Timelike Boundary Research Articles

Moving holographic boundaries

In this paper, we show that for one sign of the deformation coupling single-trace TT‾ deformation moves the holographic screen in Gödel universe radially inward. For the other sign of the coupling it moves the holographic screen radially outward. We (thus) argue, on general grounds, that in holography (single-trace) TT‾ deformation can be generally thought of as either moving the holographic boundary into the bulk or washing it away to infinity. We explain in what sense. In Anti-de Sitter this breaks the spacetime conformal symmetry. We further note that moving timelike holographic boundary into bulk creates (at onset) a curvature singularity. In the boundary the singularity is understood by states with imaginary energies. To define or make the bulk theory sensible we introduce an ultraviolet cutoff and thereby move the boundary into the bulk. However, it is not clear whether suitable boundary conditions that lead to a consistent string theory exist. In this paper we first obtain the Penrose limit of the single-trace TT‾ deformed string background and then perform T-duality along a space-like isometry to obtain a class of deformed Gödel universes. The string background we consider is AdS3×S3×M4. The single-trace TT‾ deformation is a particular example of the more general O(d,d) transformations.

Boundary Structure of the Standard Model Coupled to Gravity

AbstractIn this article a description of the reduced phase space of the standard model coupled to gravity is given. For space or time-like boundaries this is achieved as the reduction of a symplectic space with respect to a coisotropic submanifold and with the BFV formalism. For light-like boundaries the reduced phase space is described as the reduction of a symplectic manifold with respect to a set of constraints. Some results about the Poisson brackets of sums of functionals are also proved.

A new 2+1 coherent spin-foam vertex for quantum gravity

Abstract We propose an explicit spin-foam amplitude for Lorentzian gravity in three dimensions, allowing for both space- and time-like boundaries. The model is based on two main requirements: that it should be structurally similar to its well-known Euclidean analog, and that geometricity should be recovered in the semiclassical regime. To this end we introduce new coherent states for space-like boundary edges, derived from the continuous series of unitary SU ( 1 , 1 ) representations. We show that the relevant objects in the amplitude can be written in terms of the defining representation of the group, just as so happens in the Euclidean case. We derive an expression for the semiclassical amplitude at large spins, showing it relates to the Lorentzian Regge action.

On the interplay between boundary conditions and the Lorentzian Wetterich equation

In the framework of the functional renormalization group and of the perturbative, algebraic approach to quantum field theory (pAQFT), in D’Angelo et al. [Ann. Henri Poinc. 25 (2024) 2295–2352] it has been derived a Lorentzian version of a flow equation à la Wetterich, which can be used to study nonlinear, quantum scalar field theories on a globally hyperbolic spacetime. In this work, we show that the realm of validity of this result can be extended to study interacting scalar field theories on globally hyperbolic manifolds with a timelike boundary. By considering the specific examples of half-Minkowski spacetime and of the Poincaré patch of Anti-de Sitter, we show that the form of the Lorentzian Wetterich equation is strongly dependent on the boundary conditions assigned to the underlying field theory. In addition, using a numerical approach, we are able to provide strong evidences that there is a qualitative and not only a quantitative difference in the associated flow and we highlight this feature by considering Dirichlet and Neumann boundary conditions on half-Minkowski spacetime.

Cosmological observatories

Abstract We study the static patch of de Sitter space in the presence of a timelike boundary. We impose that the conformal class of the induced metric and the trace of the extrinsic curvature, K, are fixed at the boundary. We present the thermodynamic structure of de Sitter space subject to these boundary conditions, for static and spherically symmetric configurations to leading order in the semiclassical approximation. In three spacetime dimensions, and taking K constant on a toroidal Euclidean boundary, we find that the spacetime is thermally stable for all K. In four spacetime dimensions, the thermal stability depends on the value of K. It is established that for sufficiently large K, the de Sitter static patch subject to conformal boundary conditions is thermally stable. This contrasts the Dirichlet problem for which the region encompassing the cosmological horizon has negative specific heat. We present an analysis of the linearised Einstein equations subject to conformal boundary conditions. In the worldline limit of the timelike boundary, the underlying modes are linked to the quasinormal modes of the static patch. In the limit where the timelike boundary approaches the cosmological event horizon, the linearised modes are interpreted in terms of the shear and sound modes of a fluid dynamical system. Additionally, we find modes with a frequency of positive imaginary part. Measured in a local inertial reference frame, and taking the stretched cosmological horizon limit, these modes grow at most polynomially.

Heisenberg soft hair on Robinson-Trautman spacetimes

We study 4 dimensional (4d) gravitational waves (GWs) with compact wavefronts, generalizing Robinson-Trautman (RT) solutions in Einstein gravity with an arbitrary cosmological constant. We construct the most general solution of the GWs in the presence of a causal, timelike, or null boundary when the usual tensor modes are turned off. Our solution space besides the shape and topology of the wavefront which is a generic compact, smooth, and orientable 2d surface Σ, is specified by a vector over Σ satisfying the conformal Killing equation and two scalars that are arbitrary functions over the causal boundary, the boundary modes (soft hair). We work out the symplectic form over the solution space using covariant phase space formalism and analyze the boundary symmetries and charges. The algebra of surface charges is a Heisenberg algebra. Only the overall size of the compact wavefront and not the details of its shape appears in the boundary symplectic form and is canonical conjugate to the overall mass of the GW. Hence, the information about the shape of the wavefront can’t be probed by the boundary observer. We construct a boundary energy-momentum tensor and a boundary current, whose conservation yields the RT equation for both asymptotically AdS and flat spacetimes. The latter provides a hydrodynamic description for our RT solutions.

Quantum approach to bound states in field theory

It is well known that (possibly nonunique) suitable field dynamics can be prescribed in spacetimes with timelike boundaries by means of appropriate boundary conditions. In [R. M. Wald, ], Wald derived a conserved energy functional for each prescribed dynamics. This conserved energy is related to the positive self-adjoint extensions of the spatial part A of the wave equation ∂2Φ/∂t2=−AΦ (A may not be, in principle, essentially self-adjoint). This is quite surprising since the canonical energy is not conserved in these cases. In this paper, we rederive this energy functional from an action principle (with appropriate boundary terms) following [A. A. Saharian, ] and consider field dynamics arising from nonpositive self-adjoint extensions of A. The spectrum of the resulting theory fails to be positive and unstable mode solutions for classical fields come to light. By studying fields in half-Minkowski spacetime, we illustrate that these unstable classical solutions come as a consequence of an inverted parabolic potential governing their dynamics. From the quantum mechanical point of view, this leads to an effective inverted harmonic oscillator at the boundary. We then explore these unstable modes behavior, as well as their instabilities, at the quantum level. Published by the American Physical Society 2024

4D spherically symmetric time-dependent quantum gravity amplitudes

In these short notes, we compute non-perturbatively the time-dependent quantum gravity amplitudes for a four-dimensional spherically symmetric space-time with space-like and time-like boundaries. We solve the 4D classical and quantum constraints in a novel way. We identify the classical solution of the constraints as a canonical transformation, where the integration constants are the new variables. We apply this canonical transformation to the path integral representation of the amplitudes we are interested in. We use both, the canonical and the path integral formalism. This procedure allows us to reduce the action to the actual degree of freedom of the theory. In the end, we get the time-dependent amplitudes from the path integral without performing an actual path integration. From these amplitudes, we show that for most of the boundary conditions time evolution in quantum gravity is non-unitary. There is however a special case where unitary evolution could be achieved.

Lorentzian Calderón problem near the Minkowski geometry

We study a Lorentzian version of the well-known Calderón problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map on its timelike boundary. In the earlier work of the authors it was shown that zeroth order coefficients can be uniquely determined under a two-sided spacetime curvature bound and the additional assumption that there are no conjugate points along null or spacelike geodesics. In this paper we show that uniqueness for the zeroth order coefficient holds for manifolds satisfying a weaker curvature bound and for spacetime perturbations of such manifolds. This relies on a new enhanced optimal unique continuation principle for the wave equation in the exterior regions of double null cones. In particular, we solve the Lorentzian Calderón problem near the Minkowski geometry.

On the Cauchy problem for the Faraday tensor on globally hyperbolic manifolds with timelike boundary

We study the well-posedness of the Cauchy problem for the Faraday tensor on globally hyperbolic manifolds with timelike boundary. The existence of Green operators for the operator \mathrm{d}+\delta and a suitable pre-symplectic structure on the space of solutions are discussed.

Gravitational observatories

We consider four-dimensional general relativity with vanishing cosmological constant defined on a manifold with a boundary. In Lorentzian signature, the timelike boundary is of the form σ × ℝ, with σ a spatial two-manifold that we take to be either flat or S2. In Euclidean signature we take the boundary to be S2 × S1. We consider conformal boundary conditions, whereby the conformal class of the induced metric and trace K of the extrinsic curvature are fixed at the timelike boundary. The problem of linearised gravity is analysed using the Kodama-Ishibashi formalism. It is shown that for a round metric on S2 with constant K, there are modes that grow exponentially in time. We discuss a method to control the growing modes by varying K. The growing modes are absent for a conformally flat induced metric on the timelike boundary. We provide evidence that the Dirichlet problem for a spherical boundary does not suffer from non-uniqueness issues at the linearised level. We consider the extension of black hole thermodynamics to the case of conformal boundary conditions, and show that the form of the Bekenstein-Hawking entropy is retained.

Boundary conditions and infrared divergences

We review the procedure to construct quasi-free ground states, for real scalar fields whose dynamics is dictated by the Klein-Gordon equation, on standard static Lorentzian manifolds with a time-like boundary. We observe that, depending on the assigned boundary condition of Robin type, this procedure does not always lead to the existence of a suitable bi-distribution ω2∈D′(M×M) due to the presence of infrared divergences. As a concrete example we consider a Bertotti-Robinson spacetime in two different coordinate patches. In one case we show that infrared divergences do not occur only for Dirichlet boundary conditions as one might expect a priori, while, in the other case, we prove that they occur only when Neumann boundary conditions are imposed at the time-like boundary.

A half de Sitter holography

A long-standing and intriguing question is: does the holographic principle apply to cosmologies like de Sitter spacetime? In this work, we consider a half dS spacetime wherein a timelike boundary encloses the bulk spacetime, presenting a version of de Sitter holography. By analyzing the holographic entanglement entropy in this space and comparing it with that in AdS/CFT, we argue that gravity on a half dSd+1 is dual to a highly non-local field theory residing on dSd boundary. This non-locality induces a breach in the subadditivity of holographic entanglement entropy. Remarkably, this observation can be linked to another argument that time slices in global de Sitter space overestimate the degrees of freedom by redundantly counting the same Hilbert space multiple times.

Temporal vs spatial conservation and memory effect in electrodynamics

We consider the standard Maxwell’s theory in 1+3 dimensions in the presence of a timelike boundary. In this context, we show that (generalized) Ampere-Maxwell’s charge appears as a Noether charge associated with the Maxwell U(1) gauge symmetry which satisfies a spatial conservation equation. Furthermore, we also introduce the notion of spatial memory field and its corresponding memory effect. Finally, similar to the temporal case through the lens of Strominger’s triangle proposal, we show how spatial memory and conservation are related.

Black hole to cosmic horizon microstates in string/M theory: timelike boundaries and internal averaging

In this note, we resolve an apparent obstacle to string/M theory realizations of dS observer patch holography, finding a new role for averaging in quantum gravity. The solvable Toverline{T} (+Λ2) deformation recently provided a detailed microstate count of the dS3 cosmic horizon, reproducing the refined Gibbons-Hawking entropy computed by Anninos et al. along with the correct radial bulk geometry. On the gravity side, the deformation brings in the boundary to just outside a black hole horizon, where it is indistinguishable from the dS cosmic horizon, enabling a continuous passage to a bounded patch of dS. In string/M theory, the relationship between AdS/CFT and dS involves uplifts that change the internal topology, e.g. replacing an internal sphere mathbbm{S} with an internal hyperbolic space ℍ (and incorporating varying warp and conformal factors). We connect these two approaches, noting that the differences in the extra dimensions between AdS black hole and dS solutions are washed out by internal averaging in the presence of a timelike boundary skirting the horizon. This helps to motivate a detailed investigation into the possibility of such timelike boundaries in (A)dS solutions of string/M theory, and we take initial steps toward suitable generalizations of Liouville walls as one approach.

Stars and junction conditions in Einstein–Gauss–Bonnet gravity

The junction conditions for a higher dimensional spherically symmetric charged and anisotropic static star are derived in Einstein–Gauss–Bonnet (EGB) gravity with nonvanishing cosmological constant. It is shown that for a timelike boundary hypersurface of zero thickness, the generalised matching conditions across this surface in EGB gravity are satisfied. A sufficient condition is that the Israel-Darmois conditions are valid. Therefore it is possible to generate a complete stellar model in EGB gravity. The interior matches to the exterior higher dimensional charged Boulware–Deser spacetime with cosmological constant. The barotropic radial pressure has to vanish at the boundary of the star which is also the case in general relativity.

Hidden freedom in the mode expansion on static spacetimes

We review the construction of ground states focusing on a real scalar field whose dynamics is ruled by the Klein–Gordon equation on a large class of static spacetimes. As in the analysis of the classical equations of motion, when enough isometries are present, via a mode expansion the construction of two-point correlation functions boils down to solving a second order, ordinary differential equation on an interval of the real line. Using the language of Sturm–Liouville theory, most compelling is the scenario when one endpoint of such interval is classified as a limit circle, as it often happens when one is working on globally hyperbolic spacetimes with a timelike boundary. In this case, beyond initial data, one needs to specify a boundary condition both to have a well-defined classical dynamics and to select a corresponding ground state. Here, we take into account boundary conditions of Robin type by using well-known results from Sturm–Liouville theory, but we go beyond the existing literature by exploring an unnoticed freedom that emerges from the intrinsic arbitrariness of secondary solutions at a limit circle endpoint. Accordingly, we show that infinitely many one-parameter families of sensible dynamics are admissible. In other words, we emphasize that physical constraints guaranteeing the construction of ground states do not, in general, fix one such state unambiguously. In addition, we provide, in full detail, an example on (1 + 1)-half Minkowski spacetime to spell out the rationale in a specific scenario where analytic formulae can be obtained.

Escaping the interiors of pure boundary-state black holes

We consider a class of pure black hole microstates and demonstrate that they can be made escapable by turning on certain double trace deformations in the CFT. These microstates are dual to BCFT states prepared via a Euclidean path integral starting from a boundary in Euclidean time. These states are dual to black holes in the bulk with an End-of-the-World brane; a codimension one timelike boundary of the spacetime behind the horizon. We show that by tuning the sign of the coupling of the double trace operator to the boundary conditions on the brane the deformation injects negative energy into the black hole causing a time advance for signals behind the horizon. We demonstrate how the property of escapability in the considered microstates follows immediately from the traversability of deformed wormholes. We briefly comment on reconstruction of the black hole interior and state dependence.

Strictification theorems for the homotopy time-slice axiom

It is proven that the homotopy time-slice axiom for many types of algebraic quantum field theories (AQFTs) taking values in chain complexes can be strictified. This includes the cases of Haag–Kastler-type AQFTs on a fixed globally hyperbolic Lorentzian manifold (with or without time-like boundary), locally covariant conformal AQFTs in two spacetime dimensions, locally covariant AQFTs in one spacetime dimension, and the relative Cauchy evolution. The strictification theorems established in this paper prove that, under suitable hypotheses that hold true for the examples listed above, there exists a Quillen equivalence between the model category of AQFTs satisfying the homotopy time-slice axiom and the model category of AQFTs satisfying the usual strict time-slice axiom.

Physical significance of generalized boundary conditions: An Unruh-DeWitt detector viewpoint on [formula omitted

On PAdS2×S2, we construct the two-point correlation functions for the ground and thermal states of a real Klein-Gordon field admitting generalized (γ,v)-boundary conditions. We follow the prescription recently outlined in [1] for two different choices of secondary solutions. For each of them, we obtain a family of admissible boundary conditions parametrized by γ∈[0,π2]. We study how they affect the response of a static Unruh-DeWitt detector. The latter not only perceives variations of γ, but also distinguishes between the two families of secondary solutions in a qualitatively different, and rather bizarre, fashion. Our results highlight once more the existence of a freedom in choosing boundary conditions at a timelike boundary which is often neglected, though it has a notable associated physical significance.