Constant Time Slices Research Articles

Modular conjugations in 2D conformal field theory and holographic bit threads

We study the geometric action of some modular conjugations in two dimensional (2D) conformal field theories. We investigate the bipartition given by an interval when the system is in the ground state, either on the line or on the circle, and in the thermal Gibbs state on the line. We find that the restriction of the corresponding inversion maps to a spatial slice is obtained also in the gauge/gravity correspondence through the geodesic bit threads in a constant time slice of the dual static asymptotically AdS background. For a conformal field theory in the thermal state on the line, the modular conjugation suggests the occurrence of a second world which can be related through the geodesic bit threads to the horizon of the BTZ black brane background. An inversion map is constructed also for the massless Dirac fermion in the ground state and on the line bipartite by the union of two disjoint intervals.

Checking volume-law entropy with Hubeny-Rangamani-Takayanagi surfaces

We check formally that the Hubeny-Rangamani-Takayanagi prescription for holographic entanglement entropy -- when applied to a static black brane spacetime and to a wide class of subregions that do not lie on a constant time slice -- gives rise to volume-law entropy in the limit of large subregion. By volume-law entropy, we mean that the entanglement entropy scales with the volume of the projection of the boundary subregion onto a static time slice with respect to the boundary thermal state (with the same coefficient as in the volume law on the static time slice). Our result applies to subregions that have reflection symmetry as well as strips, and we also present field-theoretic arguments in support of our holographic findings.

On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization

In this work, we perform a general study of properties of a class of locally symmetric embedded hypersurfaces in spacetimes admitting a [Formula: see text] spacetime decomposition. The hypersurfaces are given by specifying the form of the Ricci tensor with respect to the induced metric. These are slices of constant time in the spacetime. First, the form of the Ricci tensor for general hypersurfaces is obtained and the conditions under which the general case reduces to those of constant time slices are specified. We provide a characterization of these hypersurfaces, with key physical quantities in the spacetime playing a role in specifying the local geometry of these hypersurfaces. Furthermore, we investigate the case where these hypersurfaces admit a Ricci soliton structure. The particular cases where the vector fields associated with the solitons are Killing or conformal Killing vector fields are analyzed. Finally, in the context of spacetimes with local rotational symmetry, it is shown that only spacetimes in this class with vanishing rotation and spatial twist can admit the hypersurface types considered, and that the hypersurfaces are necessarily flat. If such hypersurface does admit a Ricci soliton structure, the soliton is steady, with the components of the soliton field being constants.

Emergent phase space description of unitary matrix model

We show that large N phases of a 0 dimensional generic unitary matrix model (UMM) can be described in terms of topologies of two dimensional droplets on a plane spanned by eigenvalue and number of boxes in Young diagram. Information about different phases of UMM is encoded in the geometry of droplets. These droplets are similar to phase space distributions of a unitary matrix quantum mechanics (UMQM) ((0 + 1) dimensional) on constant time slices. We find that for a given UMM, it is possible to construct an effective UMQM such that its phase space distributions match with droplets of UMM on different time slices at large N . Therefore, large N phase transitions in UMM can be understood in terms of dynamics of an effective UMQM. From the geometry of droplets it is also possible to construct Young diagrams corresponding to U(N) representations and hence different large N states of the theory in momentum space. We explicitly consider two examples: single plaquette model with TrU2 terms and Chern-Simons theory on S3. We describe phases of CS theory in terms of eigenvalue distributions of unitary matrices and find dominant Young distributions for them.

Soliton Resolution for Equivariant Wave Maps on a Wormhole

In this paper, we initiate the study of finite energy equivariant wave maps from the (1+3)-dimensional spacetime $\mathbb R \times (\mathbb R \times \mathbb{S}^2) \rightarrow \mathbb{S}^3$ where the metric on $\mathbb R \times (\mathbb R \times \mathbb{S}^2)$ is given by ds^2 = -dt^2 + dr^2 + (r^2 + 1) \left ( d \theta^2 + \sin^2 \theta d \varphi^2 \right ), \quad t,r \in \mathbb{R}, (\theta,\varphi) \in \mathbb{S}^2. The constant time slices are each given by the Riemannian manifold $\mathcal M := \mathbb R \times \mathbb{S}^2$ with metric ds^2 = dr^2 + (r^2 + 1) \left ( d \theta^2 + \sin^2 \theta d \varphi^2 \right ). The Riemannian manifold $\mathcal M$ contains two asymptotically Euclidean ends at $r \rightarrow \pm \infty$ that are connected by a spherical throat of area $4 \pi^2$ at $r = 0$. The spacetime $\mathbb R \times \mathcal M$ is a simple example of a wormhole geometry in general relativity. In this work we will consider 1--equivariant or corotational wave maps. Each corotational wave map can be indexed by its topological degree $n$. For each $n$, there exists a unique energy minimizing corotational harmonic map $Q_{n} : \mathcal M \rightarrow \mathbb{S}^3$ of degree $n$. In this work, we show that modulo a free radiation term, every corotational wave map of degree $n$ converges strongly to $Q_{n}$. This resolves a conjecture made by Bizon and Kahl in the corotational case.

Holographic entanglement entropy on generic time slices

We study the holographic entanglement entropy and mutual information for Lorentz boosted subsystems. In holographic CFTs at zero and finite temperature, we find that the mutual information gets divergent in a universal way when the end points of two subsystems are light-like separated. In Lifshitz and hyperscaling violating geometries dual to non-relativistic theories, we show that the holographic entanglement entropy is not well-defined for Lorentz boosted subsystems in general. This strongly suggests that in non-relativistic theories, we cannot make a real space factorization of the Hilbert space on a generic time slice except the constant time slice, as opposed to relativistic field theories.

Tensor networks for dynamic spacetimes

Existing tensor network models of holography are limited to representing the geometry of constant time slices of static spacetimes. We study the possibility of describing the geometry of a dynamic spacetime using tensor networks. We find it is necessary to give a new definition of length in the network, and propose a definition based on the mutual information. We show that by associating a set of networks with a single quantum state and making use of the mutual information based definition of length, a network analogue of the maximin formula can be used to calculate the entropy of boundary regions.

Scalar self-force for highly eccentric equatorial orbits in Kerr spacetime

If a small "particle" of mass $\mu M$ (with $\mu \ll 1$) orbits a black hole of mass $M$, the leading-order radiation-reaction effect is an $\mathcal{O}(\mu^2)$ "self-force" acting on the particle, with a corresponding $\mathcal{O}(\mu)$ "self-acceleration" of the particle away from a geodesic. Such "extreme--mass-ratio inspiral" systems are likely to be important gravitational-wave sources for future space-based gravitational-wave detectors. Here we consider the "toy model" problem of computing the self-force for a scalar-field particle on a bound eccentric orbit in Kerr spacetime. We use the Barack-Golbourn-Vega-Detweiler effective-source regularization with a 4th order puncture field, followed by an $e^{im\phi}$ ("m-mode") Fourier decomposition and a separate time-domain numerical evolution in $2+1$ dimensions for each $m$. We introduce a finite worldtube that surrounds the particle worldline and define our evolution equations in a piecewise manner so that the effective source is only used within the worldtube. Viewed as a spatial region, the worldtube moves to follow the particle's orbital motion. We use slices of constant Boyer-Lindquist time in the region of the particle's motion, deformed to be asymptotically hyperboloidal and compactified near the horizon and $\mathcal{J}^+$. We present numerical results for a number of test cases with orbital eccentricities as high as $0.98$. In some cases we find large oscillations ("wiggles") in the self-force shortly after periastron passage.

Holographic de Sitter Geometry from Entanglement in Conformal Field Theory.

We demonstrate that, for general conformal field theories (CFTs), the entanglement for small perturbations of the vacuum is organized in a novel holographic way. For spherical entangling regions in a constant time slice, perturbations in the entanglement entropy are solutions of a Klein-Gordon equation in an auxiliary de Sitter (dS) spacetime. The role of the emergent timelike direction in dS spacetime is played by the size of the entangling sphere. For CFTs with extra conserved charges, e.g., higher-spin charges, we show that each charge gives rise to a separate dynamical scalar field in dS spacetime.

Entanglement entropy in a holographic Kondo model

We calculate entanglement and impurity entropies in a recent holographic model of a magnetic impurity interacting with a strongly coupled system. There is an RG flow to an IR fixed point where the impurity is screened, leading to a decrease in impurity degrees of freedom. This information loss corresponds to a volume decrease in our dual gravity model, which consists of a codimension one hypersurface embedded in a BTZ black hole background in three dimensions. There are matter fields defined on this hypersurface which are dual to Kondo field theory operators. In the large N limit, the formation of the Kondo cloud corresponds to the condensation of a scalar field. The entropy is calculated according to the Ryu‐Takayanagi prescription. This requires to determine the backreaction of the hypersurface on the BTZ geometry, which is achieved by solving the Israel junction conditions. We find that the larger the scalar condensate gets, the more the volume of constant time slices in the bulk is reduced, shortening the bulk geodesics and reducing the impurity entropy. This provides a new non‐trivial example of an RG flow satisfying the g‐theorem. Moreover, we find explicit expressions for the impurity entropy which are in agreement with previous field theory results for free electrons. This demonstrates the universality of perturbing about an IR fixed point.

Horizon straddling ISCOs in spherically symmetric string black holes

The causal geodesics in the equatorial plane of a static extremal charged black holes in heterotic string theory is examined with regard to their geodesic stability and compared with similar geodesics in the nonextremal situation. Extremization of the effective potential for timelike and null circular geodesics implies that in the extremal limit, the radius of ISCO (Inner-most Stable Circular Orbit) (r ISCO ), circular photon orbit (CPO) (r ph ) and marginally bound circular orbit (MBCO) (r mb ) are coincident with the event horizon (r hor ) i.e. r ISCO = r ph = r mb = r hor = 2M. Since the proper radial distance on a constant time slice both in Schwarzschild and Painlevé–Gullstrand (PG) coordinates become zero, thus these three orbits indeed coincide with the null geodesic generators of the event horizon. This strange behavior is quite different from the static, spherically symmetric extremal Reissner–Nordstrøm (RN) BH.

A note on AdS cosmology and gauge theory correlator

Using AdS/CFT prescription, we compute two point Yang-Mills correlator on a constant time slice for the Kasner background. Pushing the surface close to the initial singularity, we find, in some cases, the correlator does not develop pole. We further numerically calculate similar correlator where the bulk is a Kasner AdS soliton. We find that the qualitative behaviour of the correlator remains unchanged.

Construction of Cauchy data of vacuum Einstein field equations evolving to black holes

We show the existence of complete, asymptotically flat Cauchy initial data for the vacuum Einstein field equations, free of trapped surfaces, whose future development must admit a trapped surface. Moreover, the datum is exactly a constant time slice in Minkowski space-time inside and exactly a constant time slice in Kerr space-time outside. The proof makes use of the full strength of Christodoulou's work on the dynamical formation of black holes and Corvino-Schoen's work on the constructions of initial data set.

A simple family of analytical trumpet slices of the Schwarzschild spacetime

We describe a simple family of analytical coordinate systems for the Schwarzschild spacetime. The coordinates penetrate the horizon smoothly and are spatially isotropic. Spatial slices of constant coordinate time t feature a trumpet geometry with an asymptotically cylindrical end inside the horizon at a prescribed areal radius R0 (with 0 < R0 ⩽ M) that serves as the free parameter for the family. The slices also have an asymptotically flat end at spatial infinity. In the limit R0 = 0 the spatial slices lose their trumpet geometry and become flat—in this limit, our coordinates reduce to Painlevé–Gullstrand coordinates.

Fluctuating geometries,q-observables, and infrared growth in inflationary spacetimes

Infrared growth of geometrical fluctuations in inflationary spacetimes is investigated. The problem of gauge-invariant characterization of growth of perturbations, which is of interest also in other spacetimes such as black holes, is addressed by studying evolution of the lengths of curves in the geometry. These may either connect freely falling ``satellites,'' or wrap nontrivial cycles of geometries like the torus, and are also used in diffeomorphism-invariant constructions of two-point functions of field operators. For spacelike separations significantly exceeding the Hubble scale, no spacetime geodesic connects two events, but one may find geodesics constrained to lie within constant-time spatial slices. In inflationary geometries, metric perturbations produce significant and growing corrections to the lengths of such geodesics, as we show in both quantization on an inflating torus and in standard slow-roll inflation. These become large, signaling breakdown of a perturbative description of the geometry via such observables, and consistent with perturbative instability of de Sitter space. In particular, we show that the geodesic distance on constant time slices during inflation becomes nonperturbative a few e-folds after a given scale has left the horizon, by distances $\ensuremath{\sim}1/{H}^{3}\ensuremath{\sim}RS$, obstructing use of such geodesics in constructing IR-safe observables based on the spatial geometry. We briefly discuss other possible measures of such geometrical fluctuations.

Holographic horizons

We examine how the $(2+1)$-dimensional anti-de Sitter space is covered by the Fefferman-Graham system of coordinates for Minkowski, Rindler, and static de Sitter boundary metrics. We show that, in the last two cases, the coordinates do not cover the full anti-de Sitter space. On a constant-time slice, the line delimiting the excluded region has endpoints at the locations of the horizons of the boundary metric. Its length, after an appropriate regularization, reproduces the entropy of the dual conformal field theory on the boundary background. The horizon can be interpreted as the holographic image of the line segment delimiting the excluded region in the vicinity of the boundary.

Q-Fuzzy Spheres and Quantum Differentials on B q [SU 2] and U q (su 2)

We provide a new unified construction of the two-parameter Podleś two-spheres as characterised by a projector e with trace q (e) = 1 + λ. In our formulation the limit in which q → 1 with λ fixed is the fuzzy sphere, while the limit λ → 0 with q fixed is the standard q-deformed sphere. We show further that the non-standard Podleś spheres arise geometrically as ‘constant time slices’ of the unit hyperboloid in q-Minkowski space viewed as the braided group B q [SU 2]. Their localisations are then isomorphic to quotients of U q (su 2) at fixed values of the q-Casimir precisely q-deforming the fuzzy case. We also use transmutation and twisting theory to introduce a $${C_q[G_\mathbb {C}]}$$ -covariant differential calculus on general B q [G] and U q (g), with Ω(B q [SU 2]) and Ω(U q (su 2) given in detail. To complete the picture, we show how the covariant calculus on the 3D bicrossproduct spacetime arises from Ω(C q [SU 2]) prior to twisting.

“Thermodynamics” of minimal surfaces and entropic origin of gravity

Deformations of minimal surfaces lying in constant-time slices in static space-times are studied. An exact and universal formula for a change of the area of a minimal surface under shifts of nearby pointlike particles is found. It allows one to introduce a local temperature on the surface and represent variations of its area in a thermodynamical form by assuming that the entropy in the Planck units equals the quarter of the area. These results provide a strong support to a recent hypothesis that gravity has an entropic origin, the minimal surfaces being a sort of holographic screens. The gravitational entropy also acquires a definite physical meaning related to quantum entanglement of fundamental degrees of freedom across the screen.

Global AdS picture of 1/2 BPS Wilson loops

We study the holographic dual string configuration of 1/2 BPS circular Wilson loops in \( \mathcal{N} = 4 \) super Yang-Mills theory by using the global coordinate of AdS. The dual string worldsheet is given by the Poincaré disk AdS 2 sitting at a constant global time slice of AdS 5. We also analyze the correlator of two concentric circular Wilson loops from the global AdS perspective and study the phase transition associated with the instability of annulus worldsheet connecting the two Wilson loops.

Entanglement entropy in quantum gravity and the Plateau problem

In a quantum gravity theory the entropy of entanglement S between the fundamental degrees of freedom spatially divided by a surface is discussed. Classical gravity is considered as an emergent phenomenon and arguments are presented that (1) S is a macroscopical quantity which can be determined without knowing a real microscopical content of the fundamental theory; (2) S is given by the Bekenstein-Hawking formula in terms of the area of a codimension 2 hypesurface B; (3) in static space-times B can be defined as a minimal hypersurface of a least volume separating the system in a constant-time slice. It is shown that properties of S are in agreement with basic properties of the von Neumann entropy. Explicit variational formulae for S in different physical examples are considered.