Flat-space Equations Research Articles

Newtonian approximation in (1 + 1) dimensions

We study the possible existence of a Newtonian regime of gravity in 1 + 1 dimensions, considering metrics in both the Kerr-Schild and conformal forms In the former case, the metric gives the exact solution of the Poisson equation in flat space, but the weak-field limit of the solutions and the non-relativistic regime of geodesic motion are not trivial. We show that using harmonic coordinates, the metric is conformally flat and a weak-field expansion is straightforward. An analysis of the non-relativistic regime of geodesic motion remains non-trivial and the weak-field potential only satisfies the flat space Poisson equation approximately.

On self-dual Yang–Mills fields on special complex surfaces

We derive a generalization of the flat space equations of Yang and Newman for self-dual Yang–Mills fields to (locally) conformally Kähler Riemannian four-manifolds. The results also apply to Einstein metrics (whose full curvature is not necessarily self-dual). We analyze the possibility of hidden symmetries in the form of Bäcklund transformations, and we find a continuous group of hidden symmetries only for the case in which the geometry is conformally half-flat. No isometries are assumed.

Universe as Klein\u2013Gordon eigenstates

We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects thebeta -times t_beta :=int ^t a^{-2beta }, where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems O1/2Ψ=Λ12Ψ,O1a=-Λ3a,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} O_{1/2} \\Psi =\\frac{\\Lambda }{12}\\Psi , \\quad O_1 a =-\\frac{\\Lambda }{3} a , \\end{aligned}$$\\end{document}which is suggestive of a measurement problem. O_{beta }(rho ,p) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The O_beta ’s are also independent of the spatial curvature, labeled by k, and absorbed in Ψ=aei2kη.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Psi =\\sqrt{a} e^{\\frac{i}{2}\\sqrt{k}\\eta } . \\end{aligned}$$\\end{document}The above pair of equations is the unique possible linear form of Friedmann’s equations unless k=0, in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time eta equiv t_{1/2} among the t_beta ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space.

The Gravity Gauge Theory and Gravity Field Equation in Flat Space

In this paper, we have proposed the theory of gravity gauge, and the gravity theory has been introduced into quantum field theory. We have further given the tensor equation of gravity field in the flat space, and found the gravity field equation is the Lorentz covariant and gauge invariant. The gravity theory can be quantized and can be unified with the electroweak and strong interaction at a new gauge group .

A five-dimensional perspective on the Klein–Gordon equation

We discuss the Klein–Gordon (KG) equation using a path-integral approach in 5D space–time. We explicitly show that the KG equation in flat space–time admits a consistent probabilistic interpretation with positively defined probability density. However, the probabilistic interpretation is not covariant. In the non-relativistic limit, the formalism reduces naturally to that of the Schrödinger equation. We further discuss other interpretations of the KG equation (and their non-relativistic limits) resulting from the 5D space–time picture. Finally, we apply our results to the problem of hydrogenic spectra and calculate the canonical sum of the hydrogenic atom.

ON SPIN-STATISTICS AND BOGOLIUBOV TRANSFORMATIONS IN FLAT SPACE–TIME WITH ACCELERATION CONDITIONS

A single real scalar field of spin zero obeying the Klein–Gordon equation in flat space–time under external conditions is considered in the context of the spin-statistics connection. An imposed accelerated boundary on the field is made to become, in the far future, (1) asymptotically inertial and (2) asymptotically noninertial (with an infinite acceleration). The constant acceleration Unruh effect is also considered. The systems involving nontrivial Bogoliubov transformations contain dynamics which point to commutation relations. Particles described by in-modes obey the same statistics as particles described by out-modes. It is found in the nontrivial systems that the spin-statistics connection can be manifest from the acceleration. The equation of motion for the boundary which forever emits thermal radiation is revealed.

Investigating Initial Conditions of the WdW Equation in Flat Space in a Transition from the Pre-Planckian Physics Era to the Electroweak Regime of Space-Time

This document is due to reviewing an article by Maydanyuk and Olkhovsky, of a Nova Science conpendium as of “The big bang, theory assumptions and Problems”, as of 2012, which uses the Wheeler De Witt equation as an evolution equation assuming a closed universe. Having the value of k, not as the closed universe, but nearly zero of a nearly flat universe, which leads to serious problems of interpretation of what initial conditions are. These problems of interpretations of initial conditions tie in with difficulties in using QM as an initial driver of inflation. And argue in favor of using a different procedure as far as forming a wave function of the universe initially. The author wishes to thank Abhay Ashtekar for his well thought out criticism but asserts that limitations in space-time geometry largely due to when is formed from semi classical reasoning, i.e. Maxwell’s equation involving a close boundary value regime between Octonionic geometry and flat space non Octonionic geometry is a datum which Abhay Ashekhar may wish to consider in his quantum bounce model and in loop quantum gravity in the future.

Strong coupling BCS superconductivity and holography

We attempt to give a holographic description of the microscopic theory of a BCS superconductor. Exploiting the analogy with chiral symmetry breaking in QCD we use the Sakai–Sugimoto model of two D8 branes in a D4 brane background with finite baryon number. In this case there is a new tachyonic instability which is plausibly the bulk analog of the Cooper pairing instability. We analyze the Yang–Mills approximation to the non-Abelian Dirac–Born–Infeld action. We give some exact solutions of the non-linear Yang–Mills equations in flat space and also give a stability analysis, showing that the instability disappears in the presence of an electric field. The holographic picture also suggests a dependence of T c on the number density which is different from the usual (weak coupling) BCS. The flat space solutions are then generalized to curved space numerically and also, in an approximate way, analytically. This configuration should then correspond to the ground state of the boundary superconducting (superfluid) ground state. We also give some preliminary results on Green functions computations in the Sakai–Sugimoto model without any chemical potential.

Spectral methods for the wave equation in second-order form

Current spectral simulations of Einstein's equations require writing the equations in first-order form, potentially introducing instabilities and inefficiencies. We present a new penalty method for pseudo-spectral evolutions of second order in space wave equations. The penalties are constructed as functions of Legendre polynomials and are added to the equations of motion everywhere, not only on the boundaries. Using energy methods, we prove semi-discrete stability of the new method for the scalar wave equation in flat space and show how it can be applied to the scalar wave on a curved background. Numerical results demonstrating stability and convergence for multi-domain second-order scalar wave evolutions are also presented. This work provides a foundation for treating Einstein's equations directly in second-order form by spectral methods.

Superintegrability and higher order integrals for quantum systems

We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. The flat space equations with potentials V = α(x + iy)k − 1/(x − iy)k + 1 in Cartesian coordinates, and V = αr2 + β/r2cos 2kθ + γ/r2sin 2kθ (the Tremblay, Turbiner and Winternitz system) in polar coordinates, have each been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.

YANG–MILLS GRAVITY IN FLAT SPACE–TIME II: GRAVITATIONAL RADIATIONS AND LEE–YANG FORCE FOR ACCELERATED COSMIC EXPANSION

Within Yang–Mills gravity with translation group T(4) in flat space–time, the invariant action involving quadratic translation gauge-curvature leads to quadrupole radiations, which are shown to be consistent with experiments. The radiation power turns out to be the same as that in Einstein's gravity to the second-order approximation. We also discuss an interesting physical reason for the accelerated cosmic expansion based on the long-range Lee–Yang force of Ub(1) gauge field associated with the established conservation law of baryon number. We show that the Lee–Yang force can be related to a linear potential ∝ r, provided the gauge field satisfies a fourth-order differential equation in flat space–time. Furthermore, we consider an experimental test of the Lee–Yang force related to the accelerated cosmic expansion. The necessity of generalizing Lorentz transformations for accelerated frames of reference and accelerated Wu–Doppler effects are briefly discussed.

DERIVATION OF THE SOURCE-FREE MAXWELL AND GRAVITATIONAL RADIATION EQUATIONS BY GROUP THEORETICAL METHODS

We derive source-free Maxwell-like equations in flat space–time for any helicity j by comparing the transformation properties of the 2(2j+1) states that carry the manifestly covariant representations of the inhomogeneous Lorentz group with the transformation properties of the two helicity j states that carry the irreducible representations of this group. The set of constraints so derived involves a pair of curl equations and a pair of divergence equations. These reduce to the free-field Maxwell equations for j = 1 and the analogous equations coupling the gravito-electric and the gravito-magnetic fields for j = 2.

LOOP VARIABLES AND THE (FREE) OPEN STRING IN A CURVED BACKGROUND

Using the loop variable formalism as applied to a sigma model in curved target space, we give a systematic method for writing down gauge and generally covariant equations of motion for the modes of the free open string in curved space. The equations are obtained by covariantizing the flat space equation and then demanding gauge invariance, which introduces additional curvature couplings. As an illustration of the procedure, the spin-two case is worked out explicitly.

Discretizations of axisymmetric systems

In this paper we discuss stability properties of various discretizations for axisymmetric systems including the so called cartoon method which was proposed by Alcubierre, Brandt et.al. for the simulation of such systems on Cartesian grids. We show that within the context of the method of lines such discretizations tend to be unstable unless one takes care in the way individual singular terms are treated. Examples are given for the linear axisymmetric wave equation in flat space.

Classical black hole production in high-energy collisions

We investigate classical formation of a D-dimensional black hole in a high energy collision of two particles. The existence of an apparent horizon is related to the solution of an unusual boundary-value problem for Poisson's equation in flat space. For sufficiently small impact parameter, we construct solutions giving such apparent horizons in D=4. These supply improved estimates of the classical cross-section for black hole production, and of the mass of the resulting black holes. We also argue that a horizon can be found in a region of weak curvature, suggesting that these solutions are valid starting points for a semiclassical analysis of quantum black hole formation.

On the Algebra of Symmetry of the Dirac Equation in Flat Space and De Sitter Space

The structure of the quadratic algebras of spinor symmetry operators for the Dirac equation is studied in a four-dimensional flat space and in the de Sitter space of arbitrary signature. The algebras are shown to be standard equivalent. Linear noncommutative subalgebras meeting the conditions of the noncommutative integrability theorem are found in these algebras.

The Eikonal equation in flat space: Null surfaces and their singularities. I

The level surfaces of solutions to the eikonal equation define null or characteristic surfaces. In this paper we study, in Minkowski space, properties of these surfaces. In particular, we are interested both in the singularities of these “surfaces” (which can, in general, self-intersect and be only piecewise smooth) and in the decomposition of the null surfaces into a one-parameter family of two-dimensional wavefronts which can also have self-intersections and singularities. We first review a beautiful method for constructing the general solution to the flat-space eikonal equation; it allows for solutions either from arbitrary Cauchy data or for time-independent (stationary) solutions of the form S=t−S0(x,y,z). We then apply this method to obtain global, asymptotically spherical, null surfaces that are associated with shearing (“bad”) two-dimensional cuts of null infinity; the surfaces are defined from the normal rays to the cut. This is followed by a study of the caustics and singularities of these surfaces and those of their associated wavefronts. We then treat the same set of issues from an alternative point of view, namely from Arnold’s theory of generating families. This treatment allows one to deal (parametrically) with the regions of self-intersection and nonsmoothness of the null surfaces, regions which are difficult to treat otherwise. Finally, we generalize the analysis of the singularities to the case of families of characteristic surfaces.

Exact plane symmetric solutions of sin-Gordon equations with inherent gravitational fields taken into account

We obtain exact self-consistent plane symmetric solutions to systems of Einstein equations and nonlinear scalar field equations of sin-Gordon type. We write out, in explicit from, two specific solutions corresponding to specific choices of the nonlinearity parameter λ. We show that for each value of λ, there exist solutions of soliton and antisoliton type with topological charge Q=±1. We establish that the basic difference between the solutions we obtain and solutions of the sin-Gordon equation in flat space—time is that when we take internal gravitational fields into account, each value of λ corresponds to a specific function ϕλ(x).

Combining Cauchy and characteristic numerical evolutions in curved coordinates

In numerical relativity there would seem to be considerable advantages in combining Cauchy and characteristic methods in different regions of spacetime. Numerical experiments demonstrate, for the model problem of the scalar wave equation in flat space using general null coordinates, that there are no unforseen problems with the interface between the two regions.

Progressing waves in flat spacetime and in plane-wave spacetimes

This comment is concerned with simple progressing solutions of the wave equation Square Operator phi =0. In flat spacetime, these were all found by Friedlander in 1946. It is shown that all of the cases he enumerated are special cases of a single progressing-wave solution in a plane-wave gravitational background. Finally, progressing waves are used to obtain a generalisation, to plane-wave spacetime, of Whittaker's formula (1903) for solutions of the wave equations in flat space.