Narrow Results

We propose additional conditions (beyond those considered in our previous papers) that should be imposed on Wick products and time-ordered products of a free quantum scalar field in curved spacetime. These conditions arise from a simple "Principle of Perturbative Agreement": for interaction Lagrangians L₁ that are such that the interacting field theory can be constructed exactly — as occurs when L₁ is a "pure divergence" or when L₁ is at most quadratic in the field and contains no more than two derivatives — then time-ordered products must be defined so that the perturbative solution for interacting fields obtained from the Bogoliubov formula agrees with the exact solution. The conditions derived from this principle include a version of the Leibniz rule (or "action Ward identity") and a condition on time-ordered products that contain a factor of the free field φ or the free stress-energy tensor T_ab. The main results of our paper are (1) a proof that in spacetime dimensions greater than 2, our new conditions can be consistently imposed in addition to our previously considered conditions and (2) a proof that, if they are imposed, then for any polynomial interaction Lagrangian L₁ (with no restriction on the number of derivatives appearing in L₁), the stress-energy tensor Θ_ab of the interacting theory will be conserved. Our work thereby establishes (in the context of perturbation theory) the conservation of stress-energy for an arbitrary interacting scalar field in curved spacetimes of dimension greater than 2. Our approach requires us to view time-ordered products as maps taking classical field expressions into the quantum field algebra rather than as maps taking Wick polynomials of the quantum field into the quantum field algebra.

Lattice-Boltzmann (LB) models provide a systematic formulation of effective-field computational approaches to the calculation of multiphase flow by replacing the mathematical surface of separation between the vapor and liquid with a thin transition region, across which all magnitudes change continuously. Many existing multiphase models of this sort do not satisfy the rigorous hydrodynamic constitutive laws. Here, we extend the two-dimensional, seven-speed Swift et al. LB model¹ to rectangular grids (nine speeds) by using symbolic manipulation (Mathematica^TM) and compare the LB model predictions with benchmark problems, in order to evaluate its merits. Particular emphasis is placed on the stress tensor formulation. Comparison with the two-phase analogue of the Couette flow and with a flow involving shear and advection of a droplet surrounded by its vapor reveals that additional terms have to be introduced in the definition of the stress tensor in order to satisfy the Navier–Stokes equation in regions of high density gradients. The use of Mathematica obviates many of the difficulties with the calculations "by-hand," allowing at the same time more flexibility to the computational analyst to experiment with geometrical and physical parameters of the formulation.

In vacuum energy calculations with an ultraviolet cutoff, divergences arise that clearly are related to the physics of boundaries. We point out how to find the cut-off energy density and pressure most directly from a Green function. Then we discuss three apparent paradoxes arising in this work that are in various stages of resolution.

A finite ultraviolet cutoff near a reflecting boundary yields a stress tensor that violates the basic energy-pressure relation. Therefore, a "soft" wall described by a power-law potential, which needs no ad hoc cutoff, is being investigated by the collaboration centered at Texas A&M University and the University of Oklahoma. Progress is reported here.

In this paper, we bring out the subtleties involved in the study of a first-order relativistic field theory with auxiliary field variables playing an essential role. In particular, we discuss the nonisentropic Eulerian (or Hamiltonian) fluid model. Interactions are introduced by coupling the fluid to a dynamical Maxwell (U(1)) gauge field. This dynamical nature of the gauge field is crucial in showing the equivalence, on the physical subspace, of the stress tensor derived from two definitions, i.e. the canonical (Noether) one and the symmetric one. In the conventional equal-time formalism, we have shown that the generators of the space–time transformations obtained from these two definitions agree modulo the Gauss constraint. This equivalence in the physical sector has been achieved only because of the dynamical nature of the gauge fields. Subsequently, we have explicitly demonstrated the validity of the Schwinger condition. A detailed analysis of the model in lightcone formalism has also been done where several interesting features are revealed.

In the study of quantum vacuum energy and the Casimir effect, it is desirable to model the conductor by a potential of the form $V(z)=z^{\alpha }$. This “soft wall” model was proposed so as to avoid the violation of the principle of virtual work under ultraviolet regularization that occurs for the standard Dirichlet wall. The model was formalized for a massless scalar field, and the expectation value of the stress tensor has been expressed in terms of the reduced Green function of the equation of motion. In the limit of interest, $\alpha \gg 1$, which approximates a Dirichlet wall, a closed-form expression for the reduced Green function cannot be found, so piecewise approximations incorporating the perturbative and WKB expansions of the Green function, along with interpolating splines in the region where neither expansion is valid, have been developed. After reviewing this program, in this paper, we apply the scheme to the wall with $\alpha =6$ and use it to compute the renormalized energy density and pressure inside the cavity for various values of the conformal parameter. The consistency of the results is verified by comparison to their numerical counterparts and verification of the trace anomaly and the conservation law. Finally, we use the approximation scheme to reproduce the energy density inside the quadratic wall, which was previously calculated exactly but with some uncertainty.

The 4-dimensionally covariant approach to multiconstituent Newtonian fluid dynamics presented in the preceding paper of this series is developed by construction of the relevant 4-dimensional stress–energy tensor whose conservation in the non-dissipative variational case is shown to be interpretable as a Noether identity of the Milne spacetime structure. The formalism is illustrated by the application to homogeneously expanding cosmological models, for which appropriately generalized local Bernoulli constants are constructed. Another application is to the Iordanski type generalization of the Joukowski formula for the Magnus force on a vortex. Finally, at a global level, a new (formally simpler but more generally applicable) version of the "virial theorem" is obtained for multiconstituent — neutron or other — fluid star models as a special case within an extensive category of formulae whereby the time evolution of variously weighted mass moment integrals is determined by corresponding space integrals of stress tensor components, with the implication that all such stress integrals must vanish for any stationary equilibrium configuration.

Although there is a vast engineering literature for granular materials, as a subject for physicists it has seen great growth in recent years. This is because, when stripped down to the fundamental problem, it is quite novel, and demands a rethink of the kind of laws familiar elsewhere in physics. We consider the transmission of stress in granular materials and investigate the simplest statically determinate problem.

In this paper, we discuss about a set of geometric and physical properties of hyper-generalized quasi-Einstein spacetime. In the beginning, we discuss about pseudosymmetry over a hyper-generalized quasi-Einstein spacetime. Here, we discuss about $W_2$-Ricci pseudosymmetry, $Z$-Ricci pseudosymmetry, Ricci pseudosymmetry and projective Ricci pseudosymmetry over a hyper-generalized quasi-Einstein spacetime. Later on, we take over Ricci symmetric hyper-generalized quasi-Einstein spacetime and derive a set of important geometric and physical theorems over it. Moving further we consider some physical applications of the hyper-generalized quasi-Einstein spacetime. Last, we prove the existence of a hyper-generalized quasi-Einstein spacetime by constructing a non-trivial example.

Averaging approaches have been widely used to estimate continuum stress in granular materials to facilitate the assimilation of granular materials’ behavior into a continuum mechanics framework. At present, selection criteria for the appropriate averaging approach are still poorly understood. This paper compares the stresses computed by two popular averaging approaches, i.e., volume average approach (VAA) and boundary average approach (BAA) in order to investigate relative errors between the two approaches under various conditions. The comparisons between VAA and BAA under a static condition were first carried out based on the rigid particle analytical framework proposed by Rothenburg, L. [1980] “Micromechanics of idealized granular systems,” Ph.D. thesis, Carleton University, Ottawa, followed by discrete element modelling (DEM) simulations using PFC3D under a quasi-static condition. The results of theoretical comparison suggested that the relative error in the computed stresses by VAA with an omission of overlap could be grossly estimated by the ratio of overlap to length of contact branch, while the relative error in the computed stresses by VAA with an omission of external contacts could be grossly estimated by the ratio of maximum particle diameter to specimen size. At the ratio of maximum particle diameter to specimen size of 5, which is usually taken as the minimum recommended ratio to minimize boundary effect, the relative error caused by omitting the external contacts was as high as 20%. Numerical comparisons suggested that the relative errors of the two approaches under a quasi-static condition were significantly influenced by the strain energy ratio defined in this study. The VAA was recommended as a preferred computation method over the BAA when the strain energy ratio was considerably lower than unity.

A finite ultraviolet cutoff near a reflecting boundary yields a stress tensor that violates the basic energy-pressure relation. Therefore, a "soft" wall described by a power-law potential, which needs no ad hoc cutoff, is being investigated by the collaboration centered at Texas A&M University and the University of Oklahoma. Progress is reported here.

In his 1951 model of the effects of a star’s radiation on its gravitation, P.C. Vaidya used radiation fields of flat spacetime. Wondering why Vaidya did not include the effects of gravitation on electrodynamics, we apply the principle of general covariance to gravitation and radiation. Others have addressed this problem with numerical methods, but, like Vaidya, we seek an analytic solution. Along the way, we see how prescient was Vaidya’s choice in modeling the radiation. Pursued originally as a pedagogical exercise with undergraduate physicists, our results compare favorably to similar models, and illustrate general relativity theorems. Given recent interest in teaching general relativity to undergraduates, a system that couples gravitation and electrodynamics and is solvable with elementary integrations may hold pedagogical interest.

In the study of quantum vacuum energy and the Casimir effect, it is desirable to model the conductor by a potential of the form V (z) = z^α. This “soft wall” model was proposed so as to avoid the violation of the principle of virtual work under ultraviolet regularization that occurs for the standard Dirichlet wall. The model was formalized for a massless scalar field, and the expectation value of the stress tensor has been expressed in terms of the reduced Green function of the equation of motion. In the limit of interest, α » 1, which approximates a Dirichlet wall, a closed-form expression for the reduced Green function cannot be found, so piecewise approximations incorporating the perturbative and WKB expansions of the Green function, along with interpolating splines in the region where neither expansion is valid, have been developed. After reviewing this program, in this paper, we apply the scheme to the wall with α = 6 and use it to compute the renormalized energy density and pressure inside the cavity for various values of the conformal parameter. The consistency of the results is verified by comparison to their numerical counterparts and verification of the trace anomaly and the conservation law. Finally, we use the approximation scheme to reproduce the energy density inside the quadratic wall, which was previously calculated exactly but with some uncertainty.

Although there is a vast engineering literature for granular materials, as a subject for physicists it has seen great growth in recent years. This is because, when stripped down to the fundamental problem, it is quite novel, and demands a rethink of the kind of laws familiar elsewhere in physics. We consider the transmission of stress in granular materials and investigate the simplest statically determinate problem.

We consider a massive scalar field living on the recently found exact quantum space-time corresponding to vacuum spherically symmetric loop quantum gravity. The discreteness of the quantum space time naturally regularizes the scalar field, eliminating divergences. However, the resulting finite theory depends on the details of the micro physics. We argue that such dependence can be eliminated through a finite renormalization and discuss its nature. This is an example of how quantum field theories on quantum space times deal with the issues of divergences in quantum field theories.

In vacuum energy calculations with an ultraviolet cutoff, divergences arise that clearly are related to the physics of boundaries. We point out how to find the cut-off energy density and pressure most directly from a Green function. Then we discuss three apparent paradoxes arising in this work that are in various stages of resolution.