Conformal Spacetime Research Articles

Analytic treatment of complete geodesics in a static cylindrically symmetric conformal spacetime

We consider the motion of test particles and light rays in a static cylindrically symmetric conformal spacetime given by Said et al [1]. We derive the equations of motion and present their analytical solutions in terms of the Weierstrass {\wp} function and the Kleinian {\sigma} function. Using parametric diagrams and effective potentials we analyze the possible orbits and characterize them in terms of the energy and the angular momentum of the test particles. Finally we show some examples of orbits.

Casimir effect in spacetimes with cosmological constant

In this work, we study the influence of the gravitational field induced by the presence of a cosmological constant [Formula: see text] on the Casimir energy density. We consider two metrics with the presence of the [Formula: see text]-term, namely de Sitter and Schwarzschild-de Sitter (SdS). In the former case, we consider a conformal de Sitter spacetime and in the last one, a weak gravitational SdS spacetime.

Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space-time dS4

We seek for a pair of a well and barrier potentials such that the real parts of the complex energies of the resonances transmitted through the barrier equal the energies of the states bound within the well and find the hyperbolic Poeschl-Teller barrier, ~sech^2\rho, and the trigonometric Scarf well, ~ \sec^2\chi. The potentials are shown to be conformally symmetric by the aid of the de Sitter space time, dS4, related to flat conformal space time by a conformal map. Namely, we transform the quantum mechanical wave equations with the above potentials to free quantum motions on the respective open time like hyperbolic and the closed space like hyper spherical, S3, geodesics of dS4, the former by itself is related to Minkowski space time by a conformal map.We formulate a conformal symmetry respecting classification scheme for mesons seen either as resonances in scattering, or as states bound within a potential, according to trajectories in which the total spin of the meson, l-depends linearly on the first power of the invariant mass, M, and not as in the canonical Regge formalism on the squared mass.We analyze 71 reported mesons in this scheme and in finding good agreement with data predict the masses of 12 missing mesons. We observe that on S3 the color quantum number of mesons is limited to neutral. To the amount physics is independent of the choice of the set of coordinates, this property remains valid in the flat geometry too. We conclude on the usefulness of conformal maps of flat to curved space times as tools for modelling and interpreting the color neutrality of hadrons required by the confinement phenomenon and on the relevance of trigonometric and hyperbolic potentials for constituent quark models. Finally, all involved potentials have been motivated by Wilson loops with cusps.

Quantum twistors

We compute explicitly a star product on the Minkowski space whose Poisson bracket is quadratic. This star product corresponds to a deformation of the conformal spacetime, whose big cell is the Minkowski spacetime. The description of Minkowski space is made in the twistor formalism and the quantization follows by substituting the classical conformal group by a quantum group.

Conformal transformations in general relativistic elasticity

Conformal transformations are applied in the context of general relativistic elasticity. Expressions relating relativistic elastic quantities and tensors are obtained for two conformal spacetimes, whose material metrics are also conformally related. Non-static shear-free spherically symmetric elastic solutions of the Einstein field equations are constructed by performing conformal transformations of spacetime and material metrics of a known non-static shear-free spherically symmetric elastic solution.

Conformal transformation, near horizon symmetry, Virasoro algebra, and entropy

There are certain black hole solutions in general relativity (GR) which are conformally related to the stationary solutions in GR. It is not obvious that the horizon entropy of these spacetimes is also one quarter of the area of horizon, like the stationary ones. Here I study this topic in the context of Virasoro algebra and Cardy formula. Using the fact that the conformal spacetime admits conformal Killing vector and the horizon is determined by the vanishing of the norm of it, the diffemorphisms are obtained which keep the near horizon structure invariant. The Noether charge and a bracket among them corresponding to these vectors are calculated in this region. Finally, they are evaluated for the Sultana-Dyer (SD) black hole, which is conformal to the Schwarzschild metric. It is found that the bracket is identical to the usual Virasoro algebra with the central extension. Identifying the zero mode eigenvalue and the central charge, the entropy of the SD horizon is obtained by using Cardy formula. Interestingly, this is again one quarter of the horizon area. Only difference in this case is that the area is modified by the conformal factor compared to that of the stationary one. The analysis gives a direct proof of the earlier assumption.

Celestial ephemerides in an expanding universe

Post-Newtonian theory was instrumental in conducting the critical experimental tests of general relativity and in building the astronomical ephemerides of celestial bodies in the solar system with an unparalleled precision. The cornerstone of the theory is the postulate that the solar system is gravitationally isolated from the rest of the universe and the background spacetime is asymptotically flat. The present article extends this theoretical concept and formulates the principles of celestial dynamics of particles and light moving in gravitational field of a localized astronomical system embedded to the expanding Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. We formulate the precise mathematical concept of the Newtonian limit of Einstein's field equations in the conformally-flat FLRW spacetime and analyze the geodesic motion of massive particles and light in this limit. We prove that by doing conformal spacetime transformations, one can reduce the equations of motion of particles and light to the classical form of the Newtonian theory. However, the time arguments in the equations of motion of particles and light differ from each other in terms being proportional to the Hubble constant, H. This leads to the important conclusion that the equations of light propagation used currently by Space Navigation Centers for fitting range and Doppler-tracking observations of celestial bodies are missing some terms of the cosmological origin that are proportional to the Hubble constant, H. We also prove that the Hubble expansion does not affect the atomic time scale used in creation of astronomical ephemerides. We derive the cosmological correction to the light travel time equation and argue that their measurement opens an exciting opportunity to determine the local value of the Hubble constant, H, in the solar system independently of cosmological observations.

De sitter relativity: a natural scenario for an evolving Λ

The dispersion relation of de Sitter special relativity is obtained in a simple and compact form, which is formally similar to the dispersion relation of ordinary special relativity. It is manifestly invariant under change of scale of mass, energy and momentum, and can thus be applied at any energy scale. When applied to the universe as a whole, the de Sitter special relativity is found to provide a natural scenario for the existence of an evolving cosmological term, and agrees in particular with the present-day observed value. It is furthermore consistent with a conformal cyclic view of the universe, in which the transition between two consecutive eras occurs through a conformal invariant spacetime.

Group analysis of a conformal perfect fluid spacetime

We find new exact solutions of the Einstein field equations for a perfect fluid metric conformal to a spacetime of type D in the Petrov classification scheme. We analyse the complete system of equations using Lie group analysis. While previous work was confined to conformal factors of the form U = U(t, x), we investigate the complete situation U = U(t, x, y, z) as well as an auxiliary integrability condition. New classes of solutions are generated via the symmetry generators. The resulting solutions are examined for physical plausibility. Expressions for the energy density and pressure are obtained explicitly and empirical results suggest that these dynamical quantities are positive as expected.

Exact solutions with AdS asymptotics of Einstein and Einstein-Maxwell gravity minimally coupled to a scalar field

We propose a general method for solving exactly the static field equations of Einstein and Einstein-Maxwell gravity minimally coupled to a scalar field. Our method starts from an ansatz for the scalar field profile, and determines, together with the metric functions, the corresponding form of the scalar self-interaction potential. Using this method we prove a new no-hair theorem about the existence of hairy black-hole and black-brane solutions and derive broad classes of static solutions with radial symmetry of the theory, which may play an important role in applications of the AdS/CFT correspondence to condensed matter and strongly coupled QFTs. These solutions include: 1) four- or generic $(d+2)$-dimensional solutions with planar, spherical or hyperbolic horizon topology; 2) solutions with AdS, domain wall and Lifshitz asymptotics; 3) solutions interpolating between an AdS spacetime in the asymptotic region and a domain wall or conformal Lifshitz spacetime in the near-horizon region.

Causal topology in future and past distinguishing spacetimes

The causal structure of a strongly causal spacetime is particularly well endowed. Not only does it determine the conformal spacetime geometry when the spacetime dimension n > 2, as shown by Malament and Hawking–King–McCarthy (MHKM), but also the manifold dimension. The MHKM result, however, applies more generally to spacetimes satisfying the weaker causality condition of future and past distinguishability (FPD), and it is an important question whether the causal structure of such spacetimes can determine the manifold dimension. In this work, we show that the answer to this question is in the affirmative. We investigate the properties of future or past distinguishing spacetimes and show that their causal structures determine the manifold dimension. This gives a non-trivial generalization of the MHKM theorem and suggests that there is a causal topology for FPD spacetimes which encodes manifold dimension and which is strictly finer than the Alexandrov topology. We show that such a causal topology does exist. We construct it using a convergence criterion based on sequences of ‘chain intervals’ which are the causal analogues of null geodesic segments. We show that when the region of strong causality violation satisfies a local achronality condition, this topology is equivalent to the manifold topology in an FPD spacetime.

Amplitudes at weak coupling as polytopes in AdS5

We show that one-loop scalar box functions can be interpreted as volumes of geodesic tetrahedra embedded in a copy of AdS5 that has dual conformal spacetime as boundary. When the tetrahedron is space-like, it lies in a totally geodesic hyperbolic three-space inside AdS5, with its four vertices on the boundary. It is a classical result that the volume of such a tetrahedron is given by the Bloch–Wigner dilogarithm. We show that this agrees with the standard physics formulae for such box functions. The combinations of box functions that arise in the n-particle one-loop MHV amplitude in super Yang–Mills correspond to the volume of a three-dimensional polytope without boundary, all of whose vertices are attached to a null polygon (which in other formulations is interpreted as a Wilson loop) at infinity.

Structure, classification, and conformal symmetry of elementary particles over non-archimedean space-time

It is well known that at distances shorter than Planck length, no length measurements are possible. The Volovich hypothesis asserts that at sub-Planckian distances and times, spacetime itself has a non-Archimedean geometry. We discuss the structure of elementary particles, their classification, and their conformal symmetry under this hypothesis. Specifically, we investigate the projective representations of the p-adic Poincaré and Galilean groups, using a new variant of the Mackey machine for projective unitary representations of semidirect products of locally compact and second countable (lcsc) groups. We construct the conformal spacetime over p-adic fields and discuss the imbedding of the p-adic Poincaré group into the p-adic conformal group. Finally, we show that the massive and the so called eventually masssive particles of the Poincaré group do not have conformal symmetry. The whole picture bears a close resemblance to what happens over the field of real numbers, but with some significant variations.

Structure, Classification, and Conformal Symmetry, of Elementary Particles over Non-Archimedean Space–Time

It is known that no length or time measurements are possible in sub-Planckian regions of spacetime. The Volovich hypothesis postulates that the micro-geometry of spacetime may therefore be assumed to be non-archimedean. In this letter, the consequences of this hypothesis for the structure, classification, and conformal symmetry of elementary particles, when spacetime is a flat space over a non-archimedean field such as the p-adic numbers, is explored. Both the Poincaré and Galilean groups are treated. The results are based on a new variant of the Mackey machine for projective unitary representations of semidirect product groups which are locally compact and second countable. Conformal spacetime is constructed over p-adic fields and the impossibility of conformal symmetry of massive and eventually massive particles is proved.

BARYOGENESIS BY QUANTUM GRAVITY

A novel mechanism of baryogenesis is proposed on the basis of the phase transition from the conformal invariant spacetime to the Einstein spacetime in quantum gravity. Strong-coupling gravitational excitations with dynamical mass about 1017 GeV are generated at the transition. They eventually decay into ordinary matters. We show that the low energy effective interactions between the gravitational potential describing the excitation and the non-conserving matter currents by the axial anomalies can explain matter asymmetry out of thermal equilibrium.

Two-dimensional conformal models of space-time and their compactification

We study geometry of two-dimensional models of conformal space-time based on the group of Möbius transformation. The natural geometric invariants, called cycles, are used to linearize Möbius action. Conformal completion of the space-time is achieved through an addition of a zero-radius cycle at infinity. We pay an attention to the natural condition of nonreversibility of time arrow in order to get a correct compactification in the hyperbolic case.

CONFORMAL STRUCTURES AND TWISTORS IN THE PARAVECTOR MODEL OF SPACETIME

Some properties of the Clifford algebras [Formula: see text] and [Formula: see text] are presented, and three isomorphisms between the Dirac–Clifford algebra [Formula: see text] and [Formula: see text] are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group $pin+(2,4) is also investigated, in the light of a suitable isomorphism between [Formula: see text] and [Formula: see text]. After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of $pin+(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian ℝ4,1 spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac–Clifford algebra [Formula: see text] using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose, the Clifford algebra over ℝ4,1 is also used to describe conformal maps, instead of ℝ2,4. Our formalism sheds some new light on the use of the paravector model and generalizations.

The Casimir effect of Reissner–Nordström black hole

The stress tensor of a massless scalar field satisfying a mixed boundary condition in a (1+1)-dimensional Reissner–Nordstrom black hole background is calculated by using Wald's axiom. We find that Dirichlet stress tensor and Neumann stress tensor can be deduced by changing the coefficients of the stress tensor calculated under a mixed boundary condition. The stress tensors satisfying Dirichlet and Neumann boundary conditions are discussed. In addition, we also find that the stress tensor in conformal flat spacetime background differs from that in flat spacetime only by a constant.

Navigation of Space-Time Ships in Unified Gravitational and Electromagnetic Waves.

On the basis of a local principle of equivalence of general relativity, we consider a navigation in a kind of 4D-ocean involving measurements of conformally invariant physical properties only. Then, applying the Pfaff theory for PDE to a particular conformally equivariant system of differential equations, we show the dependency of any kind of function describing waves, with respect to 20 parametrizing functions. These latter, appearing in a linear differential Spencer sequence and determining gauge fields of deformations relatively to ship-metrics or to flat spacetime ocean metrics, may be ascribed to unified electromagnetic and gravitational waves. The present model is based neither on a classical gauge theory of gravitation or a gravitation theory with torsion, nor on any Kaluza-Klein or Weyl type unifications, but rather on a post-Newtonian approach of gravitation in a four dimensional conformal Cosserat spacetime.

Numerical evolution of axisymmetric, isolated systems in general relativity

We describe in this article a new code for evolving axisymmetric isolated systems in general relativity. Such systems are described by asymptotically flat space-times which have the property that they admit a conformal extension. We are working directly in the extended `conformal' manifold and solve numerically Friedrich's conformal field equations, which state that Einstein's equations hold in the physical space-time. Because of the compactness of the conformal space-time the entire space-time can be calculated on a finite numerical grid. We describe in detail the numerical scheme, especially the treatment of the axisymmetry and the boundary.