Four-dimensional Case Research Articles

Differential geometrical methods in the study of optical transmission (scalar theory) II Time-dependent transmission theory

By generalization of the methods presented in Part I of the study [J. Opt. Soc. Am. A 12, 600 (1994)] to the four-dimensional (4D) Riemannian manifold case, the time-dependent behavior of light transmitting in a medium is investigated theoretically by the geodesic equation and curvature in a 4D manifold. In addition, the field equation is restudied, and the 4D conserved current of the optical fluid and its conservation equation are derived and applied to deduce the time-dependent general refractive index. On this basis the forces acting on the fluid are dynamically analyzed and the self-consistency analysis is given.

Conical singularity and quantum corrections to the entropy of a black hole

For a general finite temperature different from the Hawking one there appears a well known conical singularity in the Euclidean classical solution of gravitational equations. The method of regularizing the cone by a regular surface is used to determine the curvature tensors for such a metric. This allows one to calculate the one-loop matter effective action and the corresponding one-loop quantum corrections to the entropy in the framework of the path integral approach of Gibbons and Hawking. The two-dimensional (2D) and four-dimensional cases are considered. The entropy of Rindler space is shown to be divergent logarithmically in two dimensions and quadratically in four dimensions that coincides with results obtained earlier. For the eternal 2D black hole we observe a finite, dependent on the mass, correction to the entropy. The entropy of the 4D Schwarzschild black hole is shown to possess an additional (in comparison with the 4D Rindler space) logarithmically divergent correction which does not vanish in the limit of infinite mass of the black hole. We argue that infinities of the entropy in four dimensions are renormalized by the renormalization of the gravitational coupling.

Mass inflation in 2d dilaton gravity

Recently two-dimensional dilaton gravity has received great attention. We investigate two-dimensional charged black holes. These solutions possess two horizons, an event horizon and an inner, so-called Cauchy horizon. It has been argued on general grounds that the inner horizon must be unstable. We show that a minimally coupled scalar field indeed causes a mass inflation singularity to be formed at the Cauchy horizon, in striking similarity to the four-dimensional case.

Local heterotic geometry in holomorphic coordinates

In the same spirit as for N=2 and N=4 supersymmetric nonlinear models in two spacetime dimensions by Zumino and by Alvarez-Gaumé and Freedman, we analyse the (2,0) and (4,0) heterotic geometry in holomorphic coordinates. We study the properties of the torsion tensor and give the conditions under which (2,0) geometry is conformally equivalent to a (2,2) one. Using additional isometries, we show that it is difficult to equip a manifold with a closed torsion tensor, but for the real four-dimensional case where we exhibit new examples. We show that, contrarily to Callan et al's claim for real four-dimensional manifolds, (4,0) heterotic geometry is not necessarily conformally equivalent to a (4,4) Kähler--Ricci flat geometry. We rather prove that, whatever the real dimension is, they are special quasi-Ricci flat spaces, and we exemplify our results on Eguchi--Hanson and Taub-NUT metrics with torsion.

Supersymmetric self-gravitating solitons

We show that the “instantonic” soliton of five-dimensional Yang-Mills theory and the closely related BPS monopole of four-dimensional Yang-Mills/Higgs theory continue to be exact static, and stable, solutions of these field theories even after the inclusion of gravitational, electromagnetic and, in the four-dimensional case, dilatonic interactions, provided that certain non-minimal interactions are included. With the inclusion of these interactions, which would be required by supersymmetry, these exact self-gravitating solitons saturate a gravitational version of the Bogomol'nyi bound on the energy of an arbitrary field configuration.

Geometrical string and spin systems

We formulate the geometrical string which has been proposed in earlier articles on the euclidean lattice. There are two essentially distinct cases which correspond to non-self-avoiding surfaces and to soft-self-avoiding ones. For the last case it is possible to find such spin systems with local interaction which reproduce the same surface dynamics. In the three-dimensional case this spin system is a usual Ising ferromagnet with additional diagonal antiferromagnetic interaction and with specially adjusted coupling constants. In the four-dimensional case the spin system coincides with the gauge Ising system with an additional double-plaquette interaction and also with specially tuned coupling constants. We extend this construction to random walks and random hypersurfaces (membrane and p-branes) of high dimensionality. We compare these spin systems with the eight-vertex model and BNNNI models.

Properties of asymptotically flat two-dimensional black holes

The authors investigate properties of two-dimensional asymptotically flat black holes which arise in both string theory and in scale invariant theories of gravity. By introducing matter sources in the field equations they show how such objects can arise as the endpoint of gravitational collapse. They also investigate the thermodynamic and quantum properties, which give rise to a fundamental length scale. The 't Hooft prescription for cutting off eigenmodes of particle wavefunctions is shown to be source dependent, unlike the four-dimensional case. The relationship between these black holes and those considered previously in (1+1) dimensions is discussed.

The role of anomalous singularity in hadronic form factors

By assuming chiral-symmetry breaking as the dominant dynamics in the light-hadron structure, we study the low-energy electromagnetic form factor of light hadrons in the Nambu-Jona-Lasinio type of models in the large- N limit from a dispersion relation point of view. In addition to the expected vector-meson pole dominance picture for tightly bound states such as the pion, it is found that the anomalous singularity, occuring only to loosely bound states of the constituent quarks, could severely alter the form factors at low momentum-transfers from the vector-meson dominance behavior. This mechanism is explicitly demonstrated in a two-dimensional version of the NJL type models, the Gross-Neveu model, in which all the interested quantities are calculated analytically and a detailed comparison with two-dimensional QCD is made. The two-dimensional analysis is then argued to be qualitatively similar to the realistic four-dimensional case. The relevance of the anomalous singularity to the phenomenology of the rho and the nucleons is indicated by quantitative estimates of the anomalous thresholds in their form factors.

SOME GENERAL PROBLEMS IN QUANTUM GRAVITY II: THE THREE-DIMENSIONAL CASE

The general problems of three-dimensional quantum gravity are recapitulated here, with emphasis on the mathematical problems of defining the measure of the path integral over all three-dimensional metrics. This work should be viewed as an extension of a preceding one on the four-dimensional case,12 where some general ideas were also discussed in detail. We finally put forward some suggestions on the lines where one could expect further progress in the field.

STRING COSMOLOGIES IN HIGHER DIMENSIONS

We extend to higher dimensions an earlier work of Letelier where a Bianchi type I cosmological model with massive strings as source term is studied. Exact solutions are obtained assuming either an equation of state between the energy density and tension density of the string or an explicit form of one of the metric coefficients. The dynamical behavior of the models is studied and a comparison is made with the analogous four-dimensional case where it is interestingly observed that the string dominated phase lasts a longer period in multidimensional models compared to its four-dimensional analogs. The Kantowski-Sachs type of metric is also briefly discussed.

Probabilistic representation formulas for the time evolution of quantum systemsa)

By using a new procedure representation formulas are obtained for the solution of the Cauchy problem for hyperbolic systems generalizing Schrödinger and Dirac equations. This procedure makes explicit the relation of the underlying Poisson jump process with the perturbation series. It allows one to find out a path integral representation formula for the two-dimensional Dirac equation. Further, it can be applied in the four-dimensional case with a Coulombian potential. This approach is also used to describe the time evolution of dissipative two-level systems and estimate transition probabilities.

Brick walls for black holes

The authors examine 't Hooft's prescription (1985) for cutting off eigenmodes of particle wavefunctions in the vicinity of black hole event horizons in N dimensions. They find for any N>3 that the cutoff occurs as a consequence of the causal structure of spacetime independently of the strength of the source, in agreement with the four-dimensional case. They separately discuss the special case N=2, showing why in this case the cutoff depends on the strength of the source.

Scattering in soliton models and boson-exchange descriptions.

We analyze the relationship between the conventional one-boson-exchange model (OBE) for nuclear forces with the description of the nucleon as a soliton within the context of the two-dimensional sine-Gordon model. We find that the soliton-soliton $S$ matrix contains poles and residues compatible with the ones used in the OBE. Implications of this result for the four-dimensional case are briefly discussed.

Detection of light from moving objects in six-dimensional special relativity

In a six-dimensional space-time involving three space and three time coordinates we investigate how a particle would appear to an observer when it is moving at a speedv and when its time direction lies at an angle θ to that of the observer. It is shown how, unlike the standard four-dimensional case, the world-line of the particle is able to slip off the light surface of the observer so that the particle seems to vanish. Whether or not it does so depends on the value of γ (v) cos θ. The Doppler shift of the radiation emitted by a nonvanishing particle is calculated and it is found that when θ= 0, then a value of the red-shift can be obtained with smaller values of the observed velocity than is the case in the standard theory.

Classical and quantum properties of two-dimensional black holes

We investigate the physics of black hole 1 + 1 dimensions. We demonstrate how an event horizon structure can arise in two dimensions given a variety of energy-momentum tensors, cataloguing the resultant particular solutions. We examine the motion of freely falling test particles in the vicinity of such horizons and demonstrate that they generically fall through in a finite amount of proper time and an infinite amount of coordinate time. The thermodynamic and quantum properties of these solutions are also examined, and give rise to a fundamental length scale in the theory whose origin is markedly different from that of the Planck mass in the higher-dimensional case. We show that 't Hooft's prescription for cutting off eigenmodes of particle wave functions in the vicinity of the horizon is source dependent and particle-mass dependent, unlike the four-dimensional case.

CHIRAL (SUPER) STRING ACTIONS

We present a covariant action describing chiral bosonic string theories in space-time dimensions d<26 and a covariant (1, 0) supersymmetric action describing heterotic string theories in space-time of dimension d<10. The anomaly cancellation conditions are found. In the four-dimensional case the supersymmetric action corresponds to the SO(44) heterotic string theory.

A pilot model in two dimensions for conformally invariant particle theory

The two-dimensional analog of the mathematical model for elementary particles treated in this journal vol 75, 1–57, is developed with greater explicitness than in the four-dimensional case, preliminary to quantization. In Part I the section space of the complex spannor bundle is studied. Invariant subspaces, invariant forms, and irreducible positive-energy factors under the connected conformal group are determined; transformation properties under discrete symmetries are also treated. Relations to the spinor bundle are treated; the Dirac operator is a canonical transform of the quadratic Casimir on the spannor bundle, and the section spaces have the same invariant factors; at the same time, the spinors constitute a deformation of the spannors. Two natural parallelizations of the spannor bundle and their intertwining operator are computed. The irreducible factors of the spannor section space under the conformal group are determined by their restrictions to Minkowski space, and this group acts continuously in the L 2-topology, in contrast to its action on the spinor section space.

Congruences of totally geodesic surfaces

A general theory of congruences of totally geodesic surfaces is presented. In particular their classification, based on the properties of induced affine connections, is provided. In the four-dimensional case canonical forms of the metric tensor admitting congruences of two-dimensional totally geodesic surfaces of rank one are given. Finally, congruences of two-dimensional extremal surfaces are studied.

Remark concerning the twistor formulation of a massless particle, and the division algebras

A possible generalization of the twistor description of a massless particle to spacetime dimensions other than four is investigated. While the description of a massless particle's momentum in terms of a spinor is possible in spacetime dimensions of 3, 4, 6, and 10, the description of the angular momentum of the particle cannot be generalized from the four-dimensional case to six or ten dimensions in any straightforward manner. The possible generalization sought utilizes the \({\text{SL(2,}}\mathbb{A}{\text{),}}\mathbb{A} = \mathbb{R},\mathbb{C},\mathbb{Q},\mathbb{O}\) formulation of the covering group of the Lorentz group in D=3, 4, 6, and 10 dimensions. This formulation is reviewed and further developed here.

Notes on the Hawking effect in de Sitter space. I.

We consider the Hawking effect for a quantized massless conformal scalar field in two- and four-dimensional de Sitter space. The relation among Bogoliubov coefficients is investigated without explicit integration of Klein-Gordon products. Our method presents a clear view of the property of Bogoliubov coefficients. In the two-dimensional case the thermal distribution is exactly derived. The application to the four-dimensional case is not straightforward, but we can derive the same result with some techniques.