Center for Joint Quantum Studies, Department of Physics, School of Science, Tianjin University（天津大学理学院物理系联合量子研究中心）
Pujian Mao
备注：
v3: significant improvement on the clarity of the interpretation, detailed symmetry generators from light-ray operator presented at the linearized level
Supertranslations are usually defined as asymptotic symmetries associated with spacetime boundaries, such as null infinity and black hole horizons. In this Letter, we show that supertranslations admit a natural, coordinate-independent extension into the bulk of spacetime, realized as transitions between families of null hypersurfaces. This construction applies to generic spacetimes admitting null boundaries with residual symmetries and unifies the realizations of supertranslations at null infinity and finite-distance null hypersurfaces such as black hole horizons. The bulk supertranslation is connected to boundary supertranslation by characteristic flows. The associated symmetry algebra at the linearized level can be realized by light-ray operators defined on the null hypersurface and the bulk supertranslation acts as a zero-mode operator in the context of light-cone quantization. Within this framework, the gravitational wave memory effect corresponds to a shift of null hypersurfaces in the bulk. As explicit examples, we compute bulk supertranslations in Minkowski spacetime and four-dimensional Schwarzschild spacetime, where we uncover a novel curvature-dependent memory effect with observable consequences for light propagation.
We study several properties of some new charges of asymptotically flat spacetimes. These dual supertranslation charges are akin to the magnetic large $U(1)$ charges in QED. In this paper we find the symmetries associated with these charges and show that the global dual supertranslation charge is topological because it is invariant under globally defined, smooth variations of the asymptotic metric. We also exhibit spacetimes where the charge does not vanish and we find dynamical processes that interpolate between regions with different values of these charges.
The memory effect is a prediction of general relativity on the same footing as the existence of gravitational waves. The memory effect is understood at future null infinity as a transition induced by null radiation from a Poincaré vacuum to another vacuum. Those are related by a supertranslation, which is a fundamental symmetry of asymptotically flat spacetimes. In this essay, I argue that finite supertranslation diffeomorphisms should be extended into the bulk spacetime consistently with canonical charge conservation. It then leads to fascinating geometrical features of gravitational Poincaré…
Boundary conditions for Ashtekar-Barbero variables in the context of asymptotically flat spacetimes which lead to supertranslations at spatial infinity
A Covariant Formulation of Logarithmic Supertranslations at Spatial Infinity
University of Waterloo（滑铁卢大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；SISSA, International School for Advanced Studies（国际高等研究院）；INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Collège de France（法兰西学院）
It is well known that the memory effect in flat spacetime is parametrized by the BMS supertranslation. We investigate the relation between the memory effect and diffeomorphism in de Sitter spacetime. We find that gravitational memory is parametrized by a BMS-like supertranslation in the static patch of de Sitter spacetime. We also show a diffeomorphism that corresponds to gravitational memory in the Poincare/cosmological patch. Our method does not need to assume the separation between the source and the detector to be small compared with the Hubble radius, and can potentially be applicable to …
Supertranslations, and at least in 2+1 dimensions superrotations, are asymptotic symmetries of the metric in asymptotically flat spacetimes. They are not, however, symmetries of the boundary term of the Einstein-Hilbert action, which therefore induces an action for the Goldstone-like fields that parametrize these symmetries. I show that in 2+1 dimensions, this action is closely related to a chiral Liouville action, as well as the "Schwarzian" action that appears in two-dimensional near-AdS physics.
It is shown that black hole spacetimes in classical Einstein gravity are characterized by, in addition to their ADM mass $M$, momentum $\vec P$, angular momentum $\vec J$ and boost charge $\vec K$, an infinite head of supertranslation hair. The distinct black holes are distinguished by classical superrotation charges measured at infinity. Solutions with supertranslation hair are diffeomorphic to the Schwarzschild spacetime, but the diffeomorphisms are part of the BMS subgroup and act nontrivially on the physical phase space. It is shown that a black hole can be supertranslated by throwing in a…
We review recent results on symmetries of asymptotically flat spacetimes at null infinity. In higher dimensions, the symmetry algebra realizes the Poincaré algebra. In three and four dimensions, besides the infinitesimal supertranslations that have been known since the sixties, the algebras are evenly balanced because there are also infinitesimal superrotations. We provide the classification of central extensions of the bms3 and bms4 algebras. Applications and consequences as well as directions for future work are briefly indicated.
Asymptotic symmetries of theories with gravity in d=2m+2 spacetime dimensions are reconsidered for m>1 in light of recent results concerning d=4 BMS symmetries. Weinberg's soft graviton theorem in 2m+2 dimensions is re-expressed as a Ward identity for the gravitational S-matrix. The corresponding asymptotic symmetries are identified with 2m+2-dimensional supertranslations. An alternate derivation of these asymptotic symmetries as diffeomorphisms which preserve finite-energy boundary conditions at null infinity and act non-trivially on physical data is given. Our results differ from those of pr…
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 construct the phase space of 3-dimensional asymptotically flat spacetimes that forms the bulk metric representation of the BMS group consisting of both supertranslations and superrotations. The asymptotic symmetry group is a unique copy of the BMS group at both null infinities and spatial infinity. The BMS phase space obeys a notion of holographic causality and can be parametrized by boundary null fields. This automatically leads to the antipodal identification of bulk fields between past and future null infinity in the absence of a global conical defect.
A method for detecting gravitational memory is proposed. It makes use of ingoing null geodesics instead of timelike geodesics in the original formulation by Christodoulou. It is argued that the method is applicable in the bulk of a spacetime. In addition, it is shown that BMS symmetry generators in Newman-Unti gauge have an interpretation in terms of the memory effect. This generalises the connection between BMS supertranslations and gravitational memory, discovered by Strominger and Zhiboedov at null infinity, to the bulk.
It is argued that the symmetry algebra of asymptotically flat spacetimes at null infinity in 4 dimensions should be taken as the semi-direct sum of supertranslations with infinitesimal local conformal transformations and not, as usually done, with the Lorentz algebra. As a consequence, two dimensional conformal field theory techniques will play as fundamental a role in this context of direct physical interest as they do in three dimensional anti-de Sitter gravity.
The symmetry algebra of asymptotically flat spacetimes at null infinity in three dimensions is the semi-direct sum of the infinitesimal diffeomorphisms on the circle with an abelian ideal of supertranslations. The associated charge algebra is shown to admit a non trivial classical central extension of Virasoro type closely related to that of the anti-de Sitter case.
ENS de Lyon（里昂高等师范学院）；CNRS, LPENSL, UMR 5672（法国国家科学研究中心，理论物理与统计实验室，联合研究单位5672）；Center for Joint Quantum Studies, Department of Physics, School of Science, Tianjin University（天津大学理学院物理系联合量子研究中心）
The asymptotic structure of the Pauli-Fierz theory at spatial infinity is investigated in four spacetime dimensions. Boundary conditions on the massless spin-$2$ field that are invariant under an infinite-dimensional group of non-trivial ``improper'' gauge symmetries are given. The compatibility of these boundary conditions with invariance of the theory under Lorentz boosts is a subtle issue which is investigated in depth and leads to the identification of the improper gauge symmetries with the pure BMS supertranslations. It is also shown how rigid Poincaré transformations and improper gauge s…
We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of asymptotically-flat spacetimes. These are spacetimes that obey the Ashtekar-Hansen definition of asymptotic flatness at null and spatial infinity and satisfy an additional set of conditions which we lay out explicitly. Combined with earlier results on the matching of supertranslation charges, this shows that all Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacet…
After a review of symmetries and classical solutions involved in the AdS3/CFT2 correspondence, we apply a similar analysis to asymptotically flat spacetimes at null infinity in 3 and 4 dimensions. In the spirit of two dimensional conformal field theory, the symmetry algebra of asymptotically flat spacetimes at null infinity in 4 dimensions is taken to be the semi-direct sum of supertranslations with infinitesimal local conformal transformations and not, as usually done, with the Lorentz algebra. As a first application, we derive how the symmetry algebra is realized on solution space. In partic…
We elaborate on the Green's functions that appeared in [1,2] when generalizing, from massless to massive particles, various equivalences between soft theorems and Ward identities of large gauge symmetries. We analyze these Green's functions in considerable detail and show that they form a hierarchy of functions which describe `boundary to bulk' propagators for large $U(1)$ gauge parameters, supertranslations and sphere vector fields respectively. As a consistency check we verify that the Green's functions associated to the large diffeomorphisms map the Poincare group at null infinity to the Po…
We show that for non-conformally flat bulk spacetime, there exist no bound modes for zero mass graviton on the 3-brane. The brane world model is therefore unstable for the bulk spacetime being different from the conformally flat anti - de Sitter space.
We show that the BMS-supertranslations and their associated supermomenta on past null infinity can be related to those on future null infinity, proving the conjecture of Strominger for a class of spacetimes which are asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical 3-manifold of both null and spatial directions of approach towards spatial infinity, we impose appropriate regularity conditions on the Weyl tensor near spatial infinity along null directions. The asymptotic Einstein equations on this 3-manifold and the regularity conditions imply that the relevant Weyl t…
We use geodesic probes to recover the entire bulk metric in certain asymptotically AdS spacetimes. Given a spectrum of null geodesic endpoints on the boundary, we describe two remarkably simple methods for recovering the bulk information. After examining the issues which affect their application in practice, we highlight a significant advantage one has over the other from a computational point of view, and give some illustrative examples. We go on to consider spacetimes where the methods cannot be used to recover the complete bulk metric, and demonstrate how much information can be recovered i…
We argue that the flat spacetime with inexact quantum mechanics in it is dual to the de Sitter spacetime with exact quantum mechanics in it, and the positive cosmological constant of this de Sitter spacetime is in the second order of the degree of the violation of the bulk quantum mechanics in the flat spacetime. The flat spacetime is holographic and has a dual time-contracted boundary conformal field theory with two redefined central charges at null infinity. The vanishing smallness of the observed positive cosmological constant suggests the extraordinary exactness of the bulk quantum mechani…
Light-cone cuts have recently been proposed as a method to reconstruct the conformal metric of a holographic spacetime. We explore how additional information about the bulk geometry gets encoded in the structure of these light-cone cuts. In particular, we study how the hyperbolic angle related to a cusp in the light-cone cut encodes information about the matter content of the spacetime. We provide an explicit numerical example reconstructing the metric for a 4- dimensional spacetime composed by the superposition of a boson star and a gas of radiation in AdS.
Effective gravitational field equations on a brane have been derived, when the bulk spacetime is endowed with the second rank antisymmetric Kalb-Ramond field. Since both the graviton and the Kalb-Ramond field are closed string excitations, they can propagate in the bulk. After deriving the effective gravitational field equations on the brane, we solve them for a static spherically symmetric solution. It turns out that the solution so obtained represents a black hole or naked singularity depending on the parameter space of the model. The stability of this model is also discussed. Cosmological s…
We investigate the details of the bulk-boundary correspondence in Lorentzian signature anti-de Sitter space. Operators in the boundary theory couple to sources identified with the boundary values of non-normalizable bulk modes. Such modes do not fluctuate and provide classical backgrounds on which bulk excitations propagate. Normalizable modes in the bulk arise as a set of saddlepoints of the action for a fixed boundary condition. They fluctuate and describe the Hilbert space of physical states. We provide an explicit, complete set of both types of modes for free scalar fields in global and Po…
Holographic relationships between entanglement entropy on the boundary of a spacetime and the area of minimal surfaces in the bulk provide an important entry in the bulk/boundary dictionary. While constructing the necessary causal and entanglement wedges is well understood in asymptotically AdS spacetimes, less is known about the equivalent constructions in spacetimes with different asymptotics. In particular, recent attempts to construct entanglement and causal wedges for asymptotically Lifshitz solutions in relativistic gravitational theories have proven problematic. We note a simple observa…
In this work we present a "modest" holographic reconstruction of the bulk geometry in asymptotically flat spacetime using the two-point correlators of boundary quantum field theory (QFT) in asymptotically flat spacetime. The boundary QFT lives on the null boundary of the spacetime, namely null infinity and/or the Killing horizons. The bulk reconstruction relies on two unrelated results: (i) there is a bulk-to-boundary type correspondence between free quantum fields living in the bulk manifold and free quantum fields living on its null boundary, and (ii) one can construct the metric by making u…
It is argued that free QFT can be defined on the event horizon of a Schwarzschild-like spacetime and that this theory is unitarily and algebraically equivalent to QFT in the bulk (near the horizon). Under that unitary equivalence the bulk hidden SL(2,R) symmetry found in a previous work becomes manifest on the event horizon, it being induced by a group of horizon diffeomorphisms. The class of generators of that group can be enlarged to include a full Virasoro algebra of fields which are defined on the event horizon. These generators have a quantum representation in QFT on the event horizon and…
Motivated by the need for further insight into the emergence of AdS bulk spacetime from CFT degrees of freedom, we explore the behaviour of probes represented by specific geometric quantities in the bulk. We focus on geodesics and n-dimensional extremal surfaces in a general static asymptotically AdS spacetime with spherical and planar symmetry, respectively. While our arguments do not rely on the details of the metric, we illustrate some of our findings explicitly in spacetimes of particular interest (specifically AdS, Schwarzschild-AdS and extreme Reissner-Nordstrom-AdS). In case of geodesic…
Null infinity in asymptotically flat spacetimes posses a rich mathematical structure; including the BMS group and the Bondi news tensor that allow one to study gravitational radiation rigorously. However, FLRW spacetimes are not asymptotically flat because their stress-energy tensor does not decay sufficiently fast and in fact diverges at null infinity. This class includes matter- and radiation-dominated FLRW spacetimes. We define a class of spacetimes whose structure at null infinity is similar to FLRW spacetimes: the stress-energy tensor is allowed to diverge and the conformal factor is not …
The butterfly velocity characterizes the spread of correlations in a quantum system. Recent work has provided a method of calculating the butterfly velocity of a class of boundary operators using holographic duality. Utilizing this and a presumed extension of the canonical holographic correspondence of AdS/CFT, we investigate the butterfly velocities of operators with bulk duals living in general spacetimes. We analyze some ubiquitous issues in calculating butterfly velocities using the bulk effective theory, and then extend the previously proposed method to include operators in entanglement s…
We explore the description of bulk causal structure in a dual field theory. We observe that in the spacetime dual to a spacelike non-commutative field theory, the causal structure in the boundary directions is modified asymptotically. We propose that this modification is described in the dual theory by a modification of the micro-causal light cone. Previous studies of this micro-causal light cone for spacelike non-commutativite field theories agree with the expectations from the bulk spacetime. We describe the spacetime dual to field theories with lightlike non-commutativity, and show that the…
We argue that classical spacetimes represent amplified information in the holographic theory of quantum gravity. In general, classicalization of a quantum system involves amplification of information at the cost of exponentially reducing the number of observables. In quantum gravity, the geometry of spacetime must be the analogously amplified information. Bulk local semiclassical operators probe this information without disturbing it; these correspond to logical operators acting on code subspaces of the holographic theory. From this viewpoint, we study how bulk local operators may be realized …
In this work we elaborate on an extension of the AdS/CFT framework to a subclass of gravitational theories with vanishing cosmological constant. By building on earlier ideas, we construct a correspondence between Ricci-flat spacetimes admitting asymptotically hyperbolic hypersurfaces and a family of conformal field theories on a codimension two manifold at null infinity. By truncating the gravity theory to the pure gravitational sector, we find the most general spacetime asymptotics, renormalize the gravitational action, reproduce the holographic stress tensors and Ward identities of the famil…
We examine the five-region holographic entropy cone inequalities for the special case of the AdS$_3$-Vaidya metric for a variety of boundary configurations. This is done by numerically solving the geodesic equation in the bulk for various boundary configurations. In all the cases we examine, we find that all the inequalities are satisfied when the bulk satisfies the null energy condition, while the inequalities are all violated when the bulk spacetime violates the null energy condition. A proof of the five-region holographic entropy cone inequalities for the dynamical bulk case remains an open…
The backreaction on the Randall-Sundrum warped spacetime is determined in presence of scalar field in the bulk. A general analysis shows that the stability of such a model can be achieved only if the scalar field action has non-canonical higher derivative terms. It is further shown that the gauge hierarchy problem can be resolved in such a stabilized scenario by appropriate choice of various parameters of the theory. The effective cosmological constant on the brane is shown to vanish.
We study the covariant phase space of vacuum general relativity at the null boundary of causal diamonds. The past and future components of such a null boundary each have an infinite-dimensional symmetry algebra consisting of diffeomorphisms of the $2$-sphere and boost supertranslations corresponding to angle-dependent rescalings of affine parameter along the null generators. Associated to these symmetries are charges and fluxes obtained from the covariant phase space formalism using the prescription of Wald and Zoupas. By analyzing the behavior of the spacetime metric near the corners of the c…
As an extension of arXiv:\{2002.02448, 2008.10740\} we present a physical protocol that a set of observers can use to detect a pure state in the bulk when they are spread across a small cut near $\mathcal{I}^+_-$ in flat spacetime. The protocol involves the modification of a bulk state using simple unitary operators and measurements of the energy of the state. The states that we study are constructed by acting with low energy operators on a vacuum state such that a perturbative analysis is valid. We restrict ourselves to $3+1$ dimensional spacetimes and only consider massless excitations. From…
The quantization of a scalar field in anti de Sitter spacetime using Poincaré coordinates is considered. We find a discrete spectrum that is consistent with a possible mapping between bulk and boundary quantum states.
We use the complexity equals action proposal to calculate the rate of complexity growth for field theories that are the holographic duals of asymptotically flat spacetimes. To this aim, we evaluate the on-shell action of asymptotically flat spacetime on the Wheeler-DeWitt patch. This results in the same expression as can be found by taking the flat-space limit from the corresponding formula related to the asymptotically AdS spacetimes. For the bulk dimensions that are greater than three, the rate of complexity growth at late times approaches from above to Lloyd's bound. However, for the three-…
The effective gravitational field equations on and off a 3-brane world possessing a Z_{2} mirror symmetry and embedded in a five-dimensional bulk spacetime with cosmological constant were derived by Shiromizu, Maeda and Sasaki (SMS) in the framework of the Gauss-Codazzi projective approach with the subsequent specialization to the Gaussian normal coordinates in the neighborhood of the brane. However, the Gaussian normal coordinates imply a very special slicing of spacetime and clearly, the consistent analysis of the brane dynamics would benefit from complete freedom in the slicing of spacetime…
We write down exact solutions in the collective field theory of the c=1 matrix model and in dilaton-gravity coupled to a massless scalar. Using the known correspondence between these two theories at the null boundaries of spacetime, we make a connection between scalar fields in these two theories in the bulk of spacetime. In the process, we gain insight into how a theory containing gravity can be equivalent to one without gravity. We analyze a simple time-dependent background as an example.
In this paper we present a general asymmetric brane model involving arbitrary energy transport to and from an embedded 4-D FRW universe. We derive a locally defined mass function for the 5D spacetime and describe its time evolution on the brane. We then specialise our model to the two cases of graviton production in the early universe and radiating black holes in the bulk.
