We establish a correspondence between the gravitational phase space at null infinity and the subleading phase space near a finite-distance null hypersurface, such as a black hole horizon. Within this framework, we identify the celestial $Lw_{1+\infty}$ symmetries in the subleading phase space at the horizon by constructing their canonical generators and imposing self-duality conditions. This leads to an infinite tower of conserved charges in the absence of radiation, revealing new gravitational observables relevant to black hole physics.
The aim of these Lectures is to provide a brief overview of the subject of asymptotic symmetries of gauge and gravity theories in asymptotically flat spacetimes as background material for celestial holography.
Celestial amplitudes provide holographic imprints of four-dimensional scattering processes in terms of conformal correlation functions on a two-dimensional sphere describing Minkowski space at null infinity. We construct the generators of Poincare and conformal groups in the celestial representation and discuss how these symmetries are manifest in the amplitudes.
The soft limits of scattering amplitudes have been extensively studied due to their essential role in the computation of physical observables in collider physics. The universal factorisation that occurs in these kinematic limits has been shown to be related to conservation laws associated with asymptotic, or large, gauge symmetries. This connection has led to a deeper understanding of the symmetries of gauge and gravitational theories and to a reformulation of scattering amplitudes in a basis of boost eigenstates which makes manifest the two-dimensional global conformal symmetry of the celesti…
It is shown that the infinite tower of tree-level soft graviton symmetries in asymptotically flat 4D quantum gravity can be organized into a single chiral 2D Kac-Moody symmetry based on the wedge algebra of w(1+infinity). The infinite towers of soft photon or gluon symmetries also transform irreducibly under w(1+infinity).
We explain how twistor theory represents the self-dual sector of four dimensional gravity in terms of the loop group of Poisson diffeomorphisms of the plane via Penrose's non-linear graviton construction. The symmetries of the self-dual sector are generated by the corresponding loop algebra $Lw_{1+\infty}$ of the algebra $w_{1+\infty}$ of these Poisson diffeomorphisms. We show that these coincide with the infinite tower of soft graviton symmetries in tree-level perturbative gravity recently discovered in the context of celestial amplitudes. We use a twistor sigma model for the self-dual sector…
We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension $Δ$. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series $Δ\in 1+i\mathbb{R}$ and form a complete basis. We show that conformal primaries with analyticall…
In this paper we extract from a large-$r$ expansion of the vacuum Einstein's equations a dynamical system governing the time evolution of an infinity of higher-spin charges. Upon integration, we evaluate the canonical action of these charges on the gravity phase space. The truncation of this action to quadratic order and the associated charge conservation laws yield an infinite tower of soft theorems. We show that the canonical action of the higher spin charges on gravitons in a conformal primary basis, as well as conformally soft gravitons reproduces the higher spin celestial symmetries deriv…
Celestial $Lw_{1+\infty}$ Symmetries and Subleading Phase Space of Null Hypersurfaces
Harvard University（哈佛大学）；Perimeter Institute for Theoretical Physics（珀杜姆理论物理研究所）；Université Libre de Bruxelles and International Solvay Institutes（布鲁塞尔自由大学与国际索尔维研究所）；Collège de France（法兰西学院）
We show that time intervals of width $Δτ$ in 3-dimensional conformal field theories (CFT$_3$) on the Lorentzian cylinder admit an infinite dimensional symmetry enhancement in the limit $Δτ\rightarrow 0$. The associated vector fields are approximate solutions to the conformal Killing equations in the strip labelled by a function and a conformal Killing vector on the sphere. An Inonu-Wigner contraction yields a set of symmetry generators obeying the extended BMS$_4$ algebra. We analyze the shadow stress tensor Ward identities in CFT$_d$ on the Lorentzian cylinder with all operator insertions in …
Celestial holography proposes a duality between the gravitational $\mathcal{S}$-matrix and correlators in a conformal field theory living on the celestial sphere. In this white paper, solicited for the 2022 Snowmass process, we review the motivation from asymptotic symmetries, fundamentals of the proposed holographic dictionary, potential applications to experiment and theory, and some important open questions.
Generalized BMS (gBMS) is the Lie group of the asymptotic symmetries at null infinity, and is proposed to be a symmetry of the quantum S-matrix. Despite much progress in understanding the symplectic structure at null infinity consistent with the gBMS symmetries, the construction of a radiative phase space where all the physical soft modes and their conjugate partners are identified remains elusive. We construct just such a radiative phase space for linearized gravity by a systematic constraint analysis. In addition, we highlight the difficulties that arise in extending this analysis to the non…
Local symmetries of a non-expanding horizon has been investigated in the 1st order formulation of gravity. When applied to a spherically symmetric isolated horizon only a U(1) subgroup of the Lorentz group survives as residual local symmetry that one can make use of in constructing an effective theory on the horizon.
Celestial holography proposes a duality between gravitational scattering in asymptotically flat space-time and a conformal field theory living on the celestial sphere. Its dictionary relates the infinite dimensional space-time symmetry group to Ward identities of the CFT. The spontaneous breaking of these asymptotic symmetries governs the dynamics of the soft sector in the CFT. Here we show that this sector encodes non-trivial backreaction effects that exhibit characteristics of maximal quantum chaos. A key element in the derivation is the identification of the Hilbert space of celestial CFT, …
We study exponentiated soft exchange in $d+2$ dimensional gauge and gravitational theories using the celestial CFT formalism. These models exhibit spontaneously broken asymptotic symmetries generated by gauge transformations with non-compact support, and the effective dynamics of the associated Goldstone "edge" mode is expected to be $d$-dimensional. The introduction of an infrared regulator also explicitly breaks these symmetries so the edge mode in the regulated theory is really a $d$-dimensional pseudo-Goldstone boson. Symmetry considerations determine the leading terms in the effective act…
Near-horizon symmetries are studied for static black hole solutions to Einstein equations containing supertranslation field. We consider general diffeomorphisms which preserve the gauge and the near-horizon structure of the metric. Diffeomorphisms are generated by the vector fields and form a group of near-horizon symmetries. The densities of variation of the surface charge associated to asymptotic horizon symmetries of the metric are calculated in different coordinate systems connected by "large" transformations which change supertranslation field in the metric. Integrability of the variation…
We develop basic tools and matching conditions to interpolate between asymptotic and near horizon symmetries. We focus on black holes in three dimensions. In particular, we match Brown--Henneaux boundary conditions at infinity, which yields two Virasoro algebras, to Heisenberg boundary conditions at the horizon yielding two u(1) current algebras. Our construction allows to equip BTZ black holes with soft hair excitations at the horizon invisible to the asymptotic observer.
We study hidden symmetries, the symmetries associated with the Killing tensors, of the near horizon geometry of odd-dimensional Kerr-AdS-NUT black hole in two limits: generic extremal and extremal vanishing horizon (EVH) limits. Starting from Kerr-AdS-NUT black hole in ellipsoidal coordinates which admits integrable geodesic equations, we obtain the near horizon extremal/EVH geometries and their principal and Killing tensors by taking the near horizon limit. We explicitly demonstrate that geodesic equations are separable and integrable on theses near horizon geometries. We also compute the con…
Using the exact solution to Einstein equations of Compere and Long for the Schwarzschild metric containing supertranslation field, we study the near-horizon symmetries of the metric. We consider class of metrics with supertranslation field depending only on a spherical angle $θ$. After reviewing the action of supertranslations preserving the static gauge of the metric, we study general transformations preserving the near-horizon form of the metric. We determine the asymptotic Killing vectors and calculate the charge corresponding to the asymptotic symmetries preserving the form of the metric a…
The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat gravity. Recently, Donnay et al. have derived an analogous symmetry group acting on black hole event horizons. For a certain choice of boundary conditions, it is a semidirect product of ${\rm Diff}(S^2)$, the smooth diffeomorphisms of the two-sphere, acting on $C^\infty(S^2)$, the smooth functions on the two-sphere. We observe that the same group appears in fluid dynamics as symmetries of the compressible Euler equations. We relate these two realizations of ${\rm Diff}(S^2)\ltimes C^\infty(…
Near-horizon symmetries are studied for black hole solutions to Einstein equations containing supertranslation field constructed by Compere and Long. The metric is transformed to variables in which the horizon is located at the surface $r=2M$, where $M$ is the mass of black hole. We consider general diffeomorphisms which preserve the gauge and the near-horizon structure of the metric and find the corresponding transformations of metric components. We review the action of the generators of supertranslations preserving the static gauge of the metric and determine a subgroup of supertranslations …
We investigate asymptotic symmetries regularly defined on spherically symmetric Killing horizons in the Einstein theory with or without the cosmological constant. Those asymptotic symmetries are described by asymptotic Killing vectors, along which the Lie derivatives of perturbed metrics vanish on a Killing horizon. We derive the general form of asymptotic Killing vectors and find that the group of the asymptotic symmetries consists of rigid O(3) rotations of a horizon two-sphere and supertranslations along the null direction on the horizon, which depend arbitrarily on the null coordinate as w…
Asymptotic spacetime symmetries have been conjectured to play an important role in quantum gravity. In this paper we study the breaking of asymptotic symmetries associated with a null horizon boundary. In two-dimensions, these symmetries are reparametrizations of the time parameter on the horizon. We show how this horizon reparametrization symmetry is explicitly and spontaneously broken in dilaton gravity and construct an effective action for these pseudo-Goldstone modes using the on-shell gravitational action for a null boundary. The variation of this action yields the horizon constraint equa…
We determine the asymptotic symmetry group of Killing horizons by choosing Gaussian null coordinates in the neighbourhood of the horizon and boundary conditions that respect the leading order terms in the metric. The analysis divides naturally into the two cases of subextremal and extremal horizons. In general, we find rather involved asymptotic symmetry generators that nevertheless involve supertranslations and Killing vectors on the compact horizon. The most striking observation is the difference in the dependence on the coordinate along the horizon; we relate this to the redshift effect for…
We suggest a way to study possible conformal symmetries on black hole horizons. We do this by carrying out a Kaluza-Klein like reduction of the Einstein-Hilbert action along the ignorable coordinates of stationary and axisymmetric black holes. Rigid diffeomorphism invariance of the $m$-ignorable coordinates then becomes a global $SL(m,R)$ gauge symmetry of the reduced action. Related to each non-vanishing angular velocity there is a particular $SL(2,R)$ subgroup, which can be extended to the Witt algebra on the black hole horizons. The classical Einstein-Hilbert action thus has $k$-copies of i…
It has often been suggested (especially by Carlip) that spacetime symmetries in the neighborhood of a black hole horizon may be relevant to a statistical understanding of the Bekenstein-Hawking entropy. A prime candidate for this type of symmetry is that which is exhibited by the Einstein tensor. More precisely, it is now known that this tensor takes on a strongly constrained (block-diagonal) form as it approaches any stationary, non-extremal Killing horizon. Presently, exploiting the geometrical properties of such horizons, we provide a particularly elegant argument that substantiates this hi…
We consider a BMS-type symmetry action on isolated horizons in asymptotically flat spacetimes. From the viewpoint of the non-relativistic field theory on a horizon membrane, supertranslations shift the field theory spatial momentum. The latter is related by a Ward identity to the particle number symmetry current and is spontaneously broken. The corresponding Goldstone boson shifts the horizon angular momentum and can be detected quantum mechanically. Similarly, area preserving superrotations are spontaneously broken on the horizon membrane and we identify the corresponding gapless modes. In as…
We analyze asymptotic symmetries on the Killing horizon of the four-dimensional Kerr--Newman black hole. We first derive the asymptotic Killing vectors on the Killing horizon, which describe the asymptotic symmetries, and find that the general form of these asymptotic Killing vectors is the universal one possessed by arbitrary Killing horizons. We then construct the phase space associated with the asymptotic symmetries. It is shown that the phase space of an extreme black hole either has the size comparable with a non-extreme black hole, or is small enough to exclude degeneracy, depending on w…
School of Mathematics, University of Bristol（布里斯托大学数学学院）；School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh（爱丁堡大学数学学院与麦克斯韦数学科学研究所）；Higgs Centre for Theoretical Physics, University of Edinburgh（爱丁堡大学希格斯理论物理中心）；Department of Astrophysics and High Energy Physics, S.N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心天体物理学与高能物理系）；Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University（赫瑞-瓦特大学数学学院与麦克斯韦数学科学研究所）
We study the near horizon geometry of both static and stationary extremal Oliva Tempo Troncoso (OTT) black holes. For each of these cases, a set of consistent asymptotic conditions is introduced. The canonical generator for the static configuration is shown to be regular. For the rotating OTT black hole, the asymptotic symmetry is described by the time reparametrization, the chiral Virasoro and centrally extended $u(1)$ Kac-Moody algebras.
We develop a new approach to find asymptotic symmetries in general relativity as a modification of the Lie-algebra-based approach proposed in T. Tomitsuka et al. [Classical Quantum Gravity 38, 225007 (2021)]. Those authors proposed an algorithmic protocol to investigate asymptotic symmetries. In particular, their guiding principle helps us to find a non-vanishing charge that generates an infinitesimal diffeomorphism. However, in order to check the integrability condition for the charges, it is necessary to solve differential equations to identify the integral curve of vector fields, which is u…
It is well known that, for a wide class of spacetimes with horizons, Einstein equations near the horizon can be written as a thermodynamic identity. It is also known that the Einstein tensor acquires a highly symmetric form near static, as well as stationary, horizons. We show that, for generic static spacetimes, this highly symmetric form of the Einstein tensor leads quite naturally and generically to the interpretation of the near-horizon field equations as a thermodynamic identity. We further extend this result to generic static spacetimes in Lanczos-Lovelock gravity, and show that the near…
We investigate whether supertranslation symmetry may appear in a scenario that involves black holes in AdS space. The framework we consider is massive 3D gravity, which admits a rich black hole phase space, including stationary AdS black holes with softly decaying hair. We consider a set of asymptotic conditions that permits such decaying near the boundary, and which, in addition to the local conformal symmetry, is preserved by an extra local current. The corresponding algebra of diffeomorphisms consists of two copies of Virasoro algebra in semi-direct sum with an infinite-dimensional Abelian …
We study horizon shells and soldering freedom for extreme black holes and how supertranslation-like Bondi-Metzner-Sachs (BMS) symmetries appear as soldering transformations. Further, for a null shell placed infinitesimally close to the horizon of an extreme Reissner-Nordstr$\ddot{o}$m (RN) black hole, we show superrotation-like symmetries also arise as soldering freedom. Next, considering the interaction of impulsive gravitational waves supported at the horizon shell with test particles, we study how the "memory" (or the imprints) of BMS-like symmetries gets encoded in the geodesics (test part…
We study gravitational lensing of light by hairy black holes, which, in a certain parameter regime, can possess two photon spheres of different size outside the event horizon. In particular, we focus on higher-order images of a point-like light source and a luminous celestial sphere produced by strong gravitational lensing near photon spheres. Two photon spheres usually triple the number of high-order images of a point-like light source. When a hairy black hole is illuminated by a celestial sphere, two photon spheres would give rise to two critical curves in the black hole image, and the small…
Recent work has demonstrated an attractor mechanism for extremal rotating black holes subject to the assumption of a near-horizon SO(2,1) symmetry. We prove the existence of this symmetry for any extremal black hole with the same number of rotational symmetries as known four and five dimensional solutions (including black rings). The result is valid for a general two-derivative theory of gravity coupled to abelian vectors and uncharged scalars, allowing for a non-trivial scalar potential. We prove that it remains valid in the presence of higher-derivative corrections. We show that SO(2,1)-symm…
We consider the spacetime geometry of a static but otherwise generic black hole (that is, the horizon geometry and topology are not necessarily spherically symmetric). It is demonstrated, by purely geometrical techniques, that the curvature tensors, and the Einstein tensor in particular, exhibit a very high degree of symmetry as the horizon is approached. Consequently, the stress-energy tensor will be highly constrained near any static Killing horizon. More specifically, it is shown that -- at the horizon -- the stress-energy tensor block-diagonalizes into ``transverse'' and ``parallel'' block…
Double Wick rotations between symmetries of Taub-NUT, near-horizon extreme Kerr, and swirling spacetimes
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院理论物理研究所）；Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）
For a long time it is believed that black hole horizon are thermal and quantum mechanical in nature. The microscopic origin of this thermality is the main question behind our present investigation, which reveals possible importance of near horizon symmetry. It is this symmetry which is assumed to be spontaneously broken by the background spacetime, generates the associated Goldstone modes. In this paper we construct a suitable classical action for those Goldstone modes, and show that all the momentum modes experience nearly the same inverted harmonic potential, leading to an instability. Thank…
We explicitly construct all stationary, non-static, extremal near horizon geometries in $D$ dimensions that satisfy the vacuum Einstein equations, and that have $D-3$ commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in $D=4,5$. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology $S^2 \times T^{D-4}$, or $S^3 \times T…
