Einstein-Maxwell Spacetime with Two Commuting Spacelike Killing Vector Fields and Newman-Penrose Formalism

Einstein-Maxwell spacetimes endowed with twocommuting spacelike Killing vector fields areconsidered. Subject to the hypotheses that one of thetwo null geodesic congruence orthogonal to thetwo-surface generated by the two commuting spacelikeKilling vector fields is shearfree and theelectromagnetic field is non null, it is shown that,with a specific choice of null tetrad, theNewman-Penrose equations together with the Maxwell equations for theclass of spacetime considered may be reduced to asecond-order ode of Sturm-Liouville type, from whichexact solutions of the class of spacetimes consideredmay be constructed. Examples of exact solutions arethen given. Exact solutions with distribution-valuedWeyl curvature describing the scattering ofelectromagnetic shock wave with gravitational impulsiveor shock wave of variable polarisation are also constructed.     

Mach's Principle and Minkowski Spacetime

It is shown that when the Minkowski metric is approached by a limiting process using two different static, spherically-symmetric, closed cosmological models, that although the energy-stress tensors for the Einstein-Friedmann field equations vanishes, their integral does not. Since part of this integral consists of the mass of the incoherent dust background, which is the same in both models, the Minkowski metric obtained by this limiting process cannot be regarded as anti-Machian, since there is an infinite amount of ponderable matter in the background, albeit at vanishing density. One of the models is the Einstein static universe with its cosmological term. The other model does not employ this term, but instead uses a tensor that has vanishing trace, negative energy density and negative pressure. Gravitational energy is also studied, and it is pointed out that for both models, this energy becomes infinitely negative in the Minkowski limit.     

Radiation Fields on Schwarzschild Spacetime

In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field.     

Nambu Mechanics in the Lagrangian Formalism

A Lagrangian formulation is presented as the counterpart of the Hamiltonian onefor Nambu mechanics which is a natural generalization of Hamiltonian mechanics.If we postulate the existence of plural Lagrangians corresponding to the existenceof plural Hamiltonians, we can formulate the Lagrangian formalism in Nambumechanics as well as in Hamiltonian mechanics. Here, in terms of exteriordifferentiation, Nambu mechanics can be formulated in a completely parallel wayto ordinary analytical mechanics, including generalized Legendre transformations.     

Null Cones and Einstein's Equations in Minkowski Spacetime

If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segré classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized eigenvector formalism reduce the configuration space to the causality-respecting part. In this smaller space, gauge transformations do not form a group, but only a groupoid. The flat metric removes the difficulty of defining equal-time commutation relations in quantum gravity and guarantees global hyperbolicity.     

Covariant Canonical Formalism of Fields

The canonical formalism of fields consistentwith the covariance principle of special relativity isgiven here. The covariant canonical transformations offields are affected by 4-generating functions. All dynamical equations of fields, e.g., theHamilton, Euler–Lagrange, and other fieldequations, are preserved under the covariant canonicaltransformations. The dynamical observables are alsoinvariant under these transformations. The covariantcanonical transformations are therefore fundamentalsymmetry operations on fields, such that the physicaloutcomes of each field theory must be invariant under these transformations. We give here also thecovariant canonical equations of fields. These equationsare the covariant versions of the Hamilton equations.They are defined by a density functional that is scalar under both the Lorentz and thecovariant canonical transformations of fields.     

On Lagrangian formulations for arbitrary bosonic HS fields on Minkowski backgrounds

We review the details of unconstrained Lagrangian formulations for Bose particles propagated on an arbitrary dimensional flat space-time and described by the unitary irreducible integer higher-spin representations of the Poincare group subject to Young tableaux Y(s ₁, ..., s _k ) with k rows. The procedure is based on the construction of scalar auxiliary oscillator realizations for the symplectic sp(2k) algebra which encodes the second-class operator constraints subsystem in the HS symmetry algebra. Application of an universal BRST approach reproduces gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive bosonic fields of any spin with appropriate number of auxiliary fields.     

Simultaneity in Minkowski Spacetime: From Uniqueness to Arbitrariness

Malament (No?s 11:293–300, 1977) proved a certain uniqueness theorem about standard synchrony, also known as Poincaré-Einstein simultaneity, which has generated many commentaries over the years, some of them contradictory. We think that the situation called for some clarification. After reviewing and discussing some of the literature involved, we prove two results which, hopefully, will help clarifying this debate by filling the gap between the uniquess of Malament’s theorem, which allows the observer to use very few tools, and the complete arbitrariness of a time coordinate in full-fledged Relativity theory. In the spirit of Malament’s theorem, and in opposition to most of its commentators, we emphasize explicit definability of simultaneity relations, and give only constructive proofs. We also explore what happens when we reduce to “purely local” data with respect to an observer.     

Equivariant Self-Similar Wave Maps from Minkowski Spacetime into 3-Sphere

We prove existence of a countable family of spherically symmetric self-similar wave maps from 3+1 Minkowski spacetime into the 3-sphere. These maps can be viewed as excitations of the ground state solution found previously by Shatah. The first excitation is particularly interesting in the context of the Cauchy problem since it plays the role of a critical solution sitting at the threshold for singularity formation. We analyze the linear stability of our wave maps and show that the number of unstable modes about a given map is equal to its nodal number. Finally, we formulate a condition under which these results can be generalized to higher dimensions. Received: 20 October 1999 / Accepted: 12 May 2000     

Casimir Effect for a Massless Spin-3/2 Field in Minkowski Spacetime

The Casimir effect has been studied for various quantum fields in both flat and curved spacetimes. As a further step along this line, we provide an explicit derivation of Casimir effect for massless spin-3/2 field with periodic boundary condition imposed in four-dimensional Minkowski spacetime. The corresponding results with Dirichlet ard Neumann boundary conditions are also discussed.     

Letter: The Teleparallel Lagrangian and Hamilton-Jacobi Formalism

We analyze the Teleparallel Equivalent of General Relativity (TEGR) from the point of view of Hamilton-Jacobi approach for singular systems.     

Nonlinear Spinor Fields in Bianchi type-III Spacetime

Within the scope of Bianchi type-III spacetime we study the role of spinor field on the evolution of the Universe as well as the influence of gravity on the spinor field. In doing so we have considered a polynomial type of nonlinearity. In this case the spacetime remains locally rotationally symmetric and anisotropic all the time. It is found that depending on the sign of nonlinearity the models allows both accelerated and oscillatory modes of expansion. The non-diagonal components of energy-momentum tensor though impose some restrictions on metric functions and components of spinor field, unlike Bianchi type I, V and V I ₀ cases, they do not lead to vanishing mass and nonlinear terms of the spinor field.     

Quantum Fields in the Spacetime Tangent Bundle

Maximal-acceleration invariant quantum fields are formulated in terms of the differential geometric structure of the spacetime tangent bundle. The simple special case is considered of a flat Minkowski space-time for which the bundle is also flat. The field is shown to have a physically based Planck-scale effective regularization and a spectral cutoff at the Planck mass.     

Entropy of Spin Fields in Schwarzschild Spacetime

By using the Teukolsky master equation, we consider the gravitational,electromagnetic, and neutrino fields in Schwarzschild spacetime. The free energyand entropy of the spin fields are obtained in terms of the brick-wall model. Itis shown that the entropy of all the spin fields due to the presence of the eventhorizon is proportional to the surface area of the event horizon, and the entropyof the neutrino field is the absolute minimum.     

Dirac Fields in 3D de Sitter Spacetime

We show that the Dirac equation is separable in the circularly symmetric metric in three dimensions and when the background spacetime is de Sitter we find exact solutions to the radial equations. Using these results we show that the de Sitter horizon has a cross section equal to zero for the massless Dirac field, as in the case of the scalar field. Also, using the improved brick wall model we calculate the fermionic entropy associated with the de Sitter horizon and we compare it with some results previously published.     

Nonlinear Spinor Fields in Bianchi Type-I Spacetime Reexamined

The specific behavior of spinor field in curve space-time with the exception of FRW model almost always gives rise to non-trivial non-diagonal components of the energy-momentum tensor. This non-triviality of non-diagonal components of the energy-momentum tensor imposes some severe restrictions either on the spinor field or on the metric functions. In this paper within the scope of an anisotropic Bianchi type-I Universe we study the role of spinor field in the evolution of the Universe. It is found that there exist two possibilities. In one scenario the initially anisotropic Universe evolves into an isotropic one asymptotically, but in this case the spinor field itself undergoes some severe restrictions. In the second scenario the isotropization takes places almost at the beginning of the process.     

Propagation of Scalar Fields in a Plane Symmetric Spacetime

The present article deals with solutions for a minimally coupled scalar field propagating in a static plane symmetric spacetime. The considered metric describes the curvature outside a massive infinity plate and exhibits an intrinsic naked singularity (a singular plane) that makes the accessible universe finite in extension. This solution can be interpreted as describing the spacetime of static domain walls. In this context, a first solution is given in terms of zero order Bessel functions of the first and second kind and presents a stationary pattern which is interpreted as a result of the reflection of the scalar waves at the singular plane. This is an evidence, at least for the massless scalar field, of an old interpretation given by Amundsen and Grøn regarding the behaviour of test particles near the singularity. A second solution is obtained in the limit of a weak gravitational field which is valid only far from the singularity. In this limit, it was possible to find out an analytic solution for the scalar field in terms of the Kummer and Tricomi confluent hypergeometric functions.     

A Quantum Weak Energy Inequality¶for Dirac Fields in Curved Spacetime

Quantum fields are well known to violate the weak energy condition of general relativity: the renormalised energy density at any given point is unbounded from below as a function of the quantum state. By contrast, for the scalar and electromagnetic fields it has been shown that weighted averages of the energy density along timelike curves satisfy “quantum weak energy inequalities” (QWEIs) which constitute lower bounds on these quantities. Previously, Dirac QWEIs have been obtained only for massless fields in two-dimensional spacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields of mass m≥ 0 on general four-dimensional globally hyperbolic spacetimes, averaging along arbitrary smooth timelike curves with respect to any of a large class of smooth compactly supported positive weights. Our proof makes essential use of the microlocal characterisation of the class of Hadamard states, for which the energy density may be defined by point-splitting. Received: 21 May 2001 / Accepted: 23 August 2001