The double null form of the Schwarzschild metric is usually arrived at by demanding Eddington-Finkelstein (EF) conditions at the horizon. This leads to certain logarithmic fall-offs that are too slow along null directions at $$ \mathcal{I} $$ , resulting in divergences in the covariant surface charges. These coordinates are therefore $\textit{not}$ asymptotically flat in a conventional sense. In this paper, we find a natural alternative double null form for Schwarzschild that is adapted to $$ {\mathcal{I}}^{+} $$ or $$ {\mathcal{I}}^{-} $$ instead of the horizon. In its final form, the metric has only power law fall-offs and fits into the recently introduced Special Double Null (SDN) gauge, with finite surface charges. One remarkable feature of SDN gauge is that spherical symmetry and vacuum Einstein equations allow an infinite number of asymptotic integration constants in the metric, on top of the mass. This is an apparent violation of Birkhoff’s theorem. We note however that all except two of these new parameters are absent in the charges, and therefore correspond to trivial hair. The remaining two parameters do show up in the charges, depending on the choice of allowed fall-offs. We provide an understanding of this observation — Birkhoff’s theorem fixes Schwarzschild only $\textit{up to diffeomorphisms}$, but diffeomorphisms need not vanish at infinity and can in principle become global symmetries. If such asymptotic diffeomorphisms are spherically symmetric, their associated soft modes can become Birkhoff hair. The relevant global symmetries here are certain hypertranslation shifts in the $\textit{v}$-coordinate at $$ {\mathcal{I}}^{+} $$ (and $\textit{u}$ at $$ {\mathcal{I}}^{-} $$ ), which are inaccessible in other gauges.

India

Chethan Krishnan

United States

Jude Pereira

Soft hair on Schwarzschild: a wrinkle in Birkhoff ’s theorem

We synthesize and sharpen various observations about sub-AdS holography in the literature to associate a central charge to sub-AdS scales of AdS×$\textit{X}$ geometries. A key ingredient in our proposal is the idea that the AdS length is the length of the long string in a stack of $\textit{N backreacted}$ D-branes. This allows us to make statements about sub-AdS scales by considering long strings in sub-stacks of $\textit{M}$ < $\textit{N}$ branes. Our proposal applies in general dimensions, connects with and refines previous ideas about sub-matrix deconfinement & long strings, makes crucial use of the compact space, and is consistent with the expected thermodynamics of small black holes localized on $\textit{X}$. Some of our arguments draw intuition from separating branes, and have natural connections to (heating up) the Coulomb branch of the gauge theory. We apply related ideas to non-conformal D-branes at zero and finite temperature, discuss the holographic bound and IR/UV duality in theories with 16 supercharges, and find broad consistency.

Abir Ghosh

Adinath Mahalingam

A central charge for sub-AdS holography

In 1+1 dimensions, the equations of motion of the Horowitz-Polchinski (HP) effective string have a re-writing in terms of a first order system. This is attributed to FZZ duality. In this note, we observe that a similar re-writing exists in higher dimensions, so that the degree of the dilaton-winding subsystem reduces to first order. The 1+1 first order equations emerge as a natural limit of the higher dimensional HP system in the cap region of the cigar. As a result, there is a critical value of the winding amplitude that matches with the 1+1 coset SCFT prediction. At this critical point, the cigar has a puncture at the Euclidean horizon and the black hole entropy is correctly reproduced by the winding condensate.

Sarthak Talukdar

A vestige of FZZ duality in higher dimensions

Defining finite entanglement entropy for a subregion in quantum field theory requires the introduction of two logically independent scales: an IR scale that controls the size of the subregion, and a UV cut-off. In AdS/CFT, the IR scale is the AdS lengthscale, the UV cut-off is the bulk radial cut-off, and the subregion is specified by dimensionless angles. This is the data that determines Ryu-Takayanagi surfaces and their areas in AdS/CFT. We argue that in asymptotically flat space there exists the notion of a “spi-subregion” that one can associate to spatial infinity (spi). Even though geometrically quite different from an AdS subregion, this angle data has the crucial feature that it allows an interpretation as a bi-partitioning of spi. Therefore, the area of the RT surface associated to the spi-subregion can be interpreted as the entanglement entropy of the reduced density matrix of the bulk state under this bi-partition, as in AdS/CFT. For symmetric spi-subregions, these RT surfaces are the waists of Asymptotic Causal Diamonds. In empty flat space they reduce to Rindler horizons, and are analogues of the AdS-Rindler horizons of Casini, Huerta &amp; Myers. We connect these results to previous work on minimal surfaces anchored to screens in empty space, but also generalize the discussion to the case where there are black holes in the bulk. The phases of black hole RT surfaces as the spi-subregion is varied, naturally connect with those of black holes (small and large) in AdS. A key observation is that the radial cut-off is associated to an IR scale in flat space — and in fact there are no UV divergences. We argue that this is consistent with previous suggestions that in sub-AdS scales the holographic duality is an IR/IR correspondence and that the degrees of freedom are not those of a local QFT, but those of long strings. Strings are of course, famously UV finite.

A holographic entanglement entropy at spi

Opinion is divided about the nature of state dependence in the black hole interior. Some argue that it is a necessary feature, while others argue it is a bug. In this paper, we consider the extended half-sided modular translation U (s 0) (with s 0 &gt; 0) of Leutheusser and Liu that takes us inside the horizon. We note that we can use this operator to construct a modular Hamiltonian H and a conjugation J on the infalling time-evolved wedges. The original thermofield double translates to a new cyclic and separating vector in the shifted algebra. We use these objects and the Connes’ cocycle to repeat Witten’s crossed product construction in this new setting, and to obtain a Type II ∞  algebra that is independent of the various choices, in particular that of the cyclic separating vector. Our emergent times are implicitly boundary-dressed. But if one admits an “extra” observer in the interior, we argue that the (state-independent) algebra can be Type I or Type II1 instead of Type II ∞ , depending on whether the observer’s light cone contains an entire Cauchy slice or not. Along with these general considerations, we present some specific calculations in the setting of the Poincare BTZ black hole. We identify a specific pointwise (as opposed to non-local) modular translation in BTZ-Kruskal coordinates that is analytically tractable, exploiting a connection with AdS-Rindler. This modular translation can reach the singularity.

Iceland

Vyshnav Mohan

State-independent black hole interiors from the crossed product

In 1984, ’t Hooft famously used a brickwall (aka stretched horizon) to compute black hole entropy up to a numerical pre-factor. This calculation is sometimes interpreted as due to the entanglement of the modes across the horizon, but more operationally, it is simply an indirect count of the semi-classical modes trapped between the stretched horizon and the angular momentum barrier. Because the calculation was indirect, it needed both the mass and the temperature of the black hole as inputs, to reproduce the area. A more conventional statistical mechanics calculation should be able to get the entropy, once the ensemble is specified (say via the energy, in a microcanonical setting). In this paper, we explicitly compute black hole normal modes in various examples, numerically as well as (in various regimes) analytically. The explicit knowledge of normal modes allows us to reproduce both the Hawking temperature as well as the entropy, once the charges are specified, making this a conventional statistical mechanics calculation. A quasi-degeneracy in the angular quantum numbers is directly responsible for the area scaling of the entropy, and is the key distinction between the Planckian black body calculation (volume scaling) and the ’t Hooftian calculation (area scaling). We discuss the (rotating) BTZ case in detail and match the thermodynamic quantities exactly. Schwarzschild and Kerr normal modes are discussed in less detail using near-horizon approximations. Our calculations reveal a new hierarchy in the angular quantum numbers, which we speculate is related to string theory.

Pradipta Pathak

Normal modes of the stretched horizon: a bulk mechanism for black hole microstate level spacing

Recent observations on type III algebras in AdS/CFT raise the possibility that smoothness of the black hole horizon is an emergent feature of the large-N limit. In this paper, we present a bulk model for the finite-N mechanism underlying this transition. We quantize a free scalar field on a BTZ black hole with a Planckian stretched horizon placed as a Dirichlet boundary for the field. This is a tractable model for the stretched horizon that does not ignore the angular directions, and it defines a black hole vacuum which has similarities to (but is distinct from) the Boulware state. Using analytic approximations for the normal modes, we first improve upon ’t Hooft’s brick wall calculation: we are able to match both the entropy and the temperature, exactly. Emboldened by this, we compute the boundary Wightman function of the scalar field in a typical pure state built on our stretched horizon vacuum, at an energy sliver at the mass of the black hole. A key result is that despite the manifest lack of smoothness, this single-sided pure state calculation yields precisely the Hartle-Hawking thermal correlator associated to the smooth horizon, in the small-G  N  limit. At finite G  N , there are variance corrections that are suppressed as   <math> <mi>O</mi> <mfenced> <msup> <mi>e</mi> <mrow> <mo>−</mo> <msub> <mi>S</mi> <mi>BH</mi> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfenced> </math>  $$ \mathcal{O}\left({e}^{-{S}_{BH}/2}\right) $$ . They become important at late times and resolve Maldacena’s information paradox. Highly excited typical pure states on the stretched horizon vacuum are therefore models for black hole microstates, while the smooth horizon describes the thermal state. We note that heavy excited states on the stretched horizon are better defined than the vacuum itself. These results suggest that complementarity in the bulk EFT could arise from a UV complete bulk description in which the black hole interior is not manifest.

Vaibhav Burman

Suchetan Das

A smooth horizon without a smooth horizon

We discuss conditions under which a deterministic sequence of real numbers, interpreted as the set of eigenvalues of a Hamiltonian, can exhibit features usually associated to random matrix spectra. A key diagnostic is the spectral form factor (SFF) — a linear ramp in the SFF is often viewed as a signature of random matrix behavior. Based on various explicit examples, we observe conditions for linear and power law ramps to arise in deterministic spectra. We note that a very simple spectrum with a linear ramp is E  n  ~ log n. Despite the presence of ramps, these sequences do not exhibit conventional level repulsion, demonstrating that the lore about their concurrence needs refinement. However, when a small noise correction is added to the spectrum, they lead to clear level repulsion as well as the (linear) ramp. We note some remarkable features of logarithmic spectra, apart from their linear ramps: they are closely related to normal modes of black hole stretched horizons, and their partition function with argument s = β + it is the Riemann zeta function ζ(s). An immediate consequence is that the spectral form factor is simply −ζ|(it)|2. Our observation that log spectra have a linear ramp, is closely related to the Lindelöf hypothesis on the growth of the zeta function. With elementary numerics, we check that the slope of a best fit line through |ζ(it)|2 on a log-log plot is indeed 1, to the fourth decimal. We also note that truncating the Riemann zeta function sum at a finite integer N causes the would-be-eternal ramp to end on a plateau.

Suman Das

Sumit Garg

Arnab Kundu

What is the Simplest Linear Ramp?

Black holes are believed to have the fast scrambling properties of random matrices. If the fuzzball proposal is to be a viable model for quantum black holes, it should reproduce this expectation. This is considered challenging, because it is natural for the modes on a fuzzball microstate to follow Poisson statistics. In a previous paper, we noted a potential loophole here, thanks to the modes depending not just on the n-quantum number, but also on the J-quantum numbers of the compact dimensions. For a free scalar field ϕ, by imposing a Dirichlet boundary condition ϕ = 0 at the stretched horizon, we showed that this J-dependence leads to a linear ramp in the Spectral Form Factor (SFF). Despite this, the status of level repulsion remained mysterious. In this letter, motivated by the profile functions of BPS fuzzballs, we consider a generic profile ϕ = ϕ 0(θ) instead of ϕ = 0 at the stretched horizon. For various notions of genericity (eg. when the Fourier coefficients of ϕ 0(θ) are suitably Gaussian distributed), we find that the J-dependence of the spectrum exhibits striking evidence of level repulsion, along with the linear ramp. We also find that varying the profile leads to natural interpolations between Poisson and Wigner-Dyson(WD)-like spectra. The linear ramp in our previous work can be understood as arising via an extreme version of level repulsion in such a limiting spectrum. We also explain how the stretched horizon/fuzzball is different in these aspects from simply putting a cut-off in flat space or AdS (i.e., without a horizon).

Fuzzballs and random matrices

We show that the Friedmann–Lemaître–Robertson–Walker (FLRW) framework has an instability towards the growth of fluid flow anisotropies, even if the Universe is accelerating. This flow (tilt) instability in the matter sector is invisible to Cosmic No-Hair Theorem-like arguments, which typically only flag shear anisotropies in the metric. We illustrate our claims in the setting of “dipole cosmology”, the maximally Copernican generalization of FLRW that can accommodate a flow. Simple models are sufficient to show that the cosmic flow need not track the shear, even in the presence of a positive cosmological constant. We also emphasize that the growth of the tilt hair is fairly generic if the effective equation of state  $$w \rightarrow -1$$  <math> <mrow> <mi>w</mi> <mo>→</mo> <mo>-</mo> <mn>1</mn> </mrow> </math>   at late times (as it does in standard cosmology), irrespective of the precise model of dark energy. The generality of our theoretical result puts various recent observational claims about late time anisotropies and cosmic dipoles in a new light.