Space-time Equation Research Articles

On a remarkable electromagnetic field in the Einstein Universe

We present a time-dependent solution of the Maxwell equations in the Einstein universe, whose electric and magnetic fields, as seen by the stationary observers, are aligned with the Clifford parallels of the 3-sphere S^3. The conformal equivalence between Minkowski’s spacetime and (a region of) the Einstein cylinder is then exploited in order to obtain a knotted, finite energy, radiating solution of the Maxwell equations in flat spacetime. We also discuss similar electromagnetic fields in expanding closed Friedmann models, and compute the matter content of such configurations.

Reverse Space-Time Nonlocal Sasa–Satsuma Equation and Its Solutions

The Sasa–Satsuma equation is an integrable high-order nonlinear Schrodinger (NLS) equation, and also is a complex modified KdV-type equation. It can describe the propagation of femtosecond pulses in optical fibers. Very recently, Ablowitz and Mussliman introduced a class of reverse space-time and reverse time nonlinear integrable equations, including the reverse space nonlocal NLS equation, the real and complex reverse space-time nonlocal mKdV, sine-Gordon, Davey–Stewartson equations, etc. So, what is nonlocal version of high-order NLS? In this paper, we introduce a reverse space-time nonlocal Sasa–Satsuma equation, i.e., a reverse space-time nonlocal high-order NLS equation, and derive its solutions with the binary Darboux transformation method. Periodic solutions, and some localized solutions, such as dark soliton, W-shaped soliton, M-shaped soliton and breather soliton of the reverse space-time nonlocal Sasa–Satsuma equation are constructed.

Mapping curved spacetimes into Dirac spinors

We show how to transform a Dirac equation in a curved static spacetime into a Dirac equation in flat spacetime. In particular, we show that any solution of the free massless Dirac equation in a 1 + 1 dimensional flat spacetime can be transformed via a local phase transformation into a solution of the corresponding Dirac equation in a curved static background, where the spacetime metric is encoded into the phase. In this way, the existing quantum simulators of the Dirac equation can naturally incorporate curved static spacetimes. As a first example we use our technique to obtain solutions of the Dirac equation in a particular family of interesting spacetimes in 1 + 1 dimensions.

Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and $2s$ in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for space-time nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for $(\partial_t-\Delta)^su(t,x)$ and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. H\"older and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic H\"older spaces. Our methods involve the \textit{parabolic language of semigroups} and the Cauchy Integral Theorem, which are original to define the fractional powers of $\partial_t-\Delta$. Though we mainly focus in the equation $(\partial_t-\Delta)^su=f$, applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.

The Cauchy Problem for Space-Time Monopole Equations in Temporal and Spatial Gauge

We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition A1=0 and the temporal gauge condition A0=0.

Conformally related Einstein-Langevin equations for metric fluctuations in stochastic gravity

For a conformally-coupled scalar field we obtain the conformally-related Einstein-Langevin equations, using appropriate transformations for all the quantities in the equations between two conformally-related spacetimes. In particular, we analyze the transformations of the influence action, the stress energy tensor, the noise kernel and the dissipation kernel. In due course the fluctuation-dissipation relation is also discussed. The analysis in this paper thereby facilitates a general solution to the Einstein-Langevin equation once the solution of the equation in a simpler, conformally-related spacetime is known. For example, from the Minkowski solution of Martin and Verdaguer, those of the Einstein-Langevin equations in conformally-flat spacetimes, especially for spatially-flat Friedmann-Robertson-Walker models, can be readily obtained.

Low-dimensional approach to pair production in an oscillating electric field: Application to bandgap graphene layers

The production of particle-antiparticle pairs from the quantum field theoretic ground state in the presence of an external electric field is studied. Starting with the quantum kinetic Boltzmann-Vlasov equation in four-dimensional spacetime, we obtain the corresponding equations in lower dimensionalities by way of spatial compactification. Our outcomes in $2+1$-dimensions are applied to bandgap graphene layers, where the charge carriers have the particular property of behaving like light massive Dirac fermions. We calculate the single-particle distribution function for the case of an electric field oscillating in time and show that the creation of particle-hole pairs in this condensed matter system closely resembles electron-positron pair production by the Schwinger effect.

An extended Dirac equation in noncommutative spacetime

Stabilizing, by deformation, the algebra of relativistic quantum mechanics a noncommutative spacetime geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed as well as the effects of coupling the two solutions.

Efficient and Accurate Modeling of Multiwavelength Propagation in SOAs: A Generalized Coupled-Mode Approach

We present a model for multi-wavelength mixing in semiconductor optical amplifiers (SOAs) based on coupled-mode equations. The proposed model applies to all kinds of SOA structures, takes into account the longitudinal dependence of carrier density caused by saturation, it accommodates an arbitrary functional dependencies of the material gain and carrier recombination rate on the local value of carrier density, and is computationally more efficient by orders of magnitude as compared with the standard full model based on space-time equations. We apply the coupled-mode equations model to a recently demonstrated phase-sensitive amplifier based on an integrated SOA and prove its results to be consistent with the experimental data. The accuracy of the proposed model is certified by means of a meticulous comparison with the results obtained by integrating the space-time equations.

Numerical solutions of Einstein’s equations for cosmological spacetimes with spatial topologyS3and symmetry group U(1)

In this paper we consider the single patch pseudo-spectral scheme for tensorial and spinorial evolution problems on the 2-sphere presented in [3,4] which is based on the spin-weighted spherical harmonics transform. We apply and extend this method to Einstein's equations and certain classes of spherical cosmological spacetimes. More specifically, we use the hyperbolic reductions of Einstein's equations obtained in the generalized wave map gauge formalism combined with Geroch's symmetry reduction, and focus on cosmological spacetimes with spatial S3-topologies and symmetry groups U(1) or U(1) x U(1). We discuss analytical and numerical issues related to our implementation. We test our code by reproducing the exact inhomogeneous cosmological solutions of the vacuum Einstein field equations obtained in [7].

Grand Unification Concept with Bio-force as the Fifth Fundamental Force

All the four fundamental forces of nature have been explained in terms of space time. Coupling constant of grand unification has been defined by space time. The density of space time calculated by ‘Siva’s classical equation for space time’ describes the nature of fundamental force. Thus for all fundamental forces, there exist a separate universe with a separate space time and a separate ‘maximum signal velocity’. The transformation has been explained as an expansion of a single universe for which maximum signal velocity is ‘velocity of light’. This transformation affects the concept of consciousness. Thus all other forms of space times explained as different forces of nature within the limitations of consciousness. Thus ‘Bio-Force’ defined as a form of space time like all other fundamental forces and considered as a fifth fundamental force. Consciousness is an affect of interactions of ‘Bio-Force’. Coupling Constants for ‘gravity’ and ‘Bio field’ have been calculated by space time concept and concerned maximum signal velocities. For other three forces strong force, electromagnetic force and weak forces, the space time has been defined by their coupling constants. All these calculations have been done by one single equation Vd=K. With this equation, space time parameters for all fundamental forces have been explained. The conclusions described grand unification of all five natural forces in a different way which has been extended to ‘Bio-Force’, an important aspect of consciousness.

Special generalized densities and propagators: A geometric account

Starting from a short review of spaces of generalized sections of vector bundles, we give a concise systematic description, in precise geometric terms, of Leray densities, principal value densities, propagators and elementary solutions of field equations in flat spacetime. We then sketch a partly original geometric presentation of free quantum fields and show how propagators arise from their graded commutators in the boson and fermion cases.

On a Quantum Gravity Fractal Spacetime Equation: QRG &amp;amp;simeq; HD + FG and Its Application to Dark Energy—Accelerated Cosmic Expansion

The paper suggests that quantum relativistic gravity (QRG) is basically a higher dimensionality (HD) simulating relativity and non-classical effects plus a fractal Cantorian spacetime geometry (FG) simulating quantum mechanics. This more than just a conceptual equation is illustrated by integer approximation and an exact solution of the dark energy density behind cosmic expansion.

Cosmological and astrophysical consequences from the magnetic dynamo equation in torsioned spacetime and teleparallel gravity**This observation is due to an anonymous referee who found this mistake in the first draft of the paper.

A generalized dynamo equation in the first order torsion Garcia de Andrade L C (2012 Phys. Lett. B 711 143) has previously been derived. From this equation it is shown that for the 10 kpc scale, torsion gravity is not able to help seed galactic dynamos since the dynamo time is not long enough to take into account structure formation. In this paper, the dynamo equation is extended to second-order torsion terms—but unfortunately, the situation is even worse and the torsion does not seem to help dynamo efficiency. Nevertheless, in the intergalactic magnetic field scale of 1 mpc, the efficiency of the self-induction equation with torsion changes, and even in the first-order torsion case, one obtains large-scale magnetic fields with 109 yr dynamo efficiency. Dynamo efficiency in the case of interstellar matter (ISM) reaches a diffusion time of 1013 yr. This seems to be in contrast with a recent investigation by Bamba et al (2012 J. Cosmol. Astropart. Phys. JCAP05(2010)08) where they obtained, from another type of torsion theory called teleparallelism (A Einstein, Math Annalen (1922)), a large scale intergalactic magnetic field of 10−9 G. If this is not a model-dependent result, there is an apparent contradiction that has to be addressed. It is shown that for dynamo efficiency in astrophysical flow without shear, a strong seed field of 10−11 G is obtained, which is suitable for seeding galactic dynamos. As an example of a non-parity-violating dynamo equation, a magnetic field of the order of 10−27G is obtained as a seed field for the galactic dynamo from the theory of Einstein’s unified teleparallelism. This shows that in certain gravity models, torsion is able to enhance cosmological magnetic fields in view of obtaining better dynamo efficiency. To better compare our work with Bamba et al (2012 J. Cosmol. Astropart. Phys. JCAP05(2010)08), we consider the slow decay of magnetic fields in the teleparallel model.

Dirac’s Equation in R-Minkowski Spacetime

We recently constructed the R-Poincare algebra from an appropriate deformed Poisson brackets which reproduce the Fock coordinate transformation. We showed then that the spacetime of this transformation is the de Sitter one. In this paper, we derive in the R-Minkowski spacetime the Dirac equation and show that this is none other than the Dirac equation in the de Sitter spacetime given by its conformally flat metric. Furthermore, we propose a new approach for solving Dirac’s equation in the de Sitter spacetime using the Schrodinger picture.

Exploring New Physics Frontiers Through Numerical Relativity.

The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein’s equations — along with some spectacular results — in various setups.We review techniques for solving Einstein’s equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in high-energy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology.

Almost critical local well-posedness for the space-time monopole equation in Lorenz gauge

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in Hs with [Formula: see text]. The equation is L2-critical, and hence a [Formula: see text] derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier–Lebesgue space [Formula: see text] for 1 &lt; p ≤ 2 which coincides with Hs when p = 2 but scales like lower regularity Sobolev spaces for 1 &lt; p &lt; 2. In particular, we will see that as p → 1+, the critical exponent [Formula: see text], in which case [Formula: see text] is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to p close to 1.

Similarity solution to fractional nonlinear space-time diffusion-wave equation

In this article, the so-called fractional nonlinear space-time wave-diffusion equation is presented and discussed. This equation is solved by the similarity method using fractional derivatives in the Caputo, Riesz-Feller, and Riesz senses. Some particular cases are presented and the corresponding solutions are shown by means of 2-D and 3-D plots.

Baryon spectrum from superconformal quantum mechanics and its light-front holographic embedding

We describe the observed light-baryon spectrum by extending superconformal quantum mechanics to the light front and its embedding in AdS space. This procedure uniquely determines the confinement potential for arbitrary half-integer spin. To this end, we show that fermionic wave equations in AdS space are dual to light-front supersymmetric quantum mechanical bound-state equations in physical space-time. The specific breaking of conformal invariance explains hadronic properties common to light mesons and baryons, such as the observed mass pattern in the radial and orbital excitations, from the spectrum generating algebra. The holographic embedding in AdS also explains distinctive and systematic features, such as the spin-$J$ degeneracy for states with the same orbital angular momentum, observed in the light baryon spectrum.

Schrodinger-like interpretation of scalar field equation in RW space-time

Schrodinger-like interpretation of scalar field equation in RW space-time