Four-dimensional Metrics Research Articles

Repaired coarctation of the aorta does not affect four-dimensional flow metrics in bicuspid aortic valve disease.

The objective of this study was primarily to compare four-dimensional flow magnetic resonance imaging metrics in the ascending aorta (AA) of patients with right-left fusion type bicuspid aortic valve (RL-BAV) and repaired coarctation of the aorta (CoA) to RL-BAV without CoA. Metrics of patients with RL-BAV were also compared to the matched group of patients with common tricuspid aortic valve (TAV). Eleven patients with RL-BAV and CoA, 11 patients with RL-BAV without CoA and 22 controls with TAV were investigated. Peak velocity (cm/s), peak flow (ml/s) and flow displacement (%) were analysed at 5 pre-defined AA levels. In addition, regional wall shear stress (WSS, mN/m2), circumferential WSS (WSSc) and axial WSS (WSSa) at all levels were quantified in 6 sectors of the aortic circle. Averaged WSS values on each level (WSSavg, WSSc, avg and WSSa, avg) were calculated as well. Peak velocity at the proximal tubular AA was significantly lower in BAV and CoA group (P = 0.047) compared to BAV without CoA. In addition, the WSSa, avg was found to be higher for the BAV and CoA group at proximal AA respectively (P = 0.040). No other significant differences were found between these groups. BAV group's peak velocity was higher at every level (P < 0.001-0.004) compared to TAV group. Flow displacement was significantly higher for the BAV group at every level (P < 0.001) besides at the most distal level. All averaged WSS values were significantly higher in BAV patients in distal AA (P < 0.001-0.018). Repaired CoA does not relevantly alter four-dimensional flow metrics in the AA of patients with RL-BAV. However, RL-BAV majorly alters flow dynamics in the AA when compared to patients with TAV. https://www.clinicaltrials.gov/study/NCT05065996, Unique Protocol ID 5063566.

Model-Based Predictive Control and Reinforcement Learning for Planning Vehicle-Parking Trajectories for Vertical Parking Spaces.

This paper proposes a vehicle-parking trajectory planning method that addresses the issues of a long trajectory planning time and difficult training convergence during automatic parking. The process involves two stages: finding a parking space and parking planning. The first stage uses model predictive control (MPC) for trajectory tracking from the initial position of the vehicle to the starting point of the parking operation. The second stage employs the proximal policy optimization (PPO) algorithm to transform the parking behavior into a reinforcement learning process. A four-dimensional reward function is set to evaluate the strategy based on a formal reward, guiding the adjustment of neural network parameters and reducing the exploration of invalid actions. Finally, a simulation environment is built for the parking scene, and a network framework is designed. The proposed method is compared with the deep deterministic policy gradient and double-delay deep deterministic policy gradient algorithms in the same scene. Results confirm that the MPC controller accurately performs trajectory-tracking control with minimal steering wheel angle changes and smooth, continuous movement. The PPO-based reinforcement learning method achieves shorter learning times, totaling only 30% and 37.5% of the deep deterministic policy gradient (DDPG) and twin-delayed deep deterministic policy gradient (TD3), and the number of iterations to reach convergence for the PPO algorithm with the introduction of the four-dimensional evaluation metrics is 75% and 68% shorter compared to the DDPG and TD3 algorithms, respectively. This study demonstrates the effectiveness of the proposed method in addressing a slow convergence and long training times in parking trajectory planning, improving parking timeliness.

BRST BMS4 symmetry and its cocycles from horizontality conditions

The BRST structure of the extended Bondi-Metzner-Sachs symmetry group of asymptotically flat manifolds is investigated using the recently introduced framework of the Beltrami field parametrization of four-dimensional metrics. The latter identifies geometrically the two physical degrees of freedom of the graviton as fundamental fields. The graded BRST BMS4 nilpotent differential operator relies on four horizontality conditions giving a Lagrangian reformulation of the asymptotic BMS4 symmetry. A series of cocycles is found which indicate the possibility of anomalies for three-dimensional Lagrangian theories to be built in the null boundaries of asymptotically flat spaces from the principle of BRST BMS4 invariance.

Superluminal propagation on a moving braneworld

We consider a braneworld scenario in the simplest setting, $M_4 \times S^1$, with a 4D Minkowski metric induced on the brane, and establish the possibility of superluminal propagation. If the brane is at rest, the 4D Lorentz symmetry of the brane is exact, but if the brane is in motion, it is broken globally by the compactification. By measuring bulk fields, an observer on the brane sees a slice through a higher-dimensional field profile, which carries an imprint of the extra dimensions even when the brane is at rest. If the brane is in motion we find that bulk fields can propagate outside the brane lightcone by a parametrically large amount set by the brane velocity. We mention observational tests and possible applications to cosmology.

Generalized quasi-Einstein metrics and applications on generalized Robertson–Walker spacetimes

In this paper, we study generalized quasi-Einstein manifolds (Mn, g, V, λ) satisfying certain geometric conditions on its potential vector field V whenever it is harmonic, conformal, and parallel. First, we construct some integral formulas and obtain some triviality results. Then, we find some necessary conditions to construct a quasi-Einstein structure on (Mn, g, V, λ). Moreover, we prove that for any generalized Ricci soliton (M̄=I×fM,ḡ,ξ̄,λ), where ḡ is a generalized Robertson–Walker spacetime metric and the potential field ξ̄=h∂t+ξ is conformal, (M̄,ḡ) can be considered as the model of perfect fluids in general relativity. Moreover, the fiber (M, g) also satisfies the quasi-Einstein metric condition. Therefore, the state equation of (M̄=I×fM,ḡ) is presented. We also construct some explicit examples of generalized quasi-Einstein metrics by using a four-dimensional Walker metric.

Spinors of real type as polyforms and the generalized Killing equation

We develop a new framework for the study of generalized Killing spinors, where every generalized Killing spinor equation, possibly with constraints, can be formulated equivalently as a system of partial differential equations for a polyform satisfying algebraic relations in the Kähler–Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Sigma of real type as a real algebraic variety in the Kähler–Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of text {AdS}_4 space-time.

Four-dimensional Einstein metrics from biconformal deformations

Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. One particular family of examples have ends that collapse asymptotically to ${\mathbb R}^2$.

Ultrarelativistic spinning objects in nonlocal ghost-free gravity

We study the gravitational field of ultrarelativistic spinning objects (gyratons) in a modified gravity theory with higher derivatives. In particular, we focus on a special class of such theories with an infinite number of derivatives known as "ghost-free gravity" that include a non-local form factor such as $\exp(-\Box\ell^2)$, where $\ell$ is the scale of non-locality. First, we obtain solutions of the linearized ghost-free equations for stationary spinning objects. To obtain gyraton solutions we boost these metrics and take their Penrose limit. This approach allows us to perform calculations for any number of spacetime dimensions. All solutions are regular at the gyraton axis. In four dimensions, when the scale non-locality $\ell$ tends to zero, the obtained gyraton solutions correctly reproduce the Aichelburg--Sexl metric and its generalization to spinning sources found earlier by Bonnor. We also study the properties of the obtained four-dimensional and higher-dimensional ghost-free gyraton metrics and briefly discuss their possible applications.

The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors

We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the $2$-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants $C_\rho$ and $\ell_{\mathcal C}$ are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterise the Van den Bergh metric. To understand the non-generic cases, we also find all $\Lambda$-vacuum metrics that are generic in the above sense, except that either $C_\rho$ or $\ell_{\mathcal C}$ is zero. In this way we extend the Kundu class to $\Lambda$-vacuum metrics. The results of the paper can be exploited for invariant characterisation of classes of metrics and for extension of the set of known solutions of the Einstein equations.

A note on the collapsing geometry of hyperkähler four manifolds

We make some observations concerning the one-dimensional collapsing geometry of four-dimensional hyperkahler metrics.

Four-dimensional Lorentzian plane symmetric static Ricci solitons

Our focus is to investigate the Ricci solitons of the plane symmetric and static four-dimensional Lorentzian metrics. It is found that these metrics admit shrinking and concircular potential Ricci soliton vector fields with either 6- or 10-dimensional Lie algebra. Further, it is observed that the 4-dimensional Lorentzian static Ricci soliton manifolds are Einsteinian and hence the Ricci solitons are the trivial Ricci solitons.

Type D spacetimes and the Weyl double copy

We study the double-copy relation between classical solutions in gauge theory and gravity, focusing on four-dimensional vacuum metrics of algebraic type D, a class that includes several important solutions. We present a double copy of curvatures that applies to all spacetimes of this type—the Weyl double copy—relating the curvature of the spacetime to an electromagnetic field strength. We show that the Weyl double copy is consistent with the previously known Kerr–Schild double copy, and in fact resolves certain ambiguities of the latter. The most interesting new example of the classical double copy presented here is that of the C-metric. This well-known solution, which represents a pair of uniformly accelerated black holes, is mapped to the Liénard–Wiechert potential for a pair of uniformly accelerated charges. We also present a new double-copy interpretation of the Eguchi–Hanson instanton.

Cylindrical Space–Time Model and Mirror Symmetry Violations

The symmetrical character of empty space–time is proposed as a putative reason for mirror symmetry violations in the bioorganic world and in the world of elementary particles, and a new geometrical model for the empty space–time is put forward. In contrast to flat four-dimensional Minkowski space, this model is described by a cylindrical four-dimensional space metrics. The geodesic lines of this space are helical. The helical structure of the model proposed shows that distortion of mirror symmetry may be a consequence of geometrical structure of the empty space rather than a characteristic of any specific interaction.

Geometrization conditions for perfect fluids, scalar fields, and electromagnetic fields

Rainich-type conditions giving a spacetime “geometrization” of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and electromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equations are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Geometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and formulas for constructing the scalar field from the metric are unified and extended to arbitrary dimensions, to include a cosmological constant, and to include any self-interaction potential. Necessary and sufficient conditions on a four-dimensional spacetime metric for it to be an electrovacuum and formulas for constructing the electromagnetic field from the metric are generalized to include a cosmological constant. Both null and non-null electromagnetic fields are treated. A number of examples and applications of these results are presented.

Gravity duals of $ \mathcal{N} $ = 2 superconformal field theories with no electrostatic description

Abstract We construct the first eleven-dimensional supergravity solutions, which are regular, have no smearing and possess only SO(2, 4) × SO(3) × U(1) R isometry. They are dual to four-dimensional field theories with $ \mathcal{N} $ = 2 superconformal symmetry. We utilise the Toda frame of self-dual four-dimensional Euclidean metrics with SU(2) rotational symmetry. They are obtained by transforming the Atiyah-Hitchin instanton under SL(2, $ \mathbb{R} $ ) and are expressed in terms of theta functions. The absence of any extra U(1) symmetry, even asymptotically, renders inapplicable the electrostatic description of our solution.

Black hole solutions to the [formula omitted]-model and their orbits (I)

In this paper we continue the program of the classification of nilpotent orbits using the approach developed in arXiv:1107.5986, within the study of black hole solutions in D=4 supergravities. Our goal in this work is to classify static, single center black hole solutions to a specific N=2 four-dimensional “magic” model, with special Kähler scalar manifold Sp(6,R)/U(3), as orbits of geodesics on the pseudo-quaternionic manifold F4(4)/[SL(2,R)×Sp′(6,R)] with respect to the action of the isometry group F4(4). Our analysis amounts to the classification of the orbits of the geodesic “velocity” vector with respect to the isotropy group H⁎=SL(2,R)×Sp′(6,R), which include a thorough classification of the nilpotent orbits associated with extremal solutions and reveals a richer structure than the one predicted by the β–γ-labels alone, based on the Kostant–Sekiguchi approach. We provide a general proof of the conjecture made in hep-th/0908.1742 which states that regular single center solutions belong to orbits with coinciding β–γ-labels. We also prove that the reverse is not true by finding distinct orbits with the same β–γ-labels, which are distinguished by suitably devised tensor classifiers. Only one of these is generated by regular solutions. Since regular static solutions only occur with nilpotent degree not exceeding 3, we only discuss representatives of these orbits in terms of black hole solutions. We prove that these representatives can be found in the form of a purely dilatonic four-charge solution (the generating solution in D=3) and this allows us to identify the orbit corresponding to the regular four-dimensional metrics. H⁎-orbits with degree of nilpotency greater than 3 are analyzed solely from a group theoretical point of view, leaving a systematic analysis of their possible interpretation in terms of static multicenter or stationary non-static solutions to a future work. We just limit ourselves to give (singular) single-center representatives of these orbits, to be possibly interpreted as singular limits of regular multicenter solutions. We provide the explicit transformations mapping the various H⁎-orbits and in particular BPS into non-BPS regular solutions showing that they in general belong to the complexification of the global symmetry group in D=3.

Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor

AbstractCorrespondence between torsion-free connections with nilpotent skew-symmetric curvature operator and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces.

Four-dimensional Osserman metrics of neutral signature

In the algebraic context, we show null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2, 2). We also classify the null Jordan Osserman models of signature (2, 2). In the geometric context, we show that a pseudo-Riemannian manifold with this signature is null Jordan Osserman if and only if either it has constant sectional curvature or it is locally a complex space form. © 2010 Pacific Journal of Mathematics.

New Examples of Einstein Metrics in Dimension Four

We construct new examples of four‐dimensional Einstein metrics with neutral signature and two‐dimensional holonomy Lie algebra.

Block diagonalization of four-dimensional metrics

It is shown that, in four dimensions, it is possible to introduce coordinates so that an analytic metric locally takes block diagonal form, i.e. one can find coordinates such that gαβ = 0 for (α, β) ∊ S where S = {(1, 3), (1, 4), (2, 3), (2, 4)}. We call a coordinate system in which the metric takes this form a ‘doubly biorthogonal coordinate system’. We show that all such coordinate systems are determined by a pair of coupled second-order partial differential equations.