4D Space-time Research Articles

Real-time methods in JT/SYK holography

Abstract We study the conventional holographic recipes and its real-time extensions in the context of the correspondence between SYK quantum mechanics and JT gravity. We first observe that only closed contours are allowed to have a 2d space-time holographic dual. Thus, in any real-time formulation of the duality, the boundaries of a classical connected geometry are a set of closed curves, parameterized by a complex \emph{closed} time contour as in the Schwinger-Keldysh framework. Thereby, a consistent extension of the standard holographic formulas is proposed, in order to describe the correspondence between gravity and boundary quantum models that include averaging on the coupling constants. We investigate our prescription in different AdS
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 solutions with Schwinger-Keldysh boundary condition, dual to a boundary quantum theory at finite temperature defined on a complex time contour, and consider also classical, asymptotically AdS solutions (wormholes) with two disconnected boundaries. In doing this, we revisit the so-called factorization problem, and its resolution in conventional holography by virtue of some (non-local) coupling between disconnected boundaries, and we show how in specific contexts, the averaging proposal by-passes the paradox as well, since it induces a similar effective coupling.

A new perspective on Kaluza–Klein theories

Assuming that the geometry of spacetime is uniquely determined by the energy–momentum tensor of matter alone, i.e. without any interactions, enables us to construct the Lagrangian from which the metric of higher dimensional spacetime follows. From the geodesic equations that follow, it becomes clear that the incorrect mass of elementary particles predicted by Kaluza–Klein theories arises from the assumption that in the absence of gravity the solution to the Einstein field equations reduces to the Minkowski metric. From construction of a consistent theory of 4D electromagnetism, we find that this assumption does not only result in the incorrect mass of elementary particles, but also the incorrect value of the cosmological constant. This suggests that these incorrect predictions, which are often regarded as major flaws of Kaluza–Klein theories, just reflect the inconsistency of the assumption that the solution to Einstein field equations reduces to Minkowski metric in the absence of gravity and Weyl invariance which is the symmetry of gauge theories in 4D spacetime. Abandoning this assumption results in modification of general relativity. We show that the unified description of fundamental interactions naturally incorporates the Higgs mechanism. For non-Abelian gauge fields, we find that the manifold comprising the extra dimensions has to be a group manifold and show that the standard model is realized in 16D spacetime. We show that charge and spin are the same concept, but what makes them different is that the former follows from symmetry of 4D spacetime while the latter follows from symmetry of the internal space.

Higher-spin self-dual General Relativity: 6d and 4d pictures, covariant vs. lightcone

We study the higher-spin extension of self-dual General Relativity (GR) with cosmological constant, proposed by Krasnov, Skvortsov and Tran. We show that this theory is actually a gauge-fixing of a 6d diffeomorphism-invariant Abelian theory, living on (4d spacetime)×(2d spinor space) modulo a finite group. On the other hand, we point out that the theory respects the 4d geometry of a self-dual GR solution, with no backreaction from the higher-spin fields. We also present a lightcone ansatz that reduces the covariant fields to one scalar field for each helicity. The field equations governing these scalars have only cubic vertices. We compare our lightcone ansatz to Metsaev’s lightcone formalism. We conclude with a new perspective on the lightcone formalism in (A)dS spacetime: not merely a complication of its Minkowski-space cousin, it has a built-in Lorentz covariance, and is closely related to Vasiliev’s concept of unfolding.

On simulations of 3D fractional WBBM model through mathematical and graphical analysis with the assists of fractionality and unrestricted parameters

This study retrieves some novel exact solutions to the family of 3D space–time fractional Wazwaz–Benjamin–Bona–Mahony (WBBM) equations in the context of diverse nonlinear physical phenomena resulting from water wave mechanics. The family of WBBM equations is transformed for this purpose using a space and time fractional transformation into an ordinary differential equation (ODE). The ODE then uses a strong method, namely the Unified Method. Consequently, lump solutions, dark-bright soliton, singular and multiple soliton solutions, and periodic solutions are investigated. The disparities between the current study's conclusions and previously acquired solutions via other approaches are examined. All wave solutions produced are determined to be novel in terms of fractionality, unrestricted parameters, and implemented technique sense. The impact of unrestricted parameters and fractionality on the obtained solutions are visually presented, along with physical explanations. It is observed that the wave portents are varied with the increase of unrestricted parameters as well as fractionality. We dynamically showed that the appropriate transformation and the applied Unified approach more proficient in the study of water wave dynamics and might be used in future researches to clarify the many physical phenomena. The novelty of this work validate that the proposed method seem simple and useful tools for obtaining the solutions in PDEs and it is expected to use in mathematical physics and optical engineering.

Dirac equation for photons in a fibre: Origin of polarisation

Spin is a fundamental degree of freedom, which was discovered by Dirac for an electron in his relativistic quantum mechanics, known as the Dirac equation. The origin of spin for a photon is unclear because Maxwell's equations in a vacuum are Lorentz invariant without introducing the concept of spin. Here, the propagation of coherent rays of photons in a graded-index optical fibre is considered to discuss the origin of polarisation for photons using exact solutions of the Laguerre-Gauss and Hermite-Gauss modes. The energy spectrum is massive, and the effective mass is a function of the confinement and orbital angular momentum. The propagation is described by the one-dimensional (1D) non-relativistic Schrödinger equation, which is equivalent to the 2D space-time Klein-Gordon equation by a unitary transformation. The probabilistic interpretation and the conservation law require the factorisation of the Klein-Gordon equation, leading to the 2D Dirac equation with spin. The spin expectation values of photons correspond to the polarisation state on the Poincaré sphere. As an application of the theory, a polarisation interferometer is proposed, whose energy spectrum shows a Dirac cone in the Stokes parameter space.

Gravitational fields of axially symmetric compact objects in 5D space–time–matter gravity

In the standard Einstein’s theory the exterior gravitational field of any static and axially symmetric stellar object can be described by means of a single function from which we obtain a metric into a four-dimensional space–time. In this work we present a generalization of those so called Weyl solutions to a space–time–matter metric in a five-dimensional manifold within a non-compactified Kaluza–Klein theory of gravity. The arising field equations reduce to those of vacuum Einstein’s gravity when the metric function associated to the fifth dimension is considered to be constant. The calculation of the geodesics allows to identify the existence or not of different behaviours of test particles, in orbits on a constant plane, between the two metrics. In addition, static solutions on the hypersurface orthogonal to the added dimension but with time dependence in the five-dimensional metric are also obtained. The consequences on the variation of the rest mass, if the fifth dimension is identified with it, are studied.

Spatio-Temporal Visual Analysis of Turbulent Superstructures in Unsteady Flow.

The large-scale motions in 3D turbulent channel flows, known as Turbulent Superstructures (TSS), play an essential role in the dynamics of small-scale structures within the turbulent boundary layer. However, as of today, there is no common agreement on the spatial and temporal relationships between these multiscale structures. We propose a novel space-time visualization technique for analyzing the temporal evolution of these multiscale structures in their spatial context and, thus, to further shed light on the conceptually different explanations of their dynamics. Since the temporal dynamics of TSS are believed to influence the structures in the turbulent boundary layer, we propose a combination of a 2D space-time velocity plot with an orthogonal 2D plot of projected 3D flow structures, which can interactively span the time and the space axis. Besides flow structures indicating the fluid motion, we propose showing the variations in derived fields as an additional source of explanation. The relationships between the structures in different spatial and temporal scales can be more effectively resolved by using various filtering operations and image registration algorithms. To reduce the information loss due to the non-injective nature of projection, spatial information is encoded into transparency or color. Since the proposed visualization is heavily demanding computational resources and memory bandwidth to stream unsteady flow fields and instantly compute derived 3D flow structures, the implementation exploits data compression, parallel computation capabilities, and high memory bandwidth on recent GPUs via the CUDA compute library.

The fourth dimension of space and the entanglement principle

This paper focuses on the reason behind the entanglement phenomena, which is still an unknown fundamental question. It argues that while two entangled particles are separated in the 3D space the flow of information between them is instantaneous through the 4D space. The fourth dimension of space has the unit of measurement of length like the other three space dimensions. We shall call it Q, from the word quatre (four) in French. Its magnitude is equal to the invariant proper time τ times the speed of light c. Q = cτ. Each point in the 4D space corresponds to an event. Distances between events in the four-dimensional space can be computed using the Pythagoras theorem. It is argued that time t as measured by clock in our 3D space times the speed of light gives the length of the position vector of an event in the 4D space. At time t, when an observation is made at one of the entangled particles, the length of the position vectors of both entangled particles, sharing the same wave function, in 4D space are equal to t times the speed of light, i.e., ct. The time t is also the same for all the points on the line connecting the two events in the 3D space. Consequently, the length of the position vectors of all the points on the line connecting the two events in the 3D space is equal to ct in the 4D space. The corresponding points in the 4D space make an arc of equidistance to the origin. It is conjectured that the flow of information between the two entangled particles through this arc in the 4D space is not carried by energy or mass particles. It is, therefore, not bound by the time nor the speed of light constraint in 4D space and is instantaneous.

Quantum gravity through geometric algebra

Geometric Algebra is used to construct a model of quantum gravity in which the 4D spacetime manifold is built from qubitic quantum entangled spacetime elements of geometric algebraic nature. This is achieved by first investigating the spontaneous decomposition of the spacetime continuum’s metric structure at Planck distances as one approaches what otherwise would become the Universe’s initial singularity. Considering this metric deconstruction within the model one finds that the reverse induced emergence of the spacetime continuum is generated by a coupling action triggered by the embedding action between distinct pre-spacetime manifolds. Further uncovered in this re-engineering of spacetime is a Poincaré-constrained quantum foam existing at infra-Planck scales.

Fusion rules and shrinking rules of topological orders in five dimensions

As a series of work about 5D (spacetime) topological orders, here we employ the path-integral formalism of 5D topological quantum field theory (TQFT) established in Zhang and Ye, JHEP04 (2022) 138 to explore non-Abelian fusion rules, hierarchical shrinking rules and quantum dimensions of particle-like, loop-like and membrane-like topological excitations in 5D topological orders. To illustrate, we focus on a prototypical example of twisted BF theories that comprise the twisted topological terms of the BBA type. First, we classify topological excitations by establishing equivalence classes among all gauge-invariant Wilson operators. Then, we compute fusion rules from the path-integral and find that fusion rules may be non-Abelian; that is, the fusion outcome can be a direct sum of distinct excitations. We further compute shrinking rules. Especially, we discover exotic hierarchical structures hidden in shrinking processes of 5D or higher: a membrane is shrunk into particles and loops, and the loops are subsequently shrunk into a direct sum of particles. We obtain the algebraic structure of shrinking coefficients and fusion coefficients. We compute the quantum dimensions of all excitations and find that sphere-like membranes and torus-like membranes differ not only by their shapes but also by their quantum dimensions. We further study the algebraic structure that determines anomaly-free conditions on fusion coefficients and shrinking coefficients. Besides BBA, we explore general properties of all twisted terms in 5D. Together with braiding statistics reported before, the theoretical progress here paves the way toward characterizing and classifying topological orders in higher dimensions where topological excitations consist of both particles and spatially extended objects.

Nonlocality, Superposition, and Time in the 4+1 Formalism.

The field of quantum gravity struggles with several problems related to time, quantum measurement, nonlocality, and realism. To address these issues, this study develops a 4+1 formalism featuring a flat 4D spacetime evolving with a second form of time, τ, worldlines that locally conserve momentum, and a hypersurface representing the present. As a function of τ, worldlines can spatially readjust and influences can travel backward or forward in the time dimension along these worldlines, offering a physical mechanism for retrocausality. Three theoretical models are presented, elucidating how nonlocality in an EPR experiment, the arrival time problem, and superposition in a Mach-Zehnder interferometer can be understood within this 4+1 framework. These results demonstrate that essential quantum phenomena can be reproduced in the 4+1 formalism while upholding the principles of realism, locality, and determinism at a fundamental level. Additionally, there is no measurement or collapse problem, and a natural explanation for the quantum-to-classical transition is obtained. Furthermore, observations of a 4D block universe and of the flow of time can be simultaneously understood. With these properties, the presented 4+1 formalism lays an interesting foundation for a quantum gravity theory based on intuitive principles and compatible with our observation of time.

Spherical Gauss-Laguerre beam propagation in 4D space-time.

In this paper, what we believe to be a novel class of beams, which are referred to as the spherical Gauss-Laguerre beams, are proposed. The beams propagate stably in the anomalous dispersive media, within which the second order derivative with respect to t could be combined with the two-dimensional (2D) Laplacian operator in the transverse direction and forms a three-dimensional (3D) Laplacian operator, which describes the beam propagation in the z direction within the four-dimensional (4D) x-y-z-t space-time. The wave equation is solved by the variable separation method and the analytical expression for the spherical Gauss-Laguerre beams is derived. The beams have a 3D Gaussian field distribution with a variable beam waist with respect to the propagation distance. Unlike any 2D spatial vortex beams, the 3D beams could possess either the spatial vortex or the spatiotemporal optical vortex (STOV) by choosing the vortex plane in the 3D x-y-t space-time. The derived spherical Gauss-Laguerre beam expression in the 4D space-time is verified by the numerical simulations with excellent agreement.

Anisotropic quark stars within 4D Einstein–Gauss–Bonnet gravity

By constructing models of quark stars (QSs), we use the MIT bag model as a strange quark matter equation of state (EoS). According to the strange matter hypothesis, the MIT bag model is the most successful phenomenological models of quarks confined to hadrons. This paper aims to explore the internal structure and the physical properties of QSs in the context of four-dimensional (4D) Einstein–Gauss–Bonnet (EGB) theory of gravity. The main feature of 4D EGB theory is that there is a nontrivial contribution to the gravitational dynamics in 4D spacetime. We present the essential features of QSs concerning the mass-radius relation depending on the interplay of matter parameters and the GB coupling constant. Our results show that all studied cases are found to be consistent with observations and physically relevant.

Four-Dimensional-Spacetime Atomistic Artificial Intelligence Models.

We demonstrate that AI can learn atomistic systems in the four-dimensional (4D) spacetime. For this, we introduce the 4D-spacetime GICnet model, which for the given initial conditions (nuclear positions and velocities at time zero) can predict nuclear positions and velocities as a continuous function of time up to the distant future. Such models of molecules can be unrolled in the time dimension to yield long-time high-resolution molecular dynamics trajectories with high efficiency and accuracy. 4D-spacetime models can make predictions for different times in any order and do not need a stepwise evaluation of forces and integration of the equations of motions at discretized time steps, which is a major advance over traditional, cost-inefficient molecular dynamics. These models can be used to speed up dynamics, simulate vibrational spectra, and obtain deeper insight into nuclear motions, as we demonstrate for a series of organic molecules.

Analysis of dynamic responses of circulating fuel reactors using in-house 3D space-time kinetics code ARCH

Molten Salt Reactors (MSRs) are characterized by power generation in circulating liquid fuel salt rather than in static fuel pins of traditional nuclear reactors like in water cooled reactors. The fuel flow in MSRs drift the delayed neutron precursors (DNP) and results in lower delayed neutron fraction. Thus, MSRs show added complexity in dynamic analysis due to inherently coupled neutronics and thermal hydraulics. With regard to the reactor safety analysis, accurate predictions of DNPs distribution and its effect on reactor transients turn out to be an interesting subject of research. Due to these unusual features of MSRs, the traditional safety analysis codes are not applicable in dynamic analysis of reactors with circulating fuel. Challenges in accurate modelling and analysis of dynamic responses of MSRs have been described in present work using new capabilities in 3D space-time kinetics code ARCH. The numerical results of the implemented methodology presented in this study are compared with data available from Molten Salt Reactor Experiment (MSRE), a reactor operated at ORNL.

Mirror Symmetry for New Physics beyond the Standard Model in 4D Spacetime

The two discrete generators of the full Lorentz group O(1,3) in 4D spacetime are typically chosen to be parity inversion symmetry P and time reversal symmetry T, which are responsible for the four topologically separate components of O(1,3). Under general considerations of quantum field theory (QFT) with internal degrees of freedom, mirror symmetry is a natural extension of P, while CP symmetry resembles T in spacetime. In particular, mirror symmetry is critical as it doubles the full Dirac fermion representation in QFT and essentially introduces a new sector of mirror particles. Its close connection to T-duality and Calabi–Yau mirror symmetry in string theory is clarified. Extension beyond the Standard Model can then be constructed using both left- and right-handed heterotic strings guided by mirror symmetry. Many important implications such as supersymmetry, chiral anomalies, topological transitions, Higgs, neutrinos, and dark energy are discussed.

A projection-based approach to extend digital volume correlation for 4D spacetime measurements

A projection-based approach to extend digital volume correlation for 4D spacetime measurements

Surface And Hypersurface Meshing Techniques for Space–Time Finite Element Methods

A novel method is introduced for constructing two-dimensional (2D) surface meshes embedded in three-dimensional (3D) space time and 3D hypersurface meshes embedded in four-dimensional (4D) space time. In particular, we begin by dividing the space–time domain into time slabs. Each time slab is equipped with an initial plane (hyperplane), in conjunction with an unstructured simplicial surface (hypersurface) mesh that covers the initial plane. We then obtain the vertices of the terminating plane (hyperplane) of the time slab from the vertices of the initial plane using a space–time trajectory-tracking approach. Next, these vertices are used to create an unstructured simplicial mesh on the terminating plane (hyperplane). Thereafter, the initial and terminating boundary vertices are stitched together to form simplicial meshes on the intermediate surfaces or sides of the time slab. After describing this new mesh-generation method in rigorous detail, we provide the results of multiple numerical experiments which demonstrate its validity and flexibility.

BMS field theories with u(1) symmetry

We investigate quantum field theories in two dimensions (2D) with an underlying Bondi--van der Burgh--Metzner-Sachs symmetry augmented by $\mathfrak{u}(1)$ currents. These field theories are expected to holographically capture features of charged versions of cosmological solutions in asymptotically flat 3D spacetimes called flat space cosmologies. We conduct a study of the modular properties of these field theories. The characters for the highest weight representations of the symmetry algebra are constructed, and the partition function of the theory is obtained from them. We derive the density of (primary) states and find the entropy and asymptotic values of the structure constants exploiting the modular properties of the partition function and the torus one-point function. The expression for the asymptotic structure constants shows shifts in the weights and one of the central terms and an extra phase compared to the earlier results in the literature for Bondi--van der Burgh--Metzner-Sachs invariant theories without $\mathfrak{u}(1)$ currents present. We reproduce our field results for the structure constants by a bulk computation involving a scalar probe in the background of a charged flat space cosmology.

A Method for Decoding and Representing Time in Fourdimensional Spaces via Digital Game Environment

The search for novel representation method is one of the critical components of the creative design process. Discoveries in science, such as four-dimensional (4D) spacetime, influenced artists and architects. However, existing representation techniques constrained in two-dimensional (2D) sheets for 4D spacetime for representation is evaluated as a limitation of architecture discipline. Currently, digital game environments are the potential mediums of 4D architectural representation. This study aims to decode and represent time as an entity of 4D spaces. Digital game environment provides the needed flexibility for experimenting in 4D space. Therefore, unity game engine and C# programming are used together with computer aided design (CAD) tools to generate 4D representations. 4D representations are based on two different impressions of time dimension as (1) time dilation and (2) distortion of spacetime. While time dilation is represented via motion blur (mB) script, distortion of spacetime is represented via motion trail (mT) script. As preliminary results, metrics of the time dimension in 4D spatio-temporal representations are introduced. Experimental 4D representations produced via Unity game engine and C# programming are presented to discuss the potential of game environments to be the medium of architectural representation.