Asymptotic Metric Research Articles

Three Puzzles with Covariance and Supertranslation Invariance of Angular Momentum Flux and Their Solutions.

We describe and solve three puzzles arising in covariant and supertranslation-invariant formulas for the flux of angular momentum and other Lorentz charges in asymptotically flat spacetimes: (i)Supertranslation invariance and covariance imply invariance under spacetime translations; (ii)the flux depends on redundant auxiliary degrees of freedom that cannot be set to zero in all Lorentz frames without breaking Lorentz covariance; (iii)supertranslation-invariant Lorentz charges do not generate the transformations of the Bondi mass aspect implied by the isometries of the asymptotic metric. In this Letter, we solve the first two puzzles by presenting covariant formulas that unambiguously determine the auxiliary degrees of freedom and clarify the last puzzle by explaining the different roles played by covariant and canonical charges. Our construction makes explicit the choice of reference frame underpinning seemingly unambiguous results presented in the current literature.

Asymptotic Metric on the Equator of Compact Astronomical Object with Electric Charge and Magnetic Dipole in Nonlinear Electrodynamics

Asymptotic Metric on the Equator of Compact Astronomical Object with Electric Charge and Magnetic Dipole in Nonlinear Electrodynamics

The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space

We construct an asymptotic metric on the moduli space of two centred hyperbolic monopoles by working in the point particle approximation, that is treating well-separated monopoles as point particles with an electric, magnetic and scalar charge and re-interpreting the dynamics of the 2-particle system as geodesic motion with respect to some metric. The corresponding analysis in the Euclidean case famously yields the negative mass Taub-NUT metric, which asymptotically approximates the L<sup>2</sup> metric on the moduli space of two Euclidean monopoles, the Atiyah-Hitchin metric. An important difference with the Euclidean case is that, due to the absence of Galilean symmetry, in the hyperbolic case it is not possible to factor out the centre of mass motion. Nevertheless we show that we can consistently restrict to a 3-dimensional configuration space by considering antipodal configurations. In complete parallel with the Euclidean case, the metric that we obtain is then the hyperbolic analogue of negative mass Taub-NUT. We also show how the metric obtained is related to the asymptotic form of a hyperbolic analogue of the Atiyah-Hitchin metric constructed by Hitchin.

Fleet Sizing for Reservation-Based Car-Sharing Services Allowing Cruise Parking

Fleet sizing is an important problem in the planning procedure of one-way car-sharing systems. This article investigates the fleet-sizing problem for reservation-based car-sharing systems allowing cruise parking. The system with traffic congestion is modeled as a mixed queuing network with its servers mapping the remaining road capacity. The optimal fleet size with maximum operating profit is derived, taking into account asymptotic performance metrics like reservation, vehicle availability and utilization, and occupancy of parking spaces of each of the stations. The solution is obtained based on an exact mean value analysis (MVA) algorithm and an approximate Schweitzer–Bard MVA (SB-MVA) algorithm. Numerical experiments demonstrate that the proposed method with cruise parking can meet the travel demands with greatly reduced operating cost. Another interesting finding is that a larger reservation time window implies lower parking capacity and smaller fleet size, whereas, as the time window exceeds a threshold, reducing the fleet size may damage the quality of service.

Optimization of SC-LDPC Codes for Window Decoding With Target Window Sizes

In this paper, we propose an optimization method for protograph-based spatially coupled low-density parity-check (SC-LDPC) codes under window decoding (WD). Previous works on constructing SC-LDPC codes for WD typically focused on optimizing asymptotic performance metrics such as the WD threshold. However, in this paper, it is observed that the WD threshold is not an appropriate metric to sufficiently explain the finite-length behavior of SC-LDPC codes under WD. Thus, we propose a new performance metric, called the window mean parameter, based on a scaling analysis to capture the WD performance more accurately and formulate a code optimization algorithm that optimizes the proposed performance metric. Since the proposed metric depends on the window size, the optimization algorithm can provide a code family of SC-LDPC codes optimized for various target window sizes. Simulation results confirm that the improvement in the proposed metric leads to a finite-length performance improvement, resulting in one to two orders of the frame error rate gain over the conventional SC-LDPC codes for a wide range of window sizes. Furthermore, we investigate structural characteristics of the proposed codes to provide a supplementary explanation for the performance improvement, which also promotes a better understanding of SC-LDPC codes for WD.

Holographic approach to thermalization in general anisotropic theories

We employ the holographic approach to study the thermalization in the quenched strongly-coupled field theories with very general anisotropic scalings including Lifshitz and hyperscaling violating fixed points. The holographic dual is a Vaidya-like time-dependent geometry where the asymptotic metric has general anisotropic scaling isometries. We find the Ryu-Takanayagi extremal surface and use it to calculate the time-dependent entanglement entropy between a strip region with width 2R and its outside region. In the special case with an isotropic metric, we also explore the entanglement entropy for a spherical region of radius R. The growth of the entanglement entropy characterizes the thermalization rate after a quench. We study the thermalization process in the early times and late times in both large R and small R limits. The allowed scaling parameter regions are constrained by the null energy conditions as well as the condition for the existence of the Ryu-Takanayagi extremal surfaces. This generalizes the previous works on this subject. All obtained results can be compared with experiments and other methods of probing thermalization.

Non-singular vortices with positive mass in 2+1-dimensional Einstein gravity with AdS3 and Minkowski background

In previous work, black hole vortex solutions in Einstein gravity with AdS3 background were found where the scalar matter profile had a singularity at the origin r = 0. In this paper, we find numerically static vortex solutions where the scalar and gauge fields have a non-singular profile under Einstein gravity in an AdS3 background. Vortices with different winding numbers n, VEV v and cosmological constant Λ are obtained. These vortices have positive mass and are not BTZ black holes as they have no event horizon. The mass is determined in two ways: by subtracting the numerical values of two separate asymptotic metrics and via an integral that is purely over the matter fields. The mass of the vortex increases as the cosmological constant becomes more negative and this coincides with the core of the vortex becoming smaller (compressed). We then consider the vortex with gravity in asymptotically flat spacetime for different values of the coupling α = 1/(16πG). At the origin, the spacetime has its highest curvature and there is no singularity. It transitions to an asymptotic conical spacetime with angular deficit that increases significantly as α decreases. For comparison, we also consider the vortex without gravity in flat spacetime. For this case, one cannot obtain the mass by the first method (subtracting two metrics) but remarkably, via a limiting procedure, one can obtain an integral mass formula. In the absence of gauge fields, there is a well-known logarithmic divergence in the energy of the vortex. With gravity, we present this divergence in a new light. We show that the metric acquires a logarithmic term which is the 2 + 1 dimensional realization of the Newtonian gravitational potential when General Relativity is supplemented with a scalar field. This opens up novel possibilities which we discuss in the conclusion.

Information-Theoretic Approach for Model Reduction Over Finite Time Horizon

This paper presents an information-theoretic approach for model reduction for finite time simulation. Although system models are typically used for simulation over a finite time, most of the metrics (and pseudo-metrics) used for model accuracy assessment consider asymptotic behavior e.g., Hankel singular values and Kullback-Leibler(KL) rate metric. These metrics could further be used for model order reduction. Hence, in this paper, we propose a generalization of KL divergence-based metric called n-step KL rate metric, which could be used to compare models over a finite time horizon. We then demonstrate that the asymptotic metrics for comparing dynamical systems may not accurately assess the model prediction uncertainties over a finite time horizon. Motivated by this finite time analysis, we propose a new pragmatic approach to compute the influence of a subset of states on a combination of states called information transfer (IT). Model reduction typically involves the removal or truncation of states. IT combines the concepts from the n-step KL rate metric and model reduction. Finally, we demonstrate the application of information transfer for model reduction. Although the analysis and definitions presented in this paper assume linear systems, they can be extended for nonlinear systems.

Atiyah–Hitchin in five-dimensional Einstein–Gauss–Bonnet gravity

We construct a new class of stationary exact solutions to five-dimensional Einstein–Gauss–Bonnet gravity. The solutions are based on four-dimensional self-dual Atiyah–Hitchin geometry. We find analytical solutions to the five-dimensional metric function that are regular everywhere. We find some constraints on the possible physical solutions by investigating the solutions numerically. We also study the behavior of the solutions in the extremal limits of the Atiyah–Hitchin geometry. In the extremal limits, the Atiyah–Hitchin metric reduces to a bolt structure and Euclidean Taub–NUT space, respectively. In these limits, the five-dimensional metric function approaches to a constant value and infinity, respectively. We find that the asymptotic metrics are regular everywhere.

Properties of dual supertranslation charges in asymptotically flat spacetimes

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.

Gravitational memory charges of supertranslation and superrotation on Rindler horizons

In a Rindler-type coordinate system spanned in a region outside of a black hole horizon, we have nonvanishing classical holographic charges as soft hairs on the horizon for stationary black holes. Taking a large black hole mass limit, the spacetimes with the charges are described by asymptotic Rindler metrics. We construct a general theory of gravitational holographic charges for a (1+3)-dimensional linearized gravity field in the Minkowski background with Rindler horizons. Although matter crossing a Rindler horizon causes horizon deformation and a time-dependent coordinate shift, that is, gravitational memory, the supertranslation and superrotation charges on the horizon can be defined during and after its passage through the horizon. It is generally proven that holographic states on the horizon cannot store any information about absorbed perturbative gravitational waves. However, matter crossing the horizon really excites holographic states. By using gravitational memory operators, which consist of the holographic charge operators, we suggest a resolution of the no-cloning paradox of quantum information between matter falling into the horizon and holographic charges on the horizon from the viewpoint of the contextuality of quantum measurement.

Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inverse-square potential with zero energy E. The geodesic flow on the full configuration space ℂ3 (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space ℂ2 and shape space ℝ3 (as well as 𝕊3 and the shape sphere 𝕊2 for the inverse-square potential when E = 0). The corresponding Riemannian submersions are described explicitly in “Hopf” coordinates which are particularly adapted to the isometries. For equal masses subject to inverse-square potentials, Montgomery shows that the zero-energy “pair of pants” JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on ℂ2, ℝ3, and 𝕊3 with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits, and observe that the geodesic reformulation “regularizes” pairwise and triple collisions on ℂ2 and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and zero energy, we find that the scalar curvature on ℂ2 is strictly negative though it could have either sign on ℝ3. However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.

Performance Analysis of Finite-Sized Cooperative Systems with Unreliable Backhaul Links

This paper presents a performance analysis of a finite-sized co-operative wireless system, where a group of transmitters with unreliable backhauls serve a desired receiver using noncoherent joint transmission. To facilitate analysis, an analytical expression for the distribution of the spatially averaged signal-to-interference-plus-noise ratio (SA-SINR) is derived in terms of key system and channel parameters. Leveraging the derived expression, the joint impact of node co-operation, backhaul reliability, interference, and communication range is investigated in the considered finite-sized co-operative system. Furthermore, based on the SA-SINR, closed form expressions for the average bit error rate (ABER) and average spectral efficiency (ASE) are derived. Further insights are established by analyzing the asymptotic performance in the high transmission power regime. From analytical derivations for the outage probability, ABER, and ASE and link-level simulations, it is verified that these asymptotic performance metrics are exclusively influenced by unreliable backhauls, so that the conventional diversity gains are not achievable.

Geometry of locally compact groups of polynomial growth and shape of large balls

We generalize Pansu’s thesis [27] about asymptotic cones of finitely generated nilpotent groups to arbitrary locally compact groups G of polynomial growth. We show that any such G is weakly commensurable to some simply connected solvable Lie group S , the Lie shadow of G and that balls in any reasonable left invariant metric on G admit a well-defined asymptotic shape. By-products include a formula for the asymptotics of the volume of large balls and an application to ergodic theory, namely that the Ergodic Theorem holds for all ball averages. Along the way we also answer negatively a question of Burago and Margulis [7] on asymptotic word metrics and give a geometric proof of some results of Stoll [33] of the rationality of growth series of Heisenberg groups.

Phases of five-dimensional theories, monopole walls, and melting crystals

Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimensional supersymmetric quantum field theories on $ \mathbb{R} $ 3 × T 2 and, on the other hand, with moduli spaces of local Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore the asymptotic metric of these moduli spaces and compare our results with Seiberg’s low energy description of the five-dimensional quantum theories. We also give a natural description of the phase structure of general monowall moduli spaces in terms of triangulations of Newton polygons, secondary polyhedra, and associahedral projections of secondary fans.

Geometry of periodic monopoles

Bogomol'nyi-Prasad-Sommerfield monopoles on ${\mathbb{R}}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ correspond, via the generalized Nahm transform, to certain solutions of the Hitchin equations on the cylinder $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{1}$. The moduli space $\mathcal{M}$ of two monopoles with their center of mass fixed is a four-dimensional manifold with a natural hyperk\"ahler metric, and its geodesics correspond to slow-motion monopole scattering. The purpose of this paper is to study the geometry of $\mathcal{M}$ in terms of the Nahm-Hitchin data, i.e., in terms of structures on $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{1}$. In particular, we identify the moduli, derive the asymptotic metric on $\mathcal{M}$, and discuss several geodesic surfaces and geodesics on $\mathcal{M}$. The latter include novel examples of monopole dynamics.

Absence of embedded eigenvalues for Riemannian Laplacians

In this paper we study absence of embedded eigenvalues for Schrödinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end. Another condition is an upper bound of the trace of this quantity, while a third one is a bound of the derivatives of part of the trace (some oscillatory behaviour of the trace is allowed). In addition to geometric bounds we need conditions on the potential, a regularity property of the domain of the Schrödinger operator and the unique continuation property. Examples include ends endowed with asymptotic Euclidean or hyperbolic metrics.

Outage Exponent: A Unified Performance Metric for Parallel Fading Channels

The parallel fading channel, which consists of finite number of subchannels, is very important, because it can be used to formulate many practical communication systems. The outage probability, on the other hand, is widely used to analyze the relationship among the communication efficiency, reliability, SNR, and channel fading. To the best of our knowledge, the previous works only studied the asymptotic outage performance of the parallel fading channel which are only valid for a large number of subchannels or high SNRs. In this paper, a unified performance metric, which we shall refer to as the outage exponent, will be proposed. Our approach is mainly based on the large deviations theory and the Meijer's G-function. It is shown that the proposed outage exponent is not only an accurate estimation of the outage probability for any number of subchannels, any SNR, and any target transmission rate, but also provides an easy way to compute the outage capacity, finite-SNR diversity-multiplexing tradeoff, and SNR gain. The asymptotic performance metrics, such as the delay-limited capacity, ergodic capacity, and diversity-multiplexing tradeoff can be directly obtained by letting the number of subchannels or SNR tends to infinity. Similar to Gallager's error exponent, a reliable function for parallel fading channels, which illustrates a fundamental relationship between the transmission reliability and efficiency, can also be defined from the outage exponent. Therefore, the proposed outage exponent provides a complete and comprehensive performance measure for parallel fading channels.

Scattering theory for Riemannian Laplacians

We introduce a notion of scattering theory for the Laplace–Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behavior of the metric at infinity (like asymptotic Euclidean or hyperbolic metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace–Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum.

Super Yang-Mills theory with impurity walls and instanton moduli spaces

We explore maximally supersymmetric Yang-Mills theory with walls of impurities respecting half of the supersymmetries. The walls carry fundamental or bifundamental matter multiplets. We employ three-dimensional N=2 superspace language to identify the Higgs branch of this theory. We find that the vacuum conditions determining the Higgs branch are exactly the bow equations yielding Yang-Mills instantons on a multi-Taub-NUT space. Under electric-magnetic duality, the super Yang-Mills theory describing the bulk is mapped to itself, while the fundamental- and bifundamental-carrying impurity walls are interchanged. We perform a one-loop computation on the Coulomb branch of the dual theory to find the asymptotic metric on the original Higgs branch.