Compactification of Two Dimensions in General Relativity

Motivated by Kaluza-Klein theory and modern string theories, the class of exact solutions yielding product manifolds M ₂ × S ² in general relativity is investigated. The compact submanifold homeomorphic to S ² is chosen to be a very small sphere. Choosing an anisotropic fluid as the particular physical model, it is proved that very large mass density and tension provide the mechanism of compactification. In case the transverse pressure is chosen to be zero, the corresponding spacetime is homeomorphic to ² × S ², and thus provides a tractable non-flat metric. In this simple metric, the geodesic equations are completely solved, yielding motions of massive test particles. Next, the corresponding wave mechanics (given by the Klein-Gordon equation) is explored in the same curved background. A general class of exact solutions is obtained. Four conserved quantities are explicitly computed. The scalar particles exhibit a discrete mass spectrum.     

The interpretation of the C metric. The charged case whene 2 ≤m 2

The charged C metric involves three parametersm, e andA representing mass, charge and acceleration respectively. Using a method developed in a previous paper, we show that whene ² m ² the metric may be interpreted in terms of two Reissner-Nordström particles, each of massm and with charges +e and –e, in accelerated motion and connected by a spring. The method depends on the fact that for certain regions of the coordinate space the charged C metric may be transformed into the Weyl form for a static axisymmetric system. In this form the horizons of the C metric become line sources. One of the regions leads to a Weyl metric with two line sources, one of finite length which corresponds to the outer horizon of a Reissner-Nordström particle and the other semi-infinite corresponding to a horizon associated with uniform accelerated motion. A further coordinate transformation leads to a metric valid for a larger region of space-time in which there are two charged particles in accelerated motion. WhenAm is small, the electromagnetic invariants approximate to those for the Born field for two accelerated charges in special relativity.     

Hypothesis of friedmons as dark matter particles and hypothesis of initial cosmological constant decay

Hypothesis of friedmons as dark matter particles is proposed. Friedmons are stable particles with a mass of billion nucleon masses. These particles correspond to the not yet been discovered exact symmetry group dual to the SU(2) group: for the Standard model symmetries and dual symmetries, the roles of exact and broken symmetries and corresponding stable and unstable particles change places. The hypothesis of the decay of the primordial de Sitter vacuum of the Planck density to an asymptotic state of the expanding Universe with de Sitter vacuum of the observed critical density is proposed. The T -duality and S-duality hypotheses relating subgroups SU(3)×SU(2)×U(1) and dual subgroups S??(3)× S??(2) × ??(1) with decay of the primordial symmetry group E(8) × ??(8) are proposed. In particular, these dualities relate the minimum Planck length 10^?13 cm to the primordial curvature radius 10^?13 cmof theMetagalaxy of the Planck density and its modern curvature radius of 10²⁸ cm. That is, the probable relation of the Planck mass to the Metagalaxy mass of 10⁶¹ Planck masses is indicated.     

R×S 3 special theory of relativity

A theory of relativity, along with its appropriate group of Lorentz-type transformations, is presented. The theory is developed on a metric withR×S ³ topology as compared to ordinary relativity defined on the familiar Minkowskian metric. The proposed theory is neither the ordinary special theory of relativity (since it deals with noninertial coordinate systems) nor the general theory of relativity (since it is not a dynamical theory of gravitation). The theory predicts, among other things, that finite-mass particles in nature have maximum rotational velocities, a prediction highly supported by recent experiments on 14 nuclei, such as ¹⁵⁹ Yb that survives fission with angular velocities of up to 0.9 of the predicted value but does not reach it.Address during academic year 1985/1986: Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742.     

Static and stationary solutions of the Einstein-Maxwell equations

The Weyl axially symmetric electrovac formalism for coincident gravitational and electrostatic equipotential surfaces is used to generate charged versions of some axially symmetric vacuum fields. The metric for two separated charged Curzon particles held in equilibrium by a strut is found and the condition for the removal of the strut is discussed. Kinnersley transformations applied to the two-particle metric yield spin but line singularities invariably appear along the symmetry axis and the metric is asymptotically NUT-like. It is shown that any Kinnersley transformation applied to a static axially symmetric asymptotically flat vacuum metric generates another asymptotically flat metric only if the latter is static. Moreover, the transformed metric is always undercharged (q ²<m ²) if it is asymptotically flat. A necessary and sufficient condition for asymptotic flatness in terms of the relevant parameters is found. A generalization of the Kinnersley transformation scheme is presented and illustrated by an example.     

First order formalism off(R) gravity

The first order formalism is applied to study the field equations of a general Lagrangian density for gravity of the form . These field equations correspond to theories which are a subclass of conformally metric theories in which the derivative of the metric is proportional to the metric by a Weyl vector field. The resulting geometrical structure is unique, except whenf(R)=aR ², in the sense that the Weyl field is identifiable in terms of the trace of the energy-momentum tensor and its derivatives. In the casef(R)=aR ² the metric is only defined up to a conformai factor. We discuss the matter conservation equations which are implied by the invariance of the theories under diffeomorphisms. We apply the results to the case of dust and obtain that in general the dust particles will not follow geodesic Unes. We consider the linearized field equations and apply them to obtain the weak field slow motion limit. It is found that the gravitational potential acquires a new term which depends linearly on the mass density. The importance of these new equations is briefly discussed.     

Photon Decays in the Radiation-Dominated Universe: Scalar Electrodynamics

The effect of the creation of an arbitrary number of massive pairs by a photon in the spatially flat model of the radiation-dominated Universe is considered. The process added-up probability is calculated within the framework of scalar quantum electrodynamics conformally related to the metric of a curved spacetime. The rate of photon decay in the radiation-dominated universe as well as the mean number of the created particles have been found. Comparison of the rate of the pair creation in the photon decays with the rate of the pair creation in the photon-photon collisions which take place in the Minkowski spacetime has been carried out. The estimates having been made show the number density of the particles created in the processes of the photon decays in the radiation-dominated Universe to be by a factor of 10³⁰ higher than the number density of the particles created from the vacuum of the free scalar field by the gravitational background.     

A spherically symmetric vacuum solution of the quadratic Poincaré gauge field theory of gravitation with newtonian and confinement potentials

We derive a stationary spherically symmetric vacuum solution in the framework of the Poincaré gauge field theory with a recently proposed quadratic lagrangian. We find a metric of the Schwarzschild-de Sitter type, both torsion and curvature are non vanishing, with torsion proportional to the mass and curvature proportional to the strong coupling constant κ. The metric exhibits two pieces, a newtonian potential describing the gravitational behavior of macroscopic matter, and a confining potential ～κr² presumably related to the strong-interaction properties of hadrons. To our knowledge this is a new feature of a classical solution of a Yang-Mills type gauge theory.     

Strong spin-two interaction and general relativity

The concept of short range strong spin-two (f) field (mediated by massive f-mesons) and interacting directly with hadrons was introduced along with the infinite range (g) field in early seventies. In the present review of this growing area (often referred to as strong gravity) we give a general relativistic treatment in terms of Einstein-type (non-abelian gauge) field equations with a coupling constant G_f ? 10³⁸G_N (G_N being the Newtonian constant) and a cosmological term λ_f ?;_μν (?;_μν is strong gravity metric and λ_f ～ 10²⁸ cm^? is related to the f-meson mass). The solutions of field equations linearized over de Sitter (uniformly curves) background are capable of having connections with internal symmetries of hadrons and yielding mass formulae of SU(3) or SU(6) type. The hadrons emerge as de Sitter “microuniverses” intensely curved within (radius of curvature ～10^?14 cm).The study of spinor fields in the context of strong gravity has led to Heisenberg's non-linear spinor equation with a fundamental length ～2 × 10^?14 cm. Furthermore, one finds repulsive spin-spin interaction when two identical spin-$\text{1}\text{2}$ particles are in parallel configuration and a connection between weak interaction and strong gravity.Various other consequences of strong gravity embrace black hole (solitonic) solutions representing hadronic bags with possible quark confinement, Regge-like relations between spins and masses, connection with monopoles and dyons, quantum geons and friedmons, hadronic temperature, prevention of gravitational singularities, providing a physical basis for Dirac's two metric and large numbers hypothesis and projected unification with other basic interactions through extended supergravity.     

Deriving the mean primary-particle diameter and related quantities from the size distribution and the gravimetric mass of spark generated nanoparticles

Spark generated carbon and iridium nanoparticles were characterised by their electrical-mobility diameter D and by the mass of particulate matter collected in parallel on filter. The particles exhibited slightly skewed lognormal size distributions with mean mobility diameters between 18 and 74 nm. The masses calculated from the measured distributions under the assumption that the particles were spherical (diameter D) and of bulk mass density turned out to be much higher than the gravimetric mass, by factors between 8 and as high as 340. This very pronounced difference initiated a search for an improved relation between particle size and mass. Data analysis suggested that the mass increases linearly with increasing D. Hence the measured distributions were evaluated under the assumption that the spark generated matter was composed of spherical primary nanoparticles of mean diameter d, aggregated in the form of chains of joint length βD, with β>1. Using reasonable values of β between 2 and 4, the mean diameter of carbon primary particles turned out to be 10±1.8 nm, in excellent agreement with size data recently obtained by transmission electron microscopy (TEM). The primary iridium particles were found to be distinctly smaller, with diameters between 3.5±0.6 nm and 5.4±0.9 nm. The comparatively small uncertainty is due to the fact that the primary-particle diameter is proportional to the square root of β. The calculated volume specific surface areas range between 500 and 1700 m²/cm³. These numbers are close to the ‘active’ surface areas previously measured by the BET method. The good agreement with TEM and BET data suggests that the novel approach of nanoparticle characterisation is meaningful. Accordingly, the number concentrations of all individual primary particles rather than the concentrations measured by the mobility analyser should be␣considered the correct dose metric in studies on animal exposure to spark generated nanoparticles. The␣evaluated data imply that the numbers quoted in the literature must be enlarged by factors ranging between about 10 and a maximum as high as 80. An erratum to this article can be found at     

Reciprocal Relativity of Noninertial Frames and the Quaplectic Group

The frame associated with a classical point particle is generally noninertial. The point particle may have a nonzero velocity and force with respect to an absolute inertial rest frame. In time–position–energy–momentum-space {t, q, p, e}, the group of transformations between these frames leaves invariant the symplectic metric and the classical line element ds² = d t². Special relativity transforms between inertial frames for which the rate of change of momentum is negligible and eliminates the absolute rest frame by making velocities relative but still requires the absolute inertial frame. The Lorentz group leaves invariant the symplectic metric and the line elements and . General relativity for particles under only the influence of gravity avoids the issue of noninertial frames as all particles follow geodesics and hence have locally inertial frames. For other forces, the question of the absolute inertial frame remains.) Born conjectured that the line element should be generalized to the pseudo-orthogonal metric . The group leaving this metrics and the symplectic metric invariant is the pseudo-unitary group of transformations between noninertial frames. We show that these transformations also eliminate the need for an absolute inertial frame by making forces relative and bounded by b and so embodies a relativity that is shape reciprocal in the sense of Born. The inhomogeneous version of this group is naturally the semidirect product of the pseudo-unitary group with the nonabelian Heisenberg group. This is the quaplectic group.     

A rotating mass embedded in an Einstein-Gödel universe

A metric containing a parameterε(ε ²=1) has been obtained which represents a Kerr metric in the background of a static Einstein universe whenε is put equal to +1. The same metric will represent the external field of a mass embedded in a rotating Gödel universe whenε is set equal to ?1.     

Magnetic charge,black holes,and cosmic censorship

The possibility of converting a Reissner-Nordström black hole into a naked singularity by means of test particle accretion is considered. The dually charged Reissner-Nordström metric describes a black hole only when M² > Q³ + P². The test particle equations of motion are shown to allow test particles with arbitrarily large magnetic charge/mass ratios to fall radially into electrically charged black holes. To determine the nature of the final state (black hole or naked singularity) an exact solution of Einstein's equations representing a spherical shell of magnetically charged dust falling into an electrically charged black hole is studied. Naked singularities are never formed so long as the weak energy condition is obeyed by the infalling matter. The differences between the spherical shell model and an infalling point test particle are examined and discussed.     

Gradient properties of the renormalisation group equations in multicomponent systems

The existence of a metric, which enables the renormalisation group β functions of a multicomponent field theory to be written as a gradient, has very important implications for the asymptotic behavior of the renormalisation group equations. It is shown that a very simple metric exists in a field theory with n-component Bose fields and arbitrary φ⁴ interaction, when the β functions are calculated perturbatively up to and including the 2-loop diagrams. This same metric is shown to work for all irreducible diagrams, but it must and can be modified to accommodate reducible 3-loop contributions. The prospects and outlook of this aspect of the renormalisation group are discussed.     

Electrodynamics in Dirac's Gauge: A Geometrical Equivalence

It is shown that, when classical non-relativistic electrodynamics is formulated in Dirac's gauge A A =const. and the vector potential A interpreted as a velocity field of the vacuum, the motion of a charged particle results from purely inertial effects. A metric is given for particles of a fixed charge to mass ratio.     

Mass and Charge in Brane-World and Non-Compact Kaluza-Klein Theories in 5 Dim

In classical Kaluza-Klein theory, with compactified extra dimensions and without scalar field, the rest mass as well as the electric charge of test particles are constants of motion. We show that in the case of a large extra dimension this is no longer so. We propose the Hamilton-Jacobi formalism, instead of the geodesic equation, for the study of test particles moving in a five-dimensional background metric. This formalism has a number of advantages: (i) it provides a clear and invariant definition of rest mass, without the ambiguities associated with the choice of the parameters used along the motion in 5D and 4D, (ii) the electromagnetic field can be easily incorporated in the discussion, and (iii) we avoid the difficulties associated with the splitting of the geodesic equation. For particles moving in a general 5D metric, we show how the effective rest mass, as measured by an observer in 4D, varies as a consequence of the large extra dimension. Also, the fifth component of the momentum changes along the motion. This component can be identified with the electric charge of test particles. With this interpretation, both the rest mass and the charge vary along the trajectory. The constant of motion is now a combination of these quantities. We study the cosmological variations of charge and rest mass in a five-dimensional bulk metric which is used to embed the standard k = 0 FRW universes. The time variations in the fine structure constant and the Thomson cross section are also discussed.     

Creation of fundamental particles in Wesson’s IMT

Fundamental particles, regarded as possible constituents of quarks and leptons, are described classically in the framework of the Weyl-Dirac version of Wesson’s Induced Matter Theory (IMT). There are neutral particles and particles having charge \(\pm \frac{1}{3}e\). The particles appear on the 4D brane, our universe, and are filled with an induced by the 5D bulk substance. This substance is taken to have mass density, pressure, and (if charged) charge density, and is characterized by the equation of state ρ + P = 0. The interior is separated from the surrounding vacuum by a spherical boundary surface where the components of the 4D metric tensor h ₀₀ = 1/h ₁₁ = 0. Outside of the boundary holds the Schwarzschild, or the Reissner–Nordstrøm metric, while the particles are characterized by Mass, Radius, Charge.