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In this work, we study the geodesics in different types of Carrollian RN (Reissner-Nordstr\"om) black holes, considering the motions of both neutral and charged particles. We use the geodesic equations in the weak Carrollian structure and analyze the corresponding trajectories projected onto the absolute space, and find that the geodesics are well-defined. In particular, we examine the electric-electric and magnetic-electric limit of the RN black hole, focusing on their geodesic structures. We find that the global structures of the usual RN black holes get squeezed under the ultra-relativistic limit. More precisely, the nonextreme magnetic-electric RN spacetime has two different asymptotic flat patches while the extreme black hole spacetime consists of only one patch. For the magnetic-electric RN spacetime, the Carrollian extremal surfaces (CESs) divide the spacetime into several geodesically complete regions, and the geodesics can only travel in one of these regions. For the charged particles, we extend the analysis by considering their interactions with the electromagnetic field in the Carrollian RN spacetimes and find that their trajectories are significantly different from the neutral geodesics., Comment: 27 pages, 2 figures, fixed a figure display issue

In this work, we study the quantization of Carrollian conformal scalar theories, including two-dimensional(2D) magnetic scalar and three-dimensional(3D) electric and magnetic scalars. We discuss two different quantization schemes, depending on the choice of the vacuum. We show that the standard canonical quantization corresponding to the induced vacuum yields a unitary Hilbert space and the 2-point correlation functions in this scheme match exactly with the ones computed from the path integral. In the canonical quantization, the BMS symmetry can be realized without anomaly. On the other hand, for the quantization based on the highest-weight vacuum, it does not have a unitary Hilbert space. In 2D, the correlators in the highest-weight vacuum agree with the ones obtained by taking the $c\to 0$ limit of the 2D CFT, and there is an anomalous term in the commutation relations between the Virasoso generators, whose form is similar to the one in 2D CFT. In 3D, there is no good definition of the highest-weight vacuum without breaking the rotational symmetry. In our study, we find that the usual state-operator correspondence in CFT does not hold in the Carrollian case., Comment: 44 pages, 3 figures. v2: ctations added

In this work, we study the duality symmetry group of Carrollian (nonlinear) electrodynamics and propose a family of Carrollian ModMax theories, which are invariant under Carrollian $\text{SO}(2)$ electromagnetic (EM) duality transformations and conformal transformation. We define the Carrollian $\text{SO}(2)$ EM transformations, with the help of Hodge duality in Carrollian geometry, then we rederive the Gaillard-Zumino consistency condition for EM duality of Carrollian nonlinear electrodynamics. Together with the traceless condition for the energy-momentum tensor, we are able to determine the Lagrangian of the Carrollian ModMax theories among pure electrodynamics. We furthermore study their behaviors under the $\sqrt{T\bar{T}}$ deformation flow, and show that these theories deform to each other and may reach two endpoints under the flow, with one of the endpoint being the Carrollian Maxwell theory. As a byproduct, we construct a family of two-dimensional Carrollian ModMax-like multiple scalar theories, which are closed under the $\sqrt{T\bar{T}}$ flow and may flow to a BMS free multi-scalar model., Comment: 39 pages, references added, major revised, results unchanged

In this paper, we propose a novel way to construct off-shell actions of $d$-dimensional Carrollian field theories by considering the null-reduction of the Bargmann invariant actions in $d+1$ dimensions. This is based on the fact that $d$-dimensional Carrollian symmetry is the restriction of the $(d+1)$-dimensional Bargmann symmetry to a null hyper-surface. We focus on free scalar field theory and electromagnetic field theory, and show that the electric and magnetic sectors of these theories originate from different Bargmann invariant actions in one higher dimension. In the cases of the massless free scalar field and $d=4$ electromagnetic field, we verify Carrollian conformal invariance of the resulting theories, and find that there appear naturally chain representations and staggered modules of Carrollian conformal algebra., Comment: 59 pages, major revisions, results unchanged

Safety performance evaluation is critical for developing and deploying connected and automated vehicles (CAVs). One prevailing way is to design testing scenarios using prior knowledge of CAVs, test CAVs in these scenarios, and then evaluate their safety performances. However, significant differences between CAVs and prior knowledge could severely reduce the evaluation efficiency. Towards addressing this issue, most existing studies focus on the adaptive design of testing scenarios during the CAV testing process, but so far they cannot be applied to high-dimensional scenarios. In this paper, we focus on the adaptive safety performance evaluation by leveraging the testing results, after the CAV testing process. It can significantly improve the evaluation efficiency and be applied to high-dimensional scenarios. Specifically, instead of directly evaluating the unknown quantity (e.g., crash rates) of CAV safety performances, we evaluate the differences between the unknown quantity and known quantity (i.e., control variates). By leveraging the testing results, the control variates could be well designed and optimized such that the differences are close to zero, so the evaluation variance could be dramatically reduced for different CAVs. To handle the high-dimensional scenarios, we propose the sparse control variates method, where the control variates are designed only for the sparse and critical variables of scenarios. According to the number of critical variables in each scenario, the control variates are stratified into strata and optimized within each stratum using multiple linear regression techniques. We justify the proposed method's effectiveness by rigorous theoretical analysis and empirical study of high-dimensional overtaking scenarios.

Testing and evaluation is a crucial step in the development and deployment of Connected and Automated Vehicles (CAVs). To comprehensively evaluate the performance of CAVs, it is of necessity to test the CAVs in safety-critical scenarios, which rarely happen in naturalistic driving environment. Therefore, how to purposely and systematically generate these corner cases becomes an important problem. Most existing studies focus on generating adversarial examples for perception systems of CAVs, whereas limited efforts have been put on the decision-making systems, which is the highlight of this paper. As the CAVs need to interact with numerous background vehicles (BVs) for a long duration, variables that define the corner cases are usually high dimensional, which makes the generation a challenging problem. In this paper, a unified framework is proposed to generate corner cases for the decision-making systems. To address the challenge brought by high dimensionality, the driving environment is formulated based on Markov Decision Process, and the deep reinforcement learning techniques are applied to learn the behavior policy of BVs. With the learned policy, BVs will behave and interact with the CAVs more aggressively, resulting in more corner cases. To further analyze the generated corner cases, the techniques of feature extraction and clustering are utilized. By selecting representative cases of each cluster and outliers, the valuable corner cases can be identified from all generated corner cases. Simulation results of a highway driving environment show that the proposed methods can effectively generate and identify the valuable corner cases., Comment: 23 pages, 11 figures

Microscopic traffic simulation provides a controllable, repeatable, and efficient testing environment for autonomous vehicles (AVs). To evaluate AVs' safety performance unbiasedly, the probability distributions of environment statistics in the simulated naturalistic driving environment (NDE) need to be consistent with those from the real-world driving environment. However, although human driving behaviors have been extensively investigated in the transportation engineering field, most existing models were developed for traffic flow analysis without considering the distributional consistency of driving behaviors, which could cause significant evaluation biasedness for AV testing. To fill this research gap, a distributional consistent NDE modeling framework is proposed in this paper. Using large-scale naturalistic driving data, empirical distributions are obtained to construct the stochastic human driving behavior models under different conditions. To address the error accumulation problem during the simulation, an optimization-based method is further designed to refine the empirical behavior models. Specifically, the vehicle state evolution is modeled as a Markov chain and its stationary distribution is twisted to match the distribution from the real-world driving environment. The framework is evaluated in the case study of a multi-lane highway driving simulation, where the distributional accuracy of the generated NDE is validated and the safety performance of an AV model is effectively evaluated., Comment: 13 pages, 11 figures

How to generate testing scenario libraries for connected and automated vehicles (CAVs) is a major challenge faced by the industry. In previous studies, to evaluate maneuver challenge of a scenario, surrogate models (SMs) are often used without explicit knowledge of the CAV under test. However, performance dissimilarities between the SM and the CAV under test usually exist, and it can lead to the generation of suboptimal scenario libraries. In this paper, an adaptive testing scenario library generation (ATSLG) method is proposed to solve this problem. A customized testing scenario library for a specific CAV model is generated through an adaptive process. To compensate the performance dissimilarities and leverage each test of the CAV, Bayesian optimization techniques are applied with classification-based Gaussian Process Regression and a new-designed acquisition function. Comparing with a pre-determined library, a CAV can be tested and evaluated in a more efficient manner with the customized library. To validate the proposed method, a cut-in case study was performed and the results demonstrate that the proposed method can further accelerate the evaluation process by a few orders of magnitude., Comment: 10 pages, 10 figures

Testing scenario library generation (TSLG) is a critical step for the development and deployment of connected and automated vehicles (CAVs). In Part I of this study, a general methodology for TSLG is proposed, and theoretical properties are investigated regarding the accuracy and efficiency of CAV evaluation. This paper aims to provide implementation examples and guidelines, and to enhance the proposed methodology under high-dimensional scenarios. Three typical cases, including cut-in, highway-exit, and car-following, are designed and studied in this paper. For each case, the process of library generation and CAV evaluation is elaborated. To address the challenges brought by high dimensions, the proposed methodology is further enhanced by reinforcement learning technique. For all three cases, results show that the proposed methods can accelerate the CAV evaluation process by multiple magnitudes with same evaluation accuracy, if compared with the on-road test method., Comment: 12 pages, 13 figures