called ``rigging maps'' associated with Refined A…
A global picture of quantum de Sitter space
Steven B. Giddings, Donald Marolf
Perturbative gravity about a de Sitter background motivates a global picture of quantum dynamics in `eternal de Sitter space,' the theory of states which are asymptotically de Sitt…
From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories
Donald Marolf, Onkar Parrikar, Charles Rabideau +2
We consider states of holographic conformal field theories constructed by adding sources for local operators in the Euclidean path integral, with the aim of investigating the exten…
Localized Branes and Black Holes
Sumati Surya, Donald Marolf
We address the delocalization of low dimensional D-branes and NS-branes when they are a part of a higher dimensional BPS black brane, and the homogeneity of the resulting horizon.…
The explicit form of the effective action for F1 and D-branes
Donald Marolf, Luca Martucci, Pedro J. Silva
In this work we consider the full interacting effective actions for fundamental strings and D-branes in arbitrary bosonic type II supergravity backgrounds. The explicit form of the…
Observables and a Hilbert Space for Bianchi IX
Donald Marolf
We consider a quantization of the Bianchi IX cosmological model based on taking the constraint to be a self-adjoint operator in an auxiliary Hilbert space. Using a WKB-style self-c…
Closed formula for Wilson loops for Quantum Yang-Mills Theory in two dimensions
Abhay Ashtekar, Jurek Lewandowski, Donald Marolf +2
A closed expression of the Euclidean Wilson-loop functionals is derived for pure Yang-Mills continuum theories with gauge groups and and space-time topologies $\Rl^1…
Complex Entangling Surfaces for AdS and Lifshitz Black Holes?
Sebastian Fischetti, Donald Marolf
We discuss the possible relevance of complex codimension-two extremal surfaces to the the Ryu-Takayanagi holographic entanglement proposal and its covariant Hubeny-Rangamani-Takaya…
Spacetime Embedding Diagrams for Spherically Symmetric Black Holes
John T. Giblin, Donald Marolf, Robert H. Garvey
We show that it is possible to embed the 1+1 dimensional reduction of certain spherically symmetric black hole spacetimes into 2+1 Minkowski space. The spacetimes of interest (Schw…
Brane Baldness vs. Superselection Sectors
Donald Marolf, Amanda W. Peet
The search for intersecting brane solutions in supergravity is a large and profitable industry. Recently, attention has focused on finding localized forms of known `delocalized' so…
Global structure of Witten's 2+1 gravity on
Jorma Louko, Donald Marolf
We investigate the space of classical solutions to Witten's formulation of 2+1 gravity on the manifold . is connected, but neither Hausdor…
Euclidean wormholes stability analysis revisited
Donald Marolf, Bruno Missoni
Previous studies of linearized stability of asymptotically flat Euclidean axion wormholes found that symmetric modes suffered from divergences. We show that such divergences were a…
On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT
Atsushi Higuchi, Donald Marolf, Ian A. Morrison
We study the relation between two sets of correlators in interacting quantum field theory on de Sitter space. The first are correlators computed using in-in perturbation theory in…
The IR stability of de Sitter: Loop corrections to scalar propagators
Donald Marolf, Ian A. Morrison
We compute 1-loop corrections to Lorentz-signature de Sitter-invariant 2-point functions defined by the interacting Euclidean vacuum for massive scalar quantum fields with cubic an…
Handlebody phases and the polyhedrality of the holographic entropy cone
Donald Marolf, Massimiliano Rota, Jason Wien
The notion of a holographic entropy cone has recently been introduced and it has been proven that this cone is polyhedral. However, the original definition was fully geometric and…
Covariant Counterterms and Conserved Charges in Asymptotically Flat Spacetimes
Robert B. Mann, Donald Marolf, Amitabh Virmani
Recent work has shown that the addition of an appropriate covariant boundary term to the gravitational action yields a well-defined variational principle for asymptotically flat sp…