Geodesic Analysis Research Articles

Black hole singularity from OPE

Eternal asymptotically AdS black holes are dual to thermofield double states in the boundary CFT. It has long been known that black hole singularities have certain signatures in boundary thermal two-point functions related to null geodesics bouncing off the singularities (bouncing geodesics). In this paper we shed light on the manifestations of black hole singularities in the dual CFT. We decompose the boundary CFT correlator of scalar operators using the Operator Product Expansion (OPE) and focus on the contributions from the identity, the stress tensor, and its products. We show that this part of the correlator develops singularities precisely at the points that are connected by bulk bouncing geodesics. Black hole singularities are thus encoded in the analytic behavior of the boundary correlators determined by multiple stress tensor exchanges. Furthermore, we show that in the limit where the conformal dimension of the operators is large, the sum of multi-stress-tensor contributions develops a branch point singularity as predicted by the geodesic analysis. We also argue that the appearance of complexified geodesics, which play an important role in computing the full correlator, is related to the contributions of the double-trace operators in the boundary CFT.

Real-time anomaly detection of the stochastically excited systems on spherical ([formula omitted]) manifold

Advanced analytical tools have become crucial in today’s constantly changing and complex systems. Real-time Principal Geodesic Analysis (RPGA) is a novel technique that provides a unique perspective for analyzing nonlinear data on differentiable manifolds. Traditional linear methods are often inadequate when exploring the complexities of such data. Orthogonal transformation techniques such as Principal Component Analysis (PCA) and Principal Geodesic Analysis (PGA) are highly desirable for condition monitoring stochastically excited systems in domains like mechanical, aerospace, and civil engineering. However, uncertainties and dynamic fluctuations necessitate robust analytical methods for early change detection to ensure safety, performance, and cost-effectiveness. Limitations posed by linear orthogonal transformation techniques such as PCA and its recursive counterparts make it imperative to adapt these techniques to nonlinear situations where data does not evolve in a flat Euclidean space. Significant advancements have been made in this field over recent decades, with data-driven real-time algorithms such as RPCA, RCCA, and RSSA providing reliable solutions for complex multidimensional problems. However, for curved space, the nonlinear RPGA technique takes center stage. It is known for its effectiveness in extracting meaningful information from the complex data stream. This paper explores the foundational concepts and methodologies underlying the transition from linear to nonlinear data analysis. By examining examples such as stochastic geometric oscillator on S2, and the inverted spherical pendulum cart system navigating a rough surface, we illustrate the significance of reliable, real-time damage detection techniques provided by tools such as RPGA.

Polynomial chaos expansions on principal geodesic Grassmannian submanifolds for surrogate modeling and uncertainty quantification

In this work we introduce a manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems. Our first goal is to perform data mining on the available simulation data to identify a set of low-dimensional (latent) descriptors that efficiently parameterize the response of the high-dimensional computational model. To this end, we employ Principal Geodesic Analysis on the Grassmann manifold of the response to identify a set of disjoint principal geodesic submanifolds, of possibly different dimension, that captures the variation in the data. Since operations on the Grassmann require the data to be concentrated, we propose an adaptive algorithm based on Riemannian K-means and the minimization of the sample Fréchet variance on the Grassmann manifold to identify “local” principal geodesic submanifolds that represent different system behavior across the parameter space. Polynomial chaos expansion is then used to construct a mapping between the random input parameters and the projection of the response on these local principal geodesic submanifolds. The method is demonstrated on four test cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra dynamical system, a continuous-flow stirred-tank chemical reactor system, and a two-dimensional Rayleigh-Bénard convection problem.

Noether symmetries, conservation laws and geometric properties of the Banados-Teitelboim-Zanelli black hole metric

This study provides a comprehensive analysis of the Banados-Teitelboim-Zanelli (BTZ) black hole, a significant solution in three-dimensional gravity. We begin with a detailed exploration of the BTZ black hole metric, highlighting its derivation and physical properties, such as its horizon structure and thermodynamic characteristics. The thermodynamics of BTZ black holes, including temperature, entropy, and free energy, are analyzed to understand their stability and behavior. A central focus of this work is the application of Noether symmetry principles to the BTZ black hole. We systematically compute the Noether symmetries of the BTZ metric and discuss their physical implications, including their role in conservation laws and stability analysis. This includes an examination of perturbations and their impact on the stability of the black hole, as well as a detailed geodesic analysis that leverages Noether symmetries to simplify the study of particle motion in this spacetime. Additionally, we investigate the first integrals associated with the BTZ black hole metric, elucidating their significance for the dynamics and conservation laws within this framework. The geometric analysis is further expanded through the Cartan structure equations, providing insight into the intrinsic geometric properties of the BTZ black hole and associated curvature tensors. Overall, this work integrates thermodynamic, symmetric, and geometric analyses to offer a unified understanding of the BTZ black hole’s structure and behavior, contributing to the broader field of black hole physics in lower-dimensional spacetimes.

Riemannian Geodesic Discriminant Analysis–Minimum Riemannian Mean Distance: A Robust and Effective Method Leveraging a Symmetric Positive Definite Manifold and Discriminant Algorithm for Image Set Classification

This study introduces a novel method for classifying sets of images, called Riemannian geodesic discriminant analysis–minimum Riemannian mean distance (RGDA-MRMD). This method first converts image data into symmetric positive definite (SPD) matrices, which capture important features related to the variability within the data. These SPD matrices are then mapped onto simpler, flat spaces (tangent spaces) using a mathematical tool called the logarithm operator, which helps to reduce their complexity and dimensionality. Subsequently, regularized local Fisher discriminant analysis (RLFDA) is employed to refine these simplified data points on the tangent plane, focusing on local data structures to optimize the distances between the points and prevent overfitting. The optimized points are then transformed back into a complex, curved space (SPD manifold) using the exponential operator to enhance robustness. Finally, classification is performed using the minimum Riemannian mean distance (MRMD) algorithm, which assigns each data point to the class with the closest mean in the Riemannian space. Through experiments on the ETH-80 (Eidgenössische Technische Hochschule Zürich-80 object category), AFEW (acted facial expressions in the wild), and FPHA (first-person hand action) datasets, the proposed method demonstrates superior performance, with accuracy scores of 97.50%, 37.27%, and 88.47%, respectively. It outperforms all the comparison methods, effectively preserving the unique topological structure of the SPD matrices and significantly boosting image set classification accuracy.

Class I and Class II Subdivision of the 20-meter Geodesic Dome Behavior and Analysis using Breakdown Methods 1 and 2 for Various Dome Frequencies

Class I and Class II Subdivision of the 20-meter Geodesic Dome Behavior and Analysis using Breakdown Methods 1 and 2 for Various Dome Frequencies

Data-driven surrogate model for aerodynamic design using separable shape tensor method

In the context of increasing dimensionality of design variables and the complexity of constraints, the efficacy of Surrogate-Based Optimization (SBO) is limited. The traditional linear and nonlinear dimensionality reduction algorithms are mainly to decompose the mathematical matrix composed of design variables or objective functions in various forms, the smoothness of the design space cannot be guaranteed in the process, and additional constraint functions need to be added in the optimization, which increases the calculation cost. This study presents a new parameterization method to improve both problems of SBO. The new parameterization is addressed by decoupling affine transformations (dilation, rotation, shearing, and translation) within the Grassmannian submanifold, which enables a separate representation of the physical information of the airfoil in a high-dimensional space. Building upon this, Principal Geodesic Analysis (PGA) is employed to achieve geometric control, compress the design space, reduce the number of design variables, reduce the dimensions of design variables and enhance predictive performance during the surrogate optimization process. For comparison, a dimensionality reduction space is defined using 95% of the energy, and RAE 2822 for transonic conditions are used as demonstrations. This method significantly enhances the optimization efficiency of the surrogate model while effectively enabling geometric constraints. In three-dimensional problems, it enables simultaneous design of planar shapes for various components of the aircraft and high-order perturbation deformations. Optimization was applied to the ONERA M6 wing, achieving a lift-drag ratio of 18.09, representing a 27.25% improvement compared to the baseline configuration. In comparison to conventional surrogate model optimization methods, which only achieved a 17.97% improvement, this approach demonstrates its superiority.

Geodesic analysis and steady accretion on a traversable wormhole

In this work, we analyze the behavior of light and matter as they pass near and through a traversable wormhole throat. In particular, we study the trajectories of massive and massless particles and the dust accretion around a traversable wormhole previously reported in Rueda et al. (2022). For massive particles, we integrate the trajectory equation for ingoing and outgoing geodesics and classify the orbits of particles scattered by the wormhole in accordance with their asymptotic behavior far from the throat. We represent all the time-like trajectories in an embedding surface where it is shown explicitly the trajectories of (i) particles that deviate from the throat and remain in the same universe, (ii) particles that traverse the wormhole to another universe, and (iii) particles that get trapped in the wormhole in unstable circular orbits. For the massless particles, we numerically integrate the trajectory equation to show the ray-tracing around the wormhole specifying the particles that traverse the wormhole and those that are only deviated by the throat. For the study of accretion, we consider the steady and spherically symmetric accretion of dust. Our results show that the wormhole parameters can significantly affect the behavior of light and matter near the wormhole. Some comparisons with the behavior of matter around black holes are made.

Numerical accuracy of principal geodesic analysis on the sphere in director‐based dynamics of hybrid mechanical systems

AbstractWe aim at nonlinear model order reduction (MOR) in hybrid mechanical systems by means of Principal Geodesic Analysis (PGA) on the Riemannian manifolds S2 (the sphere) and SO(3) (the rotation group). MOR requires highly accurate and efficient implementations of the logarithm maps and the resulting lifts across multiple branches. However, in our cases these maps have singularities due to periodicity. In this work we focus on the logarithm and lift maps for the sphere S2. We conduct detailed numerical experiments on mechanical systems to achieve maximal accuracy in spite of the singularities.

Qubit geodesics on the Bloch sphere from optimal-speed Hamiltonian evolutions

In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states.

Dynamic multi feature-class Gaussian process models.

In model-based medical image analysis, three relevant features are the shape of structures of interest, their relative pose, and image intensity profiles representative of some physical properties. Often, these features are modelled separately through statistical models by decomposing the object's features into a set of basis functions through principal geodesic analysis or principal component analysis. However, analysing articulated objects in an image using independent single object models may lead to large uncertainties and impingement, especially around organ boundaries. Questions that come to mind are the feasibility of building a unique model that combines all three features of interest in the same statistical space, and what advantages can be gained for image analysis. This study presents a statistical modelling method for automatic analysis of shape, pose and intensity features in medical images which we call the Dynamic multi feature-class Gaussian process models (DMFC-GPM). The DMFC-GPM is a Gaussian process (GP)-based model with a shared latent space that encodes linear and non-linear variations. Our method is defined in a continuous domain with a principled way to represent shape, pose and intensity feature-classes in a linear space, based on deformation fields. A deformation field-based metric is adapted in the method for modelling shape and intensity variation as well as for comparing rigid transformations (pose). Moreover, DMFC-GPMs inherit properties intrinsic to GPs including marginalisation and regression. Furthermore, they allow for adding additional pose variability on top of those obtained from the image acquisition process; what we term as permutation modelling. For image analysis tasks using DMFC-GPMs, we adapt Metropolis-Hastings algorithms making the prediction of features fully probabilistic. We validate the method using controlled synthetic data and we perform experiments on bone structures from CT images of the shoulder to illustrate the efficacy of the model for pose and shape prediction. The model performance results suggest that this new modelling paradigm is robust, accurate, accessible, and has potential applications in a multitude of scenarios including the management of musculoskeletal disorders, clinical decision making and image processing.

Shadow thermodynamics of AdS black hole with the nonlinear electrodynamics term

We creatively employ the shadow radius to study the thermodynamics of a charged AdS black hole with a nonlinear electrodynamics (NLED) term. First, the connection between the shadow radius and event horizon is constructed with the aid of the geodesic analysis. It turns out that the black hole shadow radius shows a positive correlation as a function of the event horizon radius. Then in the shadow context, we find that the black hole temperature and heat capacity can be presented by the shadow radius. Further analysis shows that the shadow radius can work similarly to the event horizon in revealing black hole phase transition process. In this sense, we construct the thermal profile of the charged AdS black hole with inclusion of the NLED effect. In the P < P c case, it is found that the N-type trend of the temperature given by the shadow radius is always consistent with that obtained by using the event horizon. Thus, we can conclude for the charged AdS black hole that the phase transition process can be intuitively presented as the thermal profile in the shadow context. Finally, the effects of NLED are carefully analyzed.

Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems

In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior.

Regge pole description of scattering by dirty black holes

We study the problem of plane monochromatic scalar waves impinging upon a Schwarzschild dirty black hole -- a Schwarzschild black hole surrounded by a thin spherical shell of matter -- using the complex angular momentum approach. We first recall general results concerning the differential scattering cross section in the classical limit through a null geodesic analysis by exploring different configurations of the shell. In particular, we show that dirty black hole spacetimes may exhibit various critical effects for geometrical optics. We compute the Regge pole spectrum for various shell configurations and show that it exhibits two or three distinct branches of poles, labelled inner surface waves, broad resonances and outer surface waves. In the latter, two sub-families have been identified, the surface waves associated with the outer light-ring and the creeping modes associated with the surface of the shell. We show, using WKB analysis, that the position of the shell sets the real part of the broad resonances while its energy-momentum and the discontinuity of the potential at the shell's surface set their imaginary part. Next, we provide the complex angular momentum representation of the differential scattering cross section and examine the role of the different Regge pole branches. We compute the differential scattering cross section for various configurations at several frequencies and show a very good agreement with the partial-wave calculations. Finally, we highlight the role of the critical effects, i.e., orbiting, glory, grazing and rainbow scattering, and their impact on the differential scattering cross section.

Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams).

This article presents a computational framework for the Principal Geodesic Analysis of merge trees (MT-PGA), a novel adaptation of the celebrated Principal Component Analysis (PCA) framework (K. Pearson 1901) to the Wasserstein metric space of merge trees (Pont et al. 2022). We formulate MT-PGA computation as a constrained optimization problem, aiming at adjusting a basis of orthogonal geodesic axes, while minimizing a fitting energy. We introduce an efficient, iterative algorithm which exploits shared-memory parallelism, as well as an analytic expression of the fitting energy gradient, to ensure fast iterations. Our approach also trivially extends to extremum persistence diagrams. Extensive experiments on public ensembles demonstrate the efficiency of our approach - with MT-PGA computations in the orders of minutes for the largest examples. We show the utility of our contributions by extending to merge trees two typical PCA applications. First, we apply MT-PGA to data reduction and reliably compress merge trees by concisely representing them by their first coordinates in the MT-PGA basis. Second, we present a dimensionality reduction framework exploiting the first two directions of the MT-PGA basis to generate two-dimensional layouts of the ensemble. We augment these layouts with persistence correlation views, enabling global and local visual inspections of the feature variability in the ensemble. In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.

Geodesic Logistic Analysis of Lumbar Spine Intervertebral Disc Shapes in Supine and Standing Positions.

Non-specific lower back pain (LBP) is a world-wide public health problem that affects people of all ages. Despite the high prevalence of non-specific LBP and the associated economic burdens, the pathoanatomical mechanisms for the development and course of the condition remain unclear. While intervertebral disc degeneration (IDD) is associated with LBP, there is overlapping occurrence of IDD in symptomatic and asymptomatic individuals, suggesting that degeneration alone cannot identify LBP populations. Previous work has been done trying to relate linear measurements of compression obtained from Magnetic Resonance Imaging (MRI) to pain unsuccessfully. To bridge this gap, we propose to use advanced non-Euclidean statistical shape analysis methods to develop biomarkers that can help identify symptomatic and asymptomatic adults who might be susceptible to standing-induced LBP. We scanned 4 male and 7 female participants who exhibited lower back pain after prolonged standing using an Open Upright MRI. Supine and standing MRIs were obtained for each participant. Patients reported their pain intensity every fifteen minutes within a period of 2 h. Using our proposed geodesic logistic regression, we related the structure of their lower spine to pain and computed a regression model that can delineate lower spine structures using reported pain intensities. These results indicate the feasibility of identifying individuals who may suffer from lower back pain solely based on their spinal anatomy. Our proposed spinal shape analysis methodology have the potential to provide powerful information to the clinicians so they can make better treatment decisions.

Parkinsonian gait patterns quantification from principal geodesic analysis

Parkinsonian gait patterns quantification from principal geodesic analysis

Carbon neutrality vs. neutralité carbone: A comparative study on French and English users’ perceptions and social capital on Twitter

Carbon neutrality is one of the most critical global concerns at present. As one of the largest social media, Twitter is used widely by individuals, organisations, and government agencies to share their comments and perceptions on carbon neutrality. This study collected 26425 English and 20331 French tweets to compare the differences between French and English tweets. Social network analysis found that users in the French social networks interacted more frequently than the English ones. The geodesic analysis evidenced that the connection of any two users required about five intermediate users on average in French networks, while English ones required seven intermediate users. The modularity metrics of the English network were higher, indicating that users in English networks did not communicate with different clusters and people in carbon neutrality issues. In addition, the French network of carbon neutrality activists comprised politicians, government agencies, journalists, NGOs, and companies, while those in the English network mainly included companies, media, and politicians. Sentiment analysis and independent samples t-test have confirmed that despite the types of activists and the interactions between clusters being different, negative Tweets were more than positive ones in English and French networks, especially in French networks. It may be caused by people’s dissatisfaction with the government’s current carbon neutrality policy. By analysing the social pattern on Twitter, the research results allow people to know more about the means to enhance carbon-neutral knowledge sharing, which has the policy and social significance for addressing climate change.

Charged black-bounce spacetimes: Photon rings, shadows and observational appearances

The photon ring, shadow and observational appearance of the emission originating near a charged black-bounce are investigated. Based on the geodesic analysis, we determine the upper and lower limits of critical impact parameters of a charged black-bounce. In particular, we find that the charged black-bounce shares the same critical impact parameter with the Reissner-Nordström black hole. In addition, we classify the light trajectories coming from the region near the charged black-bounce by utilizing the rays tracing procedure, and then investigate the observational appearance of the emissions from a thin disk accretion and a spherically symmetric infalling accretion. We reveal that a large charge increases the observed intensity but decreases the apparent size of shadows, and that the photon ring presents the intrinsic property of a spacetime geometry, which is independent of the types of the two accretions. Our results are in good agreement with the recent observations.

Parametrized black holes: scattering investigation

We study the scattering of light-like geodesics and massless scalar waves by a static Konoplya–Zhidenko black hole, considering the case that the parametrized black hole solution contains a single deformation parameter. By performing a geodesic analysis, we compute the classical differential scattering cross section and probe the influence of the deformation parameter on null trajectories. Moreover, we investigate the propagation of a massless scalar field in the vicinity of the static Konoplya–Zhidenko black hole and use the plane waves formalism to compute the differential scattering cross section. We confront our numerical results in the backward direction with the glory approximation, finding excellent agreement. We compare the results for the deformed black hole with the Schwarzschild case, finding that the additional parameter has an important role in the behavior of the scattering process for moderate-to-high scattering angles.