Timelike Worldlines Research Articles

(Pseudo)issue of the conformal frame revisited

The issue of the equivalence between Jordan and Einstein conformal frames in scalar-tensor gravity is revisited, with the emphasis on implementing running units in the latter. The lack of affine parametrization for timelike worldlines and the cosmological constant problem in the Einstein frame are clarified, and a paradox in the literature about cosmological singularities appearing only in one frame is solved. While, classically, the two conformal frames are physically equivalent, they seem to be inequivalent at the quantum level.

On wormholes with arbitrarily small quantities of exotic matter

Recently several models of traversable wormholes have been proposed which require only arbitrarily small amounts of negative energy to hold them open against self-collapse. If the exotic matter is assumed to be provided by quantum fields, then quantum inequalities can be used to place constraints on the negative energy densities required. In this paper, we introduce an alternative method for obtaining constraints on wormhole geometries, using a recently derived quantum inequality bound on the null-contracted stress-energy averaged over a timelike worldline. The bound allows us to perform a simplified analysis of general wormhole models, not just those with small quantities of exotic matter. We then use it to study, in particular, the models of Visser, Kar, and Dadhich (VKD) and the models of Kuhfittig. The VKD models are constrained to be either submicroscopic or to have a large discrepancy between throat size and curvature radius. A recent model of Kuhfittig is shown to be non-traversable. This is due to the fact that the throat of his wormhole flares outward so slowly that light rays and particles, starting from outside the throat, require an infinite lapse of affine parameter to reach the throat.

PASSAGE OF TIME IN A PLANCK SCALE ROOTED LOCAL INERTIAL STRUCTURE

It is argued that the "problem of time" in quantum gravity necessitates a refinement of the local inertial structure of the world, demanding a replacement of the usual Minkowski line element by a (4+2n)-dimensional pseudo-Euclidean line element, with the extra 2n being the number of internal phase space dimensions of the observed system. In the refined structure, the inverse of the Planck time takes over the role of observer-independent conversion factor usually played by the speed of light, which now emerges as an invariant but derivative quantity. In the relativistic theory based on the refined structure, energies and momenta turn out to be invariantly bounded from above, and lengths and durations similarly bounded from below, by their respective Planck scale values. Along the external timelike world-lines, the theory naturally captures the "flow of time" as a genuinely structural attribute of the world. The theory also predicts expected deviations — suppressed quadratically by the Planck energy — from the dispersion relations for free fields in the vacuum. The deviations from the special relativistic Doppler shifts predicted by the theory are also suppressed quadratically by the Planck energy. Nonetheless, in order to estimate the precision required to distinguish the theory from special relativity, an experiment with a binary pulsar emitting TeV range γ-rays is considered in the context of the predicted deviations from the second-order shifts.

Null energy conditions in quantum field theory

For the quantised, massless, minimally coupled real scalar field in four-dimensional Minkowski space, we show (by an explicit construction) that weighted averages of the null-contracted stress-energy tensor along null geodesics are unbounded from below on the class of Hadamard states. Thus there are no quantum inequalities along null geodesics in four-dimensional Minkowski spacetime. This is in contrast to the case for two-dimensional flat spacetime, where such inequalities do exist. We discuss in detail the properties of the quantum states used in our analysis, and also show that the renormalized expectation value of the stress energy tensor evaluated in these states satisfies the averaged null energy condition (as expected), despite the nonexistence of a null-averaged quantum inequality. However, we also show that in any globally hyperbolic spacetime the null-contracted stress energy averaged over a timelike worldline does satisfy a quantum inequality bound (for both massive and massless fields). We comment briefly on the implications of our results for singularity theorems.

Does the Principle of Equivalence Prohibit Trapped Surfaces from Forming in the General Relativistic Collapse Process

It has recently been shown that time-like spherical collapse of a physical fluid in General Relativity does not permit formation of “trapped surfaces.” This result followed from the fact that the formation of a trapped surface in a physical fluid would cause the time-like world lines of the collapsing fluid to become null at the would-be trapped surface, thus violating the Principle of Equivalence in General Theory of Relativity (GTR). For the case of the spherical collapse of a physical fluid, the “no trapped surface condition” 2GM(r, t)/R(r, t) c2 0, where zs is the surface red shift seen by a zero-angular momentum observer. When this condition is applied to the first integral of the time-time component of the Einstein equation, it leads to the “no trapped surface condition” 2GM(rs, t)/R(rs, t) c2<1 consistent with the condition obtained above for the interior co-moving metric. The Principle of Equivalence enforces the “no trapped surface condition” by constraining the physics of the general relativistic radiation transfer process in a manner that requires it to establish and maintain an Eddington limited secular equilibrium on the dynamics of the collapsing radiating surface so as to always keep the physical surface of the collapsing object outside of its Schwarzschild radius. The important physical implication of the “no trapped surface condition” is that galactic black hole candidates GBHC do not possess event horizons and hence do possess intrinsic magnetic fields. In this context the spectral characteristics of galactic black hole candidates offer strong evidence that their central nuclei are highly red-shifted Magnetospheric Eternally Collapsing Objects (MECO) within the framework of General Relativity.

Foundations of Gravitational Lens Theory (Geometry of Light Cones)

The main concepts of gravitational lens theory are introduced on the basis of spacetime geometry without assuming approximations. The singularities of light cones, in particular their caustics, are reviewed as examples of singularities of Lagrangian resp. Legendrian maps. It is indicated how the usual approximate lens theory may be derived from the general framework. After the discovery of the double quasar QSO O957 + 561 A, B by Walsh, Carswell and Weymann in 1979, gravitational lensing rapidly developed into a major tool of astrophysics, providing infomation about cosmological parameters, masses and mass distributions on the scales of stars, galaxies, galaxy clusters and that of the universe at large. It enables astronomers to obtain information about dark matter, the structure of quasars and very distant, early generation galaxies up to redshifts of z ≈ 5. Usually, gravitational lens theory is based on plausible assumptions and various approximations designed for astrophysical applications. Physical notions and relations are expressed essentially in the framework of classical geometrical optics, with minimal input from general relativity. While such an approach is useful for the intended pur- pose, it conceals the spacetime-geometrical origin of lensing phenomena. Moreover, by using ab initio simplifications based on intuition, one foregoes the possibility to assess the accuracy of approximations, and one may not even recognize which general relativistic relations are being approximated. Besides, such presentations may render it difficult for relativists, used to think in terms of light cones, timelike world lines and the like, to understand what it's all about. Be that as it may, here I want to outline how the basic qualitative relations of gravitational lens theory may be introduced as part of Lorentzian spacetime geometry a I dedicate this paper to George Ellis, with affection and gratitude for fourty years of many stimulating encounters and sharing of ideas.

Transport along null curves

Fermi Transport is useful for describing the behaviour of spins or gyroscopes following non-geodesic, timelike world lines. However, Fermi Transport breaks down for null world lines. We introduce a transport law for polarisation vectors along non-geodesic null curves. We show how this law emerges naturally from the geometry of null directions by comparing polarisation vectors associated with two distinct null directions. We then give a spinorial treatment of this topic and make contact with the geometric phase of quantum mechanics. There are two significant differences between the null and timelike cases. In the null case (i) The transport law does not approach a unique smooth limit as the null curve approaches a null geodesic. (ii) The transport law for vectors is integrable, i.e the result depends only on the local properties of the curve and not on the entire path taken. However, the transport of spinors is not integrable: there is a global sign of topological origin.

Exact solutions of the Yang-Mills equations with source in the form of two point color charges

We find a family of exact retarded solutions of the classical Yang—Mills equations with a source that consists of two point color charges that move along arbitrary timelike worldlines in Minkowski space. The solutions are stable with respect to small field perturbations and are therefore suitable for playing the role of a nonperturbative gluon vacuum. It is shown that a description of confinement in the framework of classical Yang—Mills theory is impossible. Confinement is ensured by quantum fluctuations over the constructed gluon vacuum.

Vortex dynamics of the nonlinear wave equation

ψ is a complex scalar field defined on (2 + 1 - D)-dimensional Minkowski space. It satisfies the nonlinear wave equation (NLWE) □ψ−(1−|ψ|)ψ = 0 We study the solutions which contain time-like world lines of zeros, with winding numbers +1 or -1. We call these particle-like defects of the ψ field vortices. We formulate an asymptotic “particle + field” vortex dynamics which derives from the NLWE. It is analogous to particle + field electrodynamics in 2 + 1 - D. In particular, the winding number plays the role of charge, and gradients, of arg ψ play the role of the electromagnetic field.

Equations of motion in a generalized theory of gravitation

The problem of the motion of test particles is studied in a theory of gravitation based on a nonsymmetric gμν. According to the conservation laws the test particles can follow two kinds of geodesies, depending on the definition of a local inertial frame in the theory. One of these geodesies is nonmaximal and leads to a timelike and null world line complete space when a new parameter l, that occurs as a constant of integration in the spherically symmetric, static solution of the field equations, satisfies [Formula: see text]. In the theory, the parameter [Formula: see text] where N is the number of fermions in a system and a is a new universal coupling constant that satisfies [Formula: see text]. The physical implications of l and the associated conservation law of fermion number is discussed in detail.

Time travel in G�del's space

We present an analysis of the motion of test particles in Godel's universe. Both geodesical and nongeodesical motions are considered; the accelerations for nongeodesical motions are given. Examples for closed timelike world lines are shown and the dynamical conditions for time travel in Godel's space-time are discussed. It is shown that these conditions alone do not suffice to exclude time travel in Godel's space-time.

General Relativity and the Conceivability of Time Travel

It has been suggested by several philosophers (Lewis 1976, Putnam 1962) that many of the so-called paradoxes of backward time travel can be resolved if we conceive of the backward time traveller as having a zig-zag or N-shaped world line in spacetime. In this I am in general agreement. But there is still a problem in conceiving of backward time travel this way. In this note I will show how we can solve this problem by conceiving of backward time travel in terms of the closed time-like world lines in certain general relativistic space-times. Indeed, it has often been claimed (cf. Dwyer 1978, Godei 1949, Tipler 1974) that such world models as Godei spacetime show (or contribute to showing) that backward time travel is conceivable. Our discussion will help to make clear just why this claim is correct.

Tentative field-theoretical interpretation of hadronic potentials

This paper is divided into two parts. In the first part we propose a generalization of d'Alembert's field equation which depends on one parameter $\ensuremath{\omega}$. For each value of $\ensuremath{\omega}$ the corresponding field equation is Poincar\'e invariant, admits causal solutions attached to timelike world lines, and is semilinear. In the static limit some of these causal solutions are the functions ${r}^{\ensuremath{\rho}}$, where the exponent $\ensuremath{\rho}$ depends on $\ensuremath{\omega}$ and where $r$ is the distance between the point where the field is calculated and the straight line defining the motion of the source. The general causal solutions are simple and interesting modifications of the preceding ones. In the second part we consider a system of $N$ pointlike particles interacting via any causal scalar field, and using the methods of relativistic predictive mechanics we calculate the Newtonian and post-Newtonian approximations of the equations of motion. The Newtonian approximation coincides with classical potential theory. The post-Newtonian approximation is unambiguously determined by the Newtonian approximation. The combined results of these two parts suggest some field-theoretical justifications to current ideas about hadronic potentials.

An Einstein-G�del universe

The metric for the standard static Einstein model of the universe can be expressed in coordinates for which a congruence of spacelike world lines of the model will be twisting. A method of “shifting the twist” has been devised by which the twist on spacelike world lines is shifted onto the timelike world lines. The model universe then becomes Godel's model. A combined Einstein-Godel model containing a parametere is obtained. Switchinge from +1 to −1 will effect the shift of twist in the world lines and lead from the Einstein model to the Godel model.

Mapping of trajectories of test bodies in the pseudo-Riemann spaces V4, and �V4

In general coordination and for identical initial conditions all time-like geodesies of the pseudo-Riemann space V4 are mapped with time-like world lines of the space ¯V4 due to a pseudo-Lorentzian force. The mapping of all time-like world lines of ¯V4 with the geodesies of the space V4, is also considered. Necessary and sufficient conditions for the existence of such mappings are found.

Relativistic treatment of multipole radiation from an extended charge-current distribution: I. Asymptotic properties of the electromagnetic field

The electromagnetic field due to a localized charge-current distribution is expanded in terms of a set of multiple moments defined with respect to a given time-like world line. The behaviour of this expansion in the asymptotic region is studied, leading to the identification of the radiation field. It is shown that the radiation field is a null field.

Begründung des Energie-Impuls-Tensors der allgemeinen Relativitätstheorie auf Grund eines kinetischen Modells der Materie

The energy-momentum tensor Tαβ of the general theory of relativity is investigated on the basis of a kinetic model of matter (point-particles). Its freedom of divergence results from the motion of the particles. After a suitable decomposition of Tαβ into its invariant components these can be interpreted in a generally covariant manner according to their microstructure. By means of the distribution of matter special time-like world-lines are designated as stream-lines. The freedom of divergence of Tαβ can be interpreted by virtue of the model as a local balance of energy and momentum. In this balance the influence of gravitational and inertial forces become immediately evident. Pressure results as a function of the particle properties (state equation).

A Class of Null Flat-Space Coordinate Systems

A new coordinate system, intrinsically attached to an arbitrary timelike world line, is investigated in flat-space time. The Maxwell field tensor associated with the field of an arbitrarily moving charged particle assumes a particularly simple form in this, its intrinsic coordinate system. This reference frame is expected to be useful in General Relativity, in asymptotic studies of radiation, and equations of motion.

Finite Universe and Mach's Principle

RECENTLY, Ozsvath and Schucking found an exact solution of Einstein's field equations of gravitation, which represents an instructive new model of a universe (finite rotating universe)1: Although in the entire cosmos matter is at rest relative to a suitably chosen co-ordinate system, nevertheless everywhere Coriolis forces do exist. This property of the Ozsvath–Schucking model is common to a solution already found by K. Godel in 19492; whereas Godel's solution represents a universe of infinite space extension and, besides, closed time-like world lines exist, the Ozsvath-Schucking model represents a spatially finite universe in which time-like world lines cut the closed space like hypersurfaces only once. Therefore, no reasons exist for excluding this model as physically meaningless.

On the theory of point-particles

It is deduced from the conservation of the energy -momentum tensor that if the flow of energy and momentum into a tube surrounding a time-like world-line, on which the field is singular, become singular as the size of the tube is contracted to zero, then the singular terms are necessarily perfect differentials of quantities on the world-line with respect to the proper time along the world-line. The same can be proved of any other tensor, as, for example, the angular-momentum tensor, which is conserved. It is proved from this that for any point -particle whatever having charge, spin or other properties, which need not be specified, it is always possible to deduce exact equations of motion which are finite. It is proved further that if the energy-momentum tensor is altered by the addition of ∂ K μvσ /∂x σ , where K μvσ is any tensor antisymmetric in v and σ , then the equations of motion are unaltered, but it is possible to choose K μvσ in such a way as to make the flow of energy and momentum into a given tube non-singular.