Space-time Formalism Research Articles

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.

Symbolic studies of Maxwell’s equations in space-time algebra formalism

Different implementations of Clifford algebra: spinors, quaternions, and geometric algebra, are used to describe physical and technical systems. The geometric algebra formalism is a relatively new approach, destined to be used primarily by engineers and applied researchers. In a number of works, the authors examined the implementation of the geometric algebra formalism for computer algebra systems. In this article, the authors extend elliptic geometric algebra to hyperbolic space-time algebra. The results are illustrated by different representations of Maxwell’s equations. Using a computer algebra system, Maxwell’s vacuum equations in the space-time algebra representation are converted to Maxwell’s equations in vector formalism. In addition to practical application, the authors would like to draw attention to the didactic significance of these studies.

On the use of multidimensional differential geometry to model covariant behaviors of viscoelastic or hyperelastic structures, illustrated with numerical simulations using spacetime finite element analysis

In the present article, a covariant spacetime formalism is used to model the behavior of viscoelastic and hyperelastic solids, within a thermodynamical framework. The latter aims to ensure the validity of thermodynamics second principle and to derive reversible or irreversible models for thermomechanics. The use of the Lie derivative is of particular interest to achieve such goals. Covariance enables to address finite deformation. Coupled to a covariant finite element analysis, it allows numerical simulations that simultaneously ensure the physical balance of energy and momentum for thermomechanical applications. Different mechanical loadings are considered, bending or uniaxial extension ones, with quasi-static or time exponential or time cyclic evolution. We also provide quantification of the different performances of the numerical simulations, and show the advantages and drawbacks of the spacetime approach.

Second-order Teukolsky formalism in Kerr spacetime: Formulation and nonlinear source

Second-order Teukolsky formalism in Kerr spacetime: Formulation and nonlinear source

Covariant spacetime formalism for applications to thermo-hyperelasticity

Covariant spacetime formalism for applications to thermo-hyperelasticity

A discrete relativistic spacetime formalism for 1 + 1-QED with continuum limits

We build a quantum cellular automaton (QCA) which coincides with 1+1 QED on its known continuum limits. It consists in a circuit of unitary gates driving the evolution of particles on a one dimensional lattice, and having them interact with the gauge field on the links. The particles are massive fermions, and the evolution is exactly U(1) gauge-invariant. We show that, in the continuous-time discrete-space limit, the QCA converges to the Kogut–Susskind staggered version of 1+1 QED. We also show that, in the continuous spacetime limit and in the free one particle sector, it converges to the Dirac equation—a strong indication that the model remains accurate in the relativistic regime.

High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid

A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion covariantly and to describe the geometry of the rotated cubed-sphere grid. The spatial discretization is done with the direct flux reconstruction method, which is an alternative formulation to the discontinuous Galerkin approach. The equations of motion are solved in differential form and the resulting discretization is free from quadrature rules. It is well known that the time step of traditional explicit methods is limited by the phase velocity of the fastest waves. Exponential integration is employed to enable integrations with significantly larger time step sizes and improve the efficiency of the overall time integration. New multistep-type exponential propagation iterative methods of orders 4, 5 and 6 are constructed and applied to integrate the shallow-water equations in time. These new schemes enable time integration with high-order accuracy but without significant increases in computational time compared to low-order methods. The exponential matrix functions-vector products used in the exponential schemes are approximated using the complex-step approximation of the Jacobian in the Krylov-based KIOPS (Krylov with incomplete orthogonalization procedure solver) algorithm. Performance of the new numerical methods is evaluated using a set of standard benchmark tests.

Two-Component Spinorial Formalism Using Quaternions for Six-Dimensional Spacetimes

In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with $SL(2;\mathbb{H})$, which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of $SL(2;\mathbb{H})$ do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of $SO(5,1)$ is presented, providing a physical interpretation for the elements of $SL(2;\mathbb{H})$. Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by $SU(4)$.

The Quantum Entanglement Roots of Newton’s Gravitational Inverse Square Law Via E-Infinity Platonic Transfinite Set Theory

Using the E-infinity Cantorian spacetime formalism, i.e. transfinite platonic set theory, we show that Newton’s gravitational inverse square law strictly implies quantum entanglement and visa versa to the extent that they are derivable from each other. Noting that the same inverse square law is quasi generic for electrical and magnetic field the unifying role of the Cantorian spacetime of E-infinity becomes evident. The implication and generalization of the above will be explored but its ramifications within physics and beyond is not easily assessed at this early stage of what may amount to a paradigm shift prompted by the simple but very effective golden mean number system, which was suspected recently of being the rationale behind the fine tuning of the universe and the standard model of high energy elementary particle physics.

Some Relativistic and Gravitational Properties of the Wolfram Model

The Wolfram Model, which is a slight generalization of the model first introduced by Stephen Wolfram in A New Kind of Science (NKS), is a discrete spacetime formalism in which space is represented by a hypergraph whose dynamics are determined by abstract replacement operations on set systems, and in which the conformal structure of spacetime is represented by a causal graph. The purpose of this article is to present rigorous mathematical derivations of many key properties of such models in the continuum limit, as first discussed in NKS, including the fact that large classes of them obey discrete forms of both special and general relativity. First, we prove that causal invariance (namely, the requirement that all causal graphs be isomorphic, irrespective of the choice of hypergraph updating order) is equivalent to a discrete version of general covariance, with changes to the updating order corresponding to discrete gauge transformations. This fact then allows one to deduce a discrete analog of Lorentz covariance, and the resultant physical consequences of discrete Lorentz transformations. We also introduce discrete notions of Riemann and Ricci curvature for hypergraphs, and prove that the correction factor for the volume of a discrete spacetime cone in a causal graph corresponding to curved spacetime of fixed dimensionality is proportional to a timelike projection of the discrete spacetime Ricci tensor, subsequently using this fact (along with the assumption that the updating rules preserve the dimensionality of the causal graph in limiting cases) to prove that the most general set of constraints on the discrete spacetime Ricci tensor corresponds to a discrete form of the Einstein field equations.

Investigation on the Use of a Spacetime Formalism for Modeling and Numerical Simulations of Heat Conduction Phenomena

Abstract The question of frame-indifference of the thermomechanical models has to be addressed to deal correctly with the behavior of matter undergoing finite transformations. In this work, we propose to test a spacetime formalism to investigate the benefits of the covariance principle for application to covariant modeling and numerical simulations for finite transformations. Several models especially for heat conduction are proposed following this framework and next compared to existing models. This article also investigates numerical simulations using the heat equation with two different thermal dissipative models for heat conduction, without thermomechanical couplings. The numerical comparison between the spacetime thermal models derived in this work and the corresponding Newtonian thermal models, which adds the time as a discretized variable, is also performed through an example to investigate their advantages and drawbacks.

Plumbing constructions and the domain of outer communication for 5-dimensional stationary black holes

The topology of the domain of outer communication for 5-dimensional stationary bi-axisymmetric black holes is classified in terms of disc bundles over the 2-sphere and plumbing constructions. In particular we find an algorithmic bijective correspondence between the plumbing of disc bundles and the rod structure formalism for such spacetimes. Furthermore, we describe a canonical fill-in for the black hole region and cap for the asymptotic region. The resulting compactified domain of outer communication is then shown to be homeomorphic to S 4 S^4 , a connected sum of S 2 × S 2 S^2\times S^2 ’s, or a connected sum of complex projective planes C P 2 \mathbb {CP}^2 . Combined with recent existence results, it is shown that all such topological types are realized by vacuum solutions. In addition, our methods treat all possible types of asymptotic ends, including spacetimes which are asymptotically flat, asymptotically Kaluza-Klein, or asymptotically locally Euclidean.

Soft rescattering in the timelike proton form factor within a spacetime scheme

The annihilation of a lepton pair into a proton-antiproton pair, and the reverse process, allow for a measurement of the timelike proton form factors. This work aims at studying the corrections due to direct soft interactions between the proton and the antiproton before their annihilation or after their creation. The analysis is carried on in the spacetime formalism used by us in a series of recent works on timelike FFs. In particular the effect of $\bar{p}p$ annihilation into many-meson states is considered. We expect it to determine similar real and imaginary parts for the form factor near the $\bar{p}p$ channel threshold. We also discuss the possibility that the timelike FF differ by a phase in the hadron pair production and annihilation.

Gauge theory of massless spin- field in de Sitter space-time

On several levels of theoretical physics, especially particle physics and early universe cosmology, de Sitter space-time has become an attractive possibility. The principle of local gauge invariance governs all known fundamental interactions of elementary particles, from electromagnetism and weak interactions to strong interactions and gravity. This paper presents a procedure for defining the gauge-covariant derivative and gauge invariant Lagrangian density in de Sitter ambient space-time formalism. The gauge invariant field equation is then explicitly calculated in detail for a massless spin- gauge field.

Hadronic bound states in SU(2) from Dyson-Schwinger equations.

By using the Dyson–Schwinger/Bethe–Salpeter formalism in Euclidean spacetime, we calculate the ground state spectrum of Jle 1 hadrons in an SU(2) gauge theory with two fundamental fermions. We show that the rainbow-ladder truncation, commonly employed in QCD studies, is unsuitable for a description of an SU(2) theory. This we remedy by truncating at the level of the quark–gluon vertex Dyson–Schwinger equation in a diagrammatic expansion. Results obtained within this novel approach show good agreement with lattice studies. These findings emphasize the need to use techniques more sophisticated than rainbow-ladder when investigating generic strongly interacting gauge theories.

Discontinuous variational time integrators for complex multibody collisions

SUMMARYThe objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time tk, representing predictor and corrector configurations of the system. The resulting sequence of configuration pairs then represents a discontinuous—or non‐classical—trajectory. Continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times. In particular, in systems subject to one‐sided contact constraints simulated via discontinuous variational time integrators, the predictor configuration is not required to satisfy the one‐sided constraints, whereas the corrector configuration is obtained by a closest‐point projection (CPP) onto the admissible set. The resulting trajectories are generally discontinuous, or non‐classical, but are expected to converge to classical or continuous solutions for decreasing time steps. We account for dissipation, including friction, by means of a discrete Lagrange–d'Alembert principle, and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems, and the right rate of energy decay in dissipative systems. The structure, range and scope of the discontinuous variational time integrators, and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics. Copyright © 2014 John Wiley & Sons, Ltd.

Type O pure radiation metrics with a cosmological constant

In this paper we complete the integration of the conformally flat pure radiation spacetimes with a non-zero cosmological constant Λ, and \(\tau \ne 0\) , by considering the case \(\Lambda +\tau\bar\tau \ne 0\) . This is a further demonstration of the power and suitability of the generalised invariant formalism (GIF) for spacetimes where only one null direction is picked out by the Riemann tensor. For these spacetimes, the GIF picks out a second null direction (from the second derivative of the Riemann tensor) and once this spinor has been identified the calculations are transferred to the simpler GHP formalism, where the tetrad and metric are determined. The whole class of conformally flat pure radiation spacetimes with a non-zero cosmological constant (those found in this paper, together with those found earlier for the case \(\Lambda +\tau\bar\tau = 0\)) have a rich variety of subclasses with zero, one, two, three, four or five Killing vectors.

Propagation of Photons in Spacetime

A quantum field theory QED formalism is systematically developed to describe photon propagation in spacetime as a time evolution process based on the actual physical process of propagation between emitters and detectors as applied to the reflection of pho- tons. This development, as well as early studies by Feynman, clearly show that a practical, computational and predictive dynamical formalism in spacetime was lacking. The present one generalizes to different experimental situations and other interacting field theories as well emphasizing the practicality of the problem treated here.

Formalism for spacetimes with a homothety

The formalism developed by Fayos and Sopuerta for isometries in vacuum spacetimes is generalized in three ways: to include the case of homotheties, to allow for a tetrad not aligned to the symmetry structure and to allow for a non-normalized tetrad. The new formalism is used to recover results by McIntosh and also applied to the particular cases of Robinson–Trautman and Kundt type N.

New Li-Yau-Hamilton Inequalities for the Ricci Flow via the Space-Time Approach

We prove Li-Yau-Hamilton inequalties that extend Hamilton's matrix inequality for solutions of the Ricci flow with nonnegative curvature operators. To obtain our extensions, we apply the space-time formalism of S.-C. Chu and one of the authors to solutions of the Ricci flow modified by a cosmological constant. Then we adjoin to the Ricci flow the evolution of a 1-form and a 2-form flowing by a system of heat-type equations. By a rescaling argument, the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of Hamilton and one of the authors.