Regge calculus and observations. I. Formalism and applications to radial motion and circular orbits

A method is developed for tracing geodesics of particles and light rays through Regge calculus space-times. In the flat interiors of the blocks, the geodesics are straight lines, and at the boundaries between blocks they are refracted according to an extremal distance law. The method is then tested for the Regge calculus version of the Schwarzschild space-time. For radial motion, it gives good agreement with the analytic solution, provided the blocks are sufficiently small. Exactly circular orbits are impossible in Regge calculus, but it is shown how to construct approximately circular ones. These examples are in preparation for an investigation of more general orbits and more general space-times, using Regge calculus.     

Equivalence of the Regge and Einstein equations for almost flat simplicial space-times

The theory of distributions is applied to almost flat simplicial space-times. Explicit expressions are given for the first-order defects. It is shown explicitly that the Riemann tensor for an almost flat simplicial space-time contains delta-functions on the bones and derivatives of delta-functions on the 3-dimensional faces of the boundary of the space-time. The latter terms have not previously been seen in the Regge calculus. It is shown that the Regge and Hilbert actions have equal values on almost fiat simplicial space-times and that the Einstein equations lead directly to the Regge field equations.     

Canonical forms for axial symmetric space-times

The general canonical forms for axial symmetric space-times are investigated. Special forms such as static, stationary and cylindrical are considered. Our results are based only upon the symmetries assumed, not upon the field equations; thus they are applicable to vacuum, electromagnetic and matter field problems. In particular, we find a term that was missing in previous work. There can be in the general canonical form a metric coefficient between the axis of symmetry and the angle about this axis. This metric coefficient survives even for static or stationary space-times.     

Meson spectrum in Regge phenomenology

Under the assumption that both light and heavy quarkonia populate approximately linear Regge trajectories with the requirements of additivity of intercepts and inverse slopes, the masses of different meson multiplets are estimated. The predictions derived from the quasi-linear Regge trajectories are in reasonable agreement with those given by many other references.Received: 22 April 2004, Revised: 10 August 2004, Published online: 7 September 2004De-Min Li: lidm@zzu.edu.cnMailing address: Department of Physics, Zhengzhou University, Zhengzhou, Henan, 450052, P.R. China     

Conformally flat space-times of locally constant connection. I

Conformally flat space-times of locally constant connection are studied. The constant connection defines a global vector field which is assumed timelike. The general solution of the geodesic equations is presented and several theorems characterizing the geometry of such space-times are proved.     

Relativistic string model of light mesons with massless quarks

The straightline string model of mesons with massless fermions (“quarks”) at the string endpoints is constructed. The spectrum of the model, consisting of four Regge trajectories weakly non-linear atM ²～0, is obtained. A comparison of the model predictions with the experimental spectrum of light mesons and prospects for future searches are discussed.     

An improved singular perturbation solution for bound geodesic orbits in the Schwarzschild metric

A modification to the Lindstedt-Poincaré method of strained parameters is applied to the differential equation of the orbit of a test particle in the Schwarzschild exterior metric. A new perturbation solution for the equation of the bound orbit, which is completely free of secular terms in the angular coordinate, is derived. The precession of the orbit per revolution is calculated using this solution and it is found to give a more accurate result than existing perturbation solutions. The method should be applicable to similar orbital problems in general relativity.     

On the Regge-pole description of nucleus-nucleus scattering

The properties of the Regge poles of theS-matrix for scattering of strongly-absorbed nuclear particles are considered. Simple formulae are obtained for describing the Regge trajectories in terms of the nuclear radius, the quasi stationary levels in the combined nuclear-Coulomb-potential and the widths of these levels. The predictions of these formulae are compared with the Regge trajectories obtained previously, for a Woods-Saxon potential, and with those required to fit¹⁶O-¹²C backward scattering.     

Regge phenomenology of inclusive spectra in the central region

In the light of new data from the ISR, initial steps are taken towards the Regge analysis of inclusive spectra in the central region. It is shown that some features in the data necessitate the introduction of a further singularity in addition to the Pomeron and the meson trajectories ? - f - ω - A₂. The resulting model yields many predictions via duality and factorisation which can easily be tested at ISR and Batavia energies.     

Relativistic perihelion precession of orbits of Venus and the Earth

Among all the theories proposed to explain the “anomalous” perihelion precession of Mercury’s orbit first announced in 1859 by Le Verrier, the general theory of relativity proposed by Einstein in November 1915 alone could calculate Mercury’s “anomalous” precession with the precision demanded by observational accuracy. Since Mercury’s precession was a directly derived result of the full general theory, it was viewed by Einstein as the most critical test of general relativity from amongst the three tests he proposed. With the advent of the space age, the level of observational accuracy has improved further and it is now possible to detect this precession for other planetary orbits of the solar system — viz., Venus and the Earth. This conclusively proved that the phenomenon of “anomalous” perihelion precession of planetary orbits is a relativistic effect. Our previous papers presented the mathematical model and the computed value of the relativistic perihelion precession of Mercury’s orbit using an alternate relativistic gravitational model, which is a remodeled form of Einstein’s relativity theories, and which retained only experimentally proven principles. In addition this model has the benefit of data from almost a century of relativity experimentation, including those that have become possible with the advent of the space age. Using this model, we present in this paper the computed values of the relativistic precession of Venus and the Earth, which compare well with the predictions of general relativity and are also in agreement with the observed values within the range of uncertainty.      

Discrete quantum gravity in the Regge calculus formalism

We discuss an approach to the discrete quantum gravity in the Regge calculus formalism that was developed in a number of our papers. The Regge calculus is general relativity for a subclass of general Riemannian manifolds called piecewise flat manifolds. The Regge calculus deals with a discrete set of variables, triangulation lengths, and contains continuous general relativity as a special limiting case where the lengths tend to zero. In our approach, the quantum length expectations are nonzero and of the order of the Plank scale, 10^?33 cm, implying a discrete spacetime structure on these scales.     

Grand Canonical Ensembles in General Relativity

We develop a formalism for general relativistic, grand canonical ensembles in space-times with timelike Killing fields. Using that, we derive ideal gas laws, and show how they depend on the geometry of the particular space-times. A systematic method for calculating Newtonian limits is given for a class of these space-times, which is illustrated for Kerr space-time. In addition, we prove uniqueness of the infinite volume Gibbs measure, and absence of phase transitions for a class of interaction potentials in anti-de Sitter space.     

Regge analysis of the ππ scattering amplitude

The theoretical predictions for the subtraction constants lead to a very accurate dispersive representation of the ππ scattering amplitude below 0.8 GeV. The extension of this representation up to the maximum energy of validity of the Roy equations (1.15 GeV) requires a more precise input at high energies. In this paper we determine the trajectories and residues of the leading Regge contributions to the ππ amplitude (Pomeron, f and ρ), using factorization, phenomenological parametrizations of the πN and NN total cross sections at high energy, and a set of sum rules which connect the high and low energy properties of ππ scattering. We find that nonleading Regge terms are necessary in order to achieve a smooth transition from the partial waves to the Regge representation at or below 2 GeV. We obtain thus a Regge representation consistent both with the experimental information at high energies and the Roy equations for the partial waves with ℓ≤4. The uncertainties in our result for the Regge parameters are sizable, but in the solutions of the Roy equations these only manifest themselves above the K[`]KK\hspace {0.03cm}\overline {\rule [0.7em]{0.6em}{0em}}\hspace {-0.8em}K threshold.     

A Fermat principle for stationary space-times and applications to light rays

We present an extension of the classical Fermat principle in optics to stationary space-times. This principle is applied to study the light rays joining an event with a timelike curve. Existence and multiplicity results of light rays are proved. Moreover, Morse Relations relating the set of rays to the topology of the space-time are obtained, by using the number of conjugate points of the ray. The results hold also for stationary space-times with boundary, in particular the Kerr space-time outside the stationary limit surface.     

Conformal Killing vectors and FRW spacetimes

Fluid space-times which admit a conformal Killing vector (CKV) are studied. It is shown that even in a perfect fluid space-time a conformal motion will not, in general, map the fluid flow lines onto fluid flow lines; consequently, perfect fluid space-times and, in particular, the simplest perfect fluid space-times known to admit a CKV, namely the Friedmann-Robertson-Walker (FRW) space-times, are studied. A direct proof that there do not exist any special CKV in FRW space-times will be given, thereby motivating the study of the physically more relevant proper CKV. Indeed, one of the principal motivations of the present work is the study of the symmetry inheritance problem for proper CKV. Since the FRW metric can, in general, satisfy the Einstein field equations for a non-comoving imperfect fluid, the relationship between the FRW models (and in particular the standard comoving perfect fluid models) and the conditions under which conformal motions (and in addition homothetic motions) map fluid flow lines onto fluid flow lines are investigated. Finally, further properties of fluid space-times which admit a proper CKV, and in particular space-times in which the CKV is parallel to the fluid four-velocity, are discussed.     

Measurement of gravitational spin-orbit coupling in a binary-pulsar system

In relativistic gravity, a spinning pulsar will precess as it orbits a compact companion star. We have measured the effect of such precession on the average shape and polarization of the radiation from PSR B1534+12. We have also detected, with limited precision, special-relativistic aberration of the revolving pulsar beam due to orbital motion. Our observations fix the system geometry, including the misalignment between the spin and orbital angular momenta, and yield a measurement of the precession time scale consistent with the predictions of general relativity.     

An application of a first-order formulation of the Regge calculus

A numerical Regge spacetime is constructed by way of a new formulation of the Regge calculus [1,2,3]. The results obtained are compared with the known exact results [4]. It will be shown that the approximations suggested in [3] are, in this instance, very accurate.     

An Entropy Functional for Riemann-Cartan Space-Times

By viewing space-time as a continuum elastic medium and introducing an entropy functional for its elastic deformations, T. Padmanabhan has shown that general relativity emerges from varying the functional and that the latter suggests holography for gravity and yields the Bekenstein-Hawking entropy formula. In this paper we extend this idea to Riemann-Cartan space-times by constructing an entropy functional for the elastic deformations of space-times with torsion. We show that varying this generalized entropy functional permits to recover the full set of field equations of the Cartan-Sciama-Kibble theory. Our generalized functional shows that the contributions to the on-shell entropy of a bulk region in Riemann-Cartan space-times come from the boundary as well as the bulk and hence does not suggest that holography would also apply for gravity with spin in space-times with torsion. It is nevertheless shown that for the specific cases of Dirac fields and spin fluids the system does become holographic. The entropy of a black hole with spin is evaluated and found to be in agreement with Bekenstein-Hawking formula.