Coordinate Patch Research Articles

Explicit construction of Penrose diagrams for black hole to white hole transition with spacelike thin shells.

In this article, we explicitly construct the coordinates associated with the Penrose diagram in spacetimes connected via a spacelike thin shell in the following two examples: the generalized black-to-white hole bounce with mass difference and the Schwarzschild-to-de Sitter transition. We point out the issue of the first junction condition in the Penrose diagram constructed by cutting and pasting analytically known metrics with spherical symmetry by a static spacelike thin shell. With the goal of a global conformal coordinate chart associated with the corresponding Penrose diagram without discontinuity at the thin shell, we give a procedure consisting of three conformal transformations that serve different purposes. The first two of them are used to generate a continuous coordinate patch covering the entire thin shell, and therefore, the Penrose diagram can be constructed properly by patches with overlapping. The third transformation removes any coordinate singularity reintroduced by the first two transformations at the event horizons.

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.

Memory effect & Carroll symmetry, 50 years later

Particles at rest before the arrival of a burst of gravitational wave move, after the wave has passed, with constant velocity along diverging geodesics. As recognized by Souriau 50 years ago and then forgotten, their motion is particularly simple in Baldwin–Jeffery–Rosen (BJR) coordinates (which are however defined only in coordinate patches): they are determined when the first integrals associated with the 5-parameter isometry group (recently identified as Lévy-Leblond’s “Carroll” group with broken rotations) are used. A global description can be given instead in terms of Brinkmann coordinates, however it requires to solve a Sturm–Liouville equation, whereas the relation between BJR and Brinkmann requires to solve yet another Sturm–Liouville equation. The theory is illustrated by geodesic motion in a linearly polarized (approximate) “sandwich” wave proposed by Gibbons and Hawking for gravitational collapse, and by circularly polarized approximate sandwich waves with Gaussian envelope.

Cosmological de Sitter Solutions of the Semiclassical Einstein Equation

Exponentially expanding space–times play a central role in contemporary cosmology, most importantly in the theory of inflation and in the dark energy driven expansion in the late universe. In this work, we give a complete list of de Sitter solutions of the semiclassical Einstein equation (SCE), where classical gravity is coupled to the expected value of a renormalized stress–energy tensor of a free quantum field in the Bunch–Davies state. To achieve this, we explicitly determine the stress–energy tensor associated with the Bunch–Davies state using the recently proposed “moment approach” on the cosmological coordinate patch of de Sitter space. From the energy component of the SCE, we thus obtain an analytic consistency equation for the model’s parameters which has to be fulfilled by solutions to the SCE. Using this equation, we then investigate the number of solutions and the structure of the solution set in dependency on the coupling parameter of the quantum field to the scalar curvature and renormalization constants using analytic arguments in combination with numerical evidence. We also identify parameter sets where multiple expansion rates separated by several orders of magnitude are possible. Potentially for such parameter settings, a fast (semi-stable) expansion in the early universe could be compatible with a late-time “Dark Energy-like” behavior of the universe.

Geometry of the black-to-white hole transition within a single asymptotic region

We write explicitly the complete Lorentzian metric of a singularity-free spacetime where a black hole transitions into a white hole located in its same asymptotic region. In particular, the metric interpolates between the black and white horizons. The metric satisfies the Einstein field equations up to the tunneling region. The matter giving rise to the black hole is described by the Oppenheimer-Snyder model, corrected with loop-quantum-cosmology techniques in the quantum region. The interior quantum geometry is fixed by a local Killing symmetry, broken at the horizon transition. At large scale, the geometry is determined by two parameters: the mass of the hole and the duration of the transition process. The latter is a global geometrical parameter. We give the full metric outside the star in a single coordinate patch.

Local Criteria for Triangulating General Manifolds

We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex {mathscr {A}}, and a map H from the underlying space of {mathscr {A}} to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of {mathscr {A}} is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.

GenASiS Mathematics: Object-oriented manifolds, operations, and solvers for large-scale physics simulations (version 2)

GenASiSMathematics provides modern Fortran classes furnishing extensible object-oriented functionality for the solution of fields governed by selected partial differential equations. The initial release included extensible object-oriented implementations of simple meshes and the evolution of generic conserved currents thereon. This revision—Version 2 of Mathematics—includes significant reorganization and streamlining of these classes, higher-order reconstruction by a different method, a Poisson solver, coarsening to avoid Courant time step limitations near coordinate singularities, and the offloading of computational kernels to GPUs. New version program summaryProgram Title:HomogeneousSpheroid, RectangularContraction, RectangularExpansion, SphericalContraction, SphericalExpansion (example problems illustrating GenASiSMathematics)CPC Library link to program files:https://doi.org/10.17632/mzvxngwtw6.2Developer's repository link:https://github.com/GenASiSCode Ocean capsule:https://codeocean.com/capsule/4106782Licensing provisions: GPLv3Programming language: Modern Fortran; OpenMP (tested with recent versions of GNU Compiler Collection (GCC), Cray Compiler Environment (CCE), IBM XL Fortran compiler)Journal reference of previous version: Computer Physics Communications 222 (2018) 384Does the new version supersede the previous version?: YesReasons for the new version: This version includes significant reorganization and streamlining, higher-order reconstruction by a different method, a Poisson solver, coarsening to avoid Courant time step limitations near coordinate singularities, and the offloading of computational kernels to GPUs.Summary of revisions: The left part of Fig. 1 shows the revised structure of GenASiSMathematics. The Algebra and Calculus divisions contain relatively simple stand-alone functionality for such tasks as root finding, interpolation, numerical integration, and the solution of ordinary differential equations. The major functionality currently provided in Mathematics, which focuses on the solution of selected classes of partial differential equations, is now collected in the CauchyProblems division.The structure of CauchyProblems is displayed in the right part of Fig. 1. The solution of partial differential equations presumes the existence of Manifolds (or ‘spaces’) and Fields thereon whose configuration we seek. The two basic types of Cauchy problems are boundary value problems, or Constraints; and initial value problems, or Evolutions.The structures of Manifolds, Fields, Constraints, and Evolutions are displayed in Fig. 2. Individual coordinate patches or Charts are combined into Atlases in order to represent Manifolds. As for Fields, the most basic functionality for a set of fields on a manifold (including I/O) is representedby classes in FieldSets. Specialized sets of fields needed for Cauchy problems are Geometries, including coarsening functionality to avoid Courant time step limitations near coordinate singularities; and CurrentSets, which represent ‘conserved’—or more properly, when source terms are present, ‘balanced’—currents, that is, densities and their corresponding fluxes. CalculusFields provides integrals of fields on manifolds, including volume and surface integrals used to tally conserved and balanced quantities, as well as spherical and azimuthal averaging. New to this version of Mathematics is the solution of PoissonEquations via multipole expansion, the first capability for Constraints or boundary value problems.The solution of balance equations is the type of initial value problem currently handled by Evolutions. As in Version 1, finite-volume discretization results in a large set of ordinary differential equations solved by the Runge-Kutta method. Now however, the right-hand sides, or Slopes, are handled by variations of a recursive class that allows the construction of hierarchies of terms. Thus complicated equations can be computed and included in I/O in pieces representing various terms, with results assembled in an extensible and quasi-automated fashion. For example, the partial derivatives, and geometric source terms from connection coefficients, can be computed and visualized separately, while being automatically combined to give the full divergence of an energy-momentum tensor. Third-order reconstruction (with limiting) to discretize cell interfaces is now included. Multi-stage Runge-Kutta Steps are then taken by Integrators in order to advance the solution of fields from their initial condition.Relative to Version 1 of Mathematics, the Manifolds and Fields classes are significantly reorganized and streamlined. Previously, classes for fields, geometry, and I/O were interleaved with those of charts and atlases. This provided for chart and atlas classes with self-contained geometry and I/O functionality, but in a manner that proved unwieldy and indeed unnecessary. This has now been disentangled: classes for fields, geometry and I/O are built entirely on top of full manifolds (i.e. classes for atlases).All computational kernels (except those specifically needed for checkpointing, that is, I/O) are coded for offloading to GPUs via OpenMP.Nature of problem: By way of illustrating GenASiSMathematics functionality, solve an example potential problem and imposed advection problems.Solution method: Multipole expansion for solution of the Poisson equation; finite-volume discretization; second- and third-order reconstruction with limiting; HLL Riemann Solver; Runge-Kutta integration.Additional comments including restrictions and unusual features: Uses the MPI [1] and Silo [2] libraries. The example problems named above are not ends in themselves, but serve to illustrate the functionality available though GenASiSMathematics. In addition to these more substantial examples, we provide individual unit test programs for the classes comprised by GenASiSMathematics.GenASiSMathematics is available in the CPC Program Library and also at https://github.com/GenASiS.

Perturbative quantum consistency near black-hole horizon formation

We study the prelude to black-hole formation using a suspended shell composed of physical matter that fulfills the dominant energy condition. Here, the collapse of the shell is brought to rest when the formation of the horizon is imminent but has not yet occurred. As the main achievement of this work, we obtain the Feynman propagator which connects the interior and the exterior of the shell within two local coordinate patches. It is derived by drawing an analogy to the propagation of light across interfaces that separate regions with different susceptibilities inside a medium. As a first application, we use this propagator to determine the vacuum persistence amplitude in the presence of external sources. On timescales much shorter than the Page time, we find that the amplitude builds up with time yet remains consistent with perturbative unitarity.

Harmonic analysis of boxed hyperoctahedral Hall-Littlewood polynomials

For positive integers n and c with c≥2n (= the Coxeter number of the hyperoctahedral group of signed permutations of degree n), we present a finite-dimensional discrete orthogonality relation for Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials of degree at most c in each of the n variables. These polynomials are labeled by partitions λ that fit inside a rectangular box of shape cn, i.e. the partitions in question are of length ≤n and have parts of size ≤c. We employ coordinate patches around the vertices of the alcove of boxed partitions Λ(n,c)={λ⊆cn} to establish the self-adjointness of a one-parameter family of commuting discrete difference operators ▪ acting on functions f:Λ(n,c)→C. By construction, the basis of hyperoctahedral Hall-Littlewood polynomials constitutes a joint eigenbasis for ▪ with simple spectrum, which gives rise to our discrete orthogonality relation. From the point of view of quantum integrable particle dynamics, the present geometric construction establishes the orthogonality of the Bethe Ansatz eigenfunctions for a recently studied q-boson system on a finite lattice with integrable open-end boundary conditions. The Bethe Ansatz equations enter in the geometric picture as compatibility conditions between coordinate patches stemming from distinct vertices. Two applications of the orthogonality relations are highlighted: (i) a cubature rule for the integration of symmetric functions with respect to the Haar measure on the compact symplectic group Sp(n), and (ii) a Verlinde formula for the structure constants of a deformation of the Wess-Zumino-Witten fusion ring associated with the affine Lie algebra of type Cˆn (= Cn(1)).

Closed Affine Manifolds with an Invariant Line

A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.

Geometries with mismatched branes

We study Randall-Sundrum two brane setups with mismatched brane tensions. For the vacuum solutions, boundary conditions demand that the induced metric on each of the branes is either de Sitter, Anti-de Sitter, or Minkowski. For incompatible boundary conditions, the bulk metric is necessarily time-dependent. This introduces a new class of time-dependent solutions with the potential to address cosmological issues and provide alternatives to conventional inflationary (or contracting) scenarios. We take a first step in this paper toward such solutions. One important finding is that the resulting solutions can be very succinctly described in terms of an effective action involving only the induced metric on either one of the branes and the radion field. But the full geometry cannot necessarily be simply described with a single coordinate patch. We concentrate here on the time- dependent solutions but argue that supplemented with a brane stabilization mechanism one can potentially construct interesting cosmological models this way. This is true both with and without a brane stabilization mechanism.

Cosmic footballs from superrotations

Superrotations arise from singular vector fields on the celestial sphere in asymptotically flat space, and their finite integrated versions have been argued by Strominger and Zhiboedov to insert cosmic strings into the spacetime. In this work, we argue for an alternative definition of the action of superrotations on Minkowski space that avoids introducing any defects. This involves realizing the finite superrotation not as a diffeomorphism between spaces, but as a mapping of Minkowski space to itself that may be multivalued or non-surjective. This eliminates any defects in the bulk spacetime at the expense of allowing for defects in the boundary celestial sphere metric. We further explore the geometry of the spatial surfaces in the superrotated spaces, and note that they intersect null infinity at the singularity of the superrotation, causing a breakdown in the large r asymptotic expansion there. To determine how these surfaces embed into Minkowski space, a derivation of the finite superrotation transformation is presented in both Bondi and Newman–Unti gauges. The latter is particularly interesting, since the superrotations are shown to preserve the hyperbolic slicing of Minkowski space in Newman–Unti gauge, and this gauge also provides a means for extending the geometry beyond the Bondi coordinate patch. We argue that the new interpretation for the action of superrotations on spacetime motivates consideration of a wider class of celestial sphere metrics and asymptotic symmetry groups.

Implications of Symmetry and Pressure in Friedmann Cosmology. I. Formalism

Abstract We show that derivation of Friedmann’s equations from the Einstein–Hilbert action, paying attention to the requirements of isotropy and homogeneity during the variation, leads to a different interpretation of pressure than what is typically adopted. Our derivation follows if we assume that the unapproximated metric and Einstein tensor have convergent perturbation series representations on a sufficiently large Robertson–Walker coordinate patch. We find the source necessarily averages all pressures, everywhere, including the interiors of compact objects. We demonstrate that our considerations apply (on appropriately restricted spacetime domains) to the Kerr solution, the Schwarzschild constant-density sphere, and the static de-Sitter sphere. From conservation of stress–energy, it follows that material contributing to the averaged pressure must shift locally in energy. We show that these cosmological energy shifts are entirely negligible for non-relativistic material. In relativistic material, however, the effect can be significant. We comment on the implications of this study for the dark energy problem.

Adaptive strategies for solving parameterized systems using homotopy continuation

Three aspects of applying homotopy continuation, which is commonly used to solve parameterized systems of polynomial equations, are investigated. First, for parameterized systems which are homogeneous, we investigate options for performing computations on an adaptively chosen affine coordinate patch. Second, for parameterized systems which are overdetermined, we investigate options for adaptively selecting a well-constrained subsystem to restore numerical stability. Finally, since one is typically interested in only computing real solutions for parameterized problems which arise from applications, we investigate a scheme for heuristically identifying solution paths which appear to be ending at nonreal solutions and truncating them. We demonstrate these three aspects on two problems arising in computer vision.

Delaunay Triangulation of Manifolds

We present an algorithmic framework for producing Delaunay triangulations of manifolds. The input to the algorithm is a set of sample points together with coordinate patches indexed by those points. The transition functions between nearby coordinate patches are required to be bi-Lipschitz with a constant close to 1. The primary novelty of the framework is that it can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. The output is a manifold simplicial complex that is the Delaunay complex of a perturbed set of points on the manifold. The guarantee of a manifold output complex demands no smoothness requirement on the transition functions, beyond the bi-Lipschitz constraint. In the smooth setting, when the transition functions are defined by common coordinate charts, such as the exponential map on a Riemannian manifold, the output manifold is homeomorphic to the original manifold, when the sampling is sufficiently dense.

Explicit reconstruction of the entanglement wedge

The problem of how the boundary encodes the bulk in AdS/CFT is still a subject of study today. One of the major issues that needs more elucidation is the problem of subregion duality; what information of the bulk a given boundary subregion encodes. Although the proof given by Dong, Harlow, and Wall [14] states that the entanglement wedge of the bulk should be encoded in boundary subregions, no explicit procedure for reconstructing the entanglement wedge was given so far. In this paper, mode sum approach to obtaining smearing functions for a single bulk scalar is generalised to include bulk reconstruction in the entanglement wedge of boundary subregions. It is generally expectated that solutions to the wave equation on a complicated coordinate patch are needed, but this hard problem has been transferred to a less hard but tractable problem of matrix inversion.

Simulating the universe(s) III: observables for the full bubble collision spacetime

This is the third paper in a series establishing a quantitative relation between inflationary scalar field potential landscapes and the relic perturbations left by the collision between bubbles produced during eternal inflation. We introduce a new method for computing cosmological observables from numerical relativity simulations of bubble collisions in one space and one time dimension. This method tiles comoving hypersurfaces with locally-perturbed Friedmann-Robertson-Walker coordinate patches. The method extends previous work, which was limited to the spacetime region just inside the future light cone of the collision, and allows us to explore the full bubble-collision spacetime. We validate our new methods against previous work, and present a full set of predictions for the comoving curvature perturbation and local negative spatial curvature produced by identical and non-identical bubble collisions, in single scalar field models of eternal inflation. In both collision types, there is a non-zero contribution to the spatial curvature and cosmic microwave background quadrupole. Some collisions between non-identical bubbles excite wall modes, giving extra structure to the predicted temperature anisotropies. We comment on the implications of our results for future observational searches. For non-identical bubble collisions, we also find that the surfaces of constant field can readjust in the presence of a collision to produce spatially infinite sections that become nearly homogeneous deep into the region affected by the collision. Contrary to previous assumptions, this is true even in the bubble into which the domain wall is accelerating.

Tensor-multi-scalar theories: relativistic stars and 3 + 1 decomposition

Gravitational theories with multiple scalar fields coupled to the metric and each other—a natural extension of the well studied single-scalar-tensor theories—are interesting phenomenological frameworks to describe deviations from general relativity in the strong-field regime. In these theories, the N-tuple of scalar fields takes values in a coordinate patch of an N-dimensional Riemannian target-space manifold whose properties are poorly constrained by weak-field observations. Here we introduce for simplicity a non-trivial model with two scalar fields and a maximally symmetric target-space manifold. Within this model we present a preliminary investigation of spontaneous scalarization for relativistic, perfect fluid stellar models in spherical symmetry. We find that the scalarization threshold is determined by the eigenvalues of a symmetric scalar-matter coupling matrix, and that the properties of strongly scalarized stellar configurations additionally depend on the target-space curvature radius. In preparation for numerical relativity simulations, we also write down the 3 + 1 decomposition of the field equations for generic tensor-multi-scalar theories.

Initial Study of Normal Isocurvature Surfaces and Their Relation to Partial Derivatives of Plumb Line Curvature

This work aimed to study isocurvature surfaces of Earth’s normal gravity field and their relation to partial derivatives of a plumb line curvature. An isocurvature surface of a gravity field is a surface along which the value of the plumb line curvature is constant. The normal gravity field is a symmetrical gravity field; therefore, isocurvature surfaces are surfaces of revolution. To study an isocurvature surface, special assumptions are made to form a vector equation, which will hold only for a small coordinate patch of the isocurvature surface. The gradient of a normal plumb line curvature is vertical to the isocurvature surface pointing to the direction along which the curvature of the plumb line decreases or increases the most. In order to show the significance of isocurvature surfaces, it was shown that it is possible to determine the value of the surface derivative of a plumb line’s curvature without differentiating the original complicated function of a plumb line curvature. KeywordsPlumb Lines; Curvature; Normal Gravity Field; Plumbline Curvature; Isocurvature Surface

Fisher information of Markovian decay modes

We introduce the Fisher information in the basis of decay modes of Markovian dynamics, arguing that it encodes important information about the behavior of nonequilibrium systems. In particular we generalize an orthonormality relation between decay eigenmodes of detailed balanced systems to normal generators that commute with their time-reversal. Viewing such modes as tangent vectors to the manifold of statistical distributions, we relate the result to the choice of a coordinate patch that makes the Fisher-Rao metric Euclidean at the steady distribution, realizing a sort of statistical equivalence principle. We then classify nonequilibrium systems according to their spectrum, showing that a degenerate Fisher matrix is the signature of the insurgence of a class of dynamical phase transitions between nonequilibrium regimes, characterized by level crossing and power-law decay in time of suitable order parameters. An important consequence is that normal systems cannot manifest critical behavior. Finally, we study the Fisher matrix of systems with time-scale separation.