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With respect to irreversible, non-homeomorphic maps, contravariant and covariant tensor fields have distinctly natural covariance and transformational behavior. For thermodynamic processes which are non-adiabatic, the fact that the process cannot be represented by a homeomorphic map emphasizes the logical arrow of time, an idea which encompasses a principle of retrodictive determinism for covariant tensor fields.
The formulation of mathematical models of aeronautical systems for simulation or other purposes, involves the transformation of aerodynamic stability derivatives. It is shown that these derivatives transform like the components of a second order tensor having one index of covariance and one index of contravariance. Moreover, due to the equivalence of covariant and contravariant transformations in orthogonal Cartesian systems of coordinates, the transformations can be treated as doubly covariant or doubly contravariant, if this simplifies the formulation. It is shown that the tensor properties of these derivatives can be used to facilitate their transformation by symbolic mathematical computation, and the use of digital computers equipped with formula manipulation compilers. When the tensor transformations are mechanised in the manner described, man-hours are saved and the errors to which human operators are prone can be avoided.
The Brans-Dicke theory of gravitation is modified by assuming that the divergence of the energy-momentum tensor is proportional to the covariant derivative of the scalar curvature. No ad hoc additions to the usual Brans-Dicke field equations are required as in Rastall's modification of the Einstein theory or as in the steady-state theories, of which this is a natural possibility. Three parameters emerge from the theory - namely, the unnormalized gravitational constant, the usual Brans-Dicke parameter, and the proportionality constant. In the post-Newtonian approximation, these parameters can be fixed by experiment. However, there exists a certain choice of the parameters for which the theory reduces to an Einstein theory with constant scalar curvature.
A new theorem on a class of four-dimensional skew-symmetric tensors is demonstrated. Coupled with the relativistic covariant form of Maxwell's equations, this theorem consolidates the classifications of guided waves by combining the three types - TE, TM, TEM - under a uniform condition applied to the generating four-potential which is Lorentz invariant. Each type corresponds to a potential of which a pair of the four components vanishes in a particular frame. Through appropriate normalization conditions, the resulting time-domain equations for the field amplitudes are readily reduced to modified telegraphist equations, which in turn lead to distributed network representations for each of the three types. The ambiguity of distributed network formalisms in general is elucidated and the concept of network parameter densities such as traditionally employed in TEM transmission line theory is questioned.
This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.
Demonstration that the conventional theory of the mass-zero spin-two field with sources introduces extraneous nonspin-two field components in source regions and fails to be covariant under the full or restricted conformal group. A modified theory is given, expressed in terms of the physical components of mass-zero spin-two field rather than in terms of 'potentials,' which has no extraneous components inside or outside sources, and which is covariant under the full conformal group. For a proper choice of source term, this modified theory has the correct Newtonian limit and automatically implies that a symmetric second-rank source tensor has zero divergence. It is shown that possibly a generally covariant form of the spin-two theory derived here can be constructed to agree with general relativity in all currently accessible experimental situations.
Covariant Noether identities in covariant field theories
The layered random medium model is used to investigate the fully polarimetric scattering of electromagnetic waves from vegetation. The vegetation canopy is modeled as an anisotropic random medium containing nonspherical scatterers with preferred alignment. The underlying medium is considered as a homogeneous half space. The scattering effect of the vegetation canopy are characterized by 3-D correlation functions with variances and correlation lengths respectively corresponding to the fluctuation strengths and the physical geometries of the scatterers. The strong fluctuation theory is used to calculate the anisotropic effective permittivity tensor of the random medium and the distorted Born approximation is then applied to obtain the covariance matrix which describes the fully polarimetric scattering properties of the vegetation field. This model accounts for all the interaction processes between the boundaries and the scatterers and includes all the coherent effects due to wave propagation in different directions such as the constructive and destructive interferences. For a vegetation canopy with low attenuation, the boundary between the vegetation and the underlying medium can give rise to significant coherent effects.
The kinematic problem of calculating second-order velocity moments from given values of the vorticity covariance is examined. Integral representation formulas for second-order velocity moments in terms of the two-point vorticity correlation tensor are derived. The special relationships existing between velocity moments in isotropic turbulence are expressed in terms of the integral formulas yielding several kinematic constraints on the two-point vorticity correlation tensor in isotropic turbulence. Numerical evaluation of these constraints suggests that a Gaussian curve may be the only form of the longitudinal velocity correlation coefficient which is consistent with the requirement of isotropy. It is shown that if this is the case, then a family of exact solutions to the decay of isotropic turbulence may be obtained which contains Batchelor's final period solution as a special case. In addition, the computed results suggest a method of approximating the integral representation formulas in general turbulent shear flows.
A set of higher-order boundary-layer equations is derived valid for three-dimensional compressible flows. The equations are written in a generalized curvilinear coordinate system, in which the surface coordinates are nonorthogonal; the third axis is restricted to be normal to the surface. Also, higher-order viscous terms which are retained depend on the surface curvature of the body. Thus, the equations are suitable for the calculation of the boundary layer about arbitrary vehicles. As a starting point, the Navier-Stokes equations are derived in a tensorian notation. Then by means of an order-of-magnitude analysis, the boundary-layer equations are developed. To provide an interface between the analytical partial differentiation notation and the compact tensor notation, a brief review of the most essential theorems of the tensor analysis related to the equations of the fluid dynamics is given. Many useful quantities, such as the contravariant and the covariant metrics and the physical velocity components, are written in both notations.
To illustrate that a relativistic field theory need not be manifestly covariant, Lorentz-invariant Lagrangian densities are constructed that yield the equation satisfied by an interacting (two-component) Sakata-Taketani spin-0 field. Six types of external field couplings are considered, two scalars, two vectors, an antisymmetric second-rank tensor, and a symmetric second-rank tensor, with the results specialized to electromagnetic interactions. For either of the two second-rank couplings, the equation is found to describe noncausal wave propagation, a property that is apparent from the dependence of the coefficients of the space derivatives on the external field; in contrast, the noncausality of the corresponding manifestly covariant Duffin-Kemmer-Petiau spin-0 equation is not so obvious. The possibilities for generalizing the results to higher spin theories involving only the essential 2(2J + 1) components for a particle with a definite spin J and mass m are discussed in considerable detail.
The two-component Sakata-Taketani (ST) spin-0 theory and the single-component Klein-Gordon theory are obtained from the five-component Duffin-Kemmer-Petiau (DKP) theory with six types of external field interactions by means of a Peirce decomposition. Whereas the DKP equation manifests the covariance, the ST equation manifests the causal properties. In particular, the presence of noncausal wave propagation when there is coupling to a second-rank tensor field is apparent from the form of the ST equation, in which the coefficients of all the space derivatives depend on the external field. The results indicate that the causal properties of higher-spin equations should also be obvious when they are expressed in 2(2J + 1)-component Schroedinger form
The metric tensor is given for describing the dynamical effects of the sun on a solar probe and the gravitational redshift of a probe-based clock in a covariance analysis for the detectability of the solar quadrupole moment and the solar angular momentum. Unknown parameters were determined by least squares fit to the probe tracking data. A polar orbit was assumed with perihelion at 5 solar radii and with an earth-sun-probe angle of 135 deg at perihelion. Tracking was assumed to be Doppler only, with a basic uncertainity of .1 mm/sec in the radial velocity. A drag-free system which could reduce nongravitational forces below the level of 2 times 10 to the minus 12th power was found necessary to preserve this same level of accuracy. Both one way and two way Doppler were supposed in order to determine the gravitational redshift as well as the pure spacecraft dynamics.
A prototype of Rastall's theory of gravity, in which the divergence of the energy-momentum tensor is proportional to the gradient of the scalar curvature, is shown to be derivable from a variational principle. Both the proportionality factor and the unrenormalized gravitational constant are found to be covariantly constant, but not necessarily constant. The prototype theory is, therefore, a gravitational theory with variable gravitational constant.
The objective of the present research is to develop a general mathematical model and solution methodologies for analyzing the structural response of thin, metallic shell structures under large transient, cyclic, or static thermomechanical loads. Among the system responses associated with these loads and conditions are thermal buckling, creep buckling, and ratcheting. Thus geometric and material nonlinearities (of high order) can be anticipated and must be considered in developing the mathematical model. A complete, true ab-initio rate theory of kinematics and kinetics for continuum and curved thin structures, without any restriction on the magnitude of the strains or the deformations, was formulated. The time dependence and large strain behavior are incorporated through the introduction of the time rates of metric and curvature in two coordinate systems: fixed (spatial) and convected (material). The relations between the time derivative and the covariant derivative (gradient) were developed for curved space and motion, so the velocity components supply the connection between the equations of motion and the time rates of change of the metric and curvature tensors.
A mathematical model and solution methodologies for analyzing structural response of thin, metallic shell-type structures under large transient, cyclic or static thermomechanical loads is disussed. Among the system responses, which are associated with these load conditions, are thermal buckling and creep buckling. Thus, geometric as well as material-type nonlinearities (of high order) can be anticipated and have been considered in the development of the mathematical model. Furthermore, this was accommodated in the solution procedures. A complete true ab-inito rate theory of kinematics and kinetics for continuum and curved thin structures, without any restriction on the magnitude of the strains or the deformation, was formulated. The time dependence and large strain behavior are incorporated through the introduction of the time rates of the metric and curvature in two coordinate systems, a fixed (spatial) one and a convected (material) coordinate system. The relations between the time derivative and the covariant derivatives (gradients) have been developed for curved space and motion, so that the velocity components supply the connection between the equations of motion and the time rate of change of the metric and curvature tensors.
Consideration of the construction of suitable Lagrangian functions for the dynamics of a cold plasma in such a way as to retain the relativistically covariant formalism. In one method, this is achieved by the introduction of a set of three variables which label the world lines of the particles. A second method results in a Clebsch-type representation. Sturrock's relativistic Lagrangian and Low's hot plasma Lagrangian are also briefly discussed in the context of the present work. The behavior of the canonical stress tensor is considered. The applicability of many of the general results in part I (Dougherty, 1970) is ensured by establishing the existence of the Lagrangian function.
An autonomous spaceborne gravity gradiometer mission is being considered as a post Geopotential Research Mission project. The introduction of satellite diometry data to geodesy is expected to improve solid earth gravity models. The possibility of utilizing gradiometer data for the determination of pertinent gravimetric quantities on a local basis is explored. The analytical technique of least squares collocation is investigated for its usefulness in local solutions of this type. It is assumed, in the error analysis, that the vertical gravity gradient component of the gradient tensor is used as the raw data signal from which the corresponding reference gradients are removed to create the centered observations required in the collocation solution. The reference gradients are computed from a high degree and order geopotential model. The solution can be made in terms of mean or point gravity anomalies, height anomalies, or other useful gravimetric quantities depending on the choice of covariance types. Selected for this study were 30 x 30 foot mean gravity and height anomalies. Existing software and new software are utilized to implement the collocation technique. It was determined that satellite gradiometry data at an altitude of 200 km can be used successfully for the determination of 30 x 30 foot mean gravity anomalies to an accuracy of 9.2 mgal from this algorithm. It is shown that the resulting accuracy estimates are sensitive to gravity model coefficient uncertainties, data reduction assumptions and satellite mission parameters.