Curvature tensor fields

In the present context of thin films on substrates, the quantity of interest is the local curvature. The curvature tensor field, Hap, can be determined directly from the CGS patterns recorded in reflection by differentiating the fringes of the in-plane gradient ... 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

To consider magnetic flux density components of IAIV, Q must have the units of weber and R, the scalar curvature, must have units of inverse square meters. In the flat spacetime limit, R 0, so it is clear that the non-Abelian part of the field tensor, Eq. (6), vanishes in special relativity. The complete field tensor F vanishes [1] in flat spacetime because the curvature tensor vanishes. These considerations refute the Maxwell-Heaviside theory, which is developed in flat spacetime, and show that 0(3) electrodynamics is a theory of conformally curved spacetime. Most generally, the Sachs theory is a closed field theory that, in principle, unifies all four fields gravitational, electromagnetic, weak, and strong. 

The quaternion-valued vector potential and the 4-current J both depend directly on the curvature tensor. The electromagnetic field tensor in the Sachs theory has the form... 

If 0(3) electromagnetism [denoted e.m. in Eq. (640)] and gravitation are both to be seen as phenomena of curved spacetime, then both fields are derived ultimately from the same Riemann curvature tensor as follows ... 

The 0(3) field equations can be obtained from the fundamental definition of the Riemann curvature tensor, Eq. (631), by defining the 0(3) field tensor using covariant derivatives of the Poincare group. 

There are well known similarities between the Riemann curvature tensor of general relativity and the field tensor in non-Abelian electrodynamics. The Riemann tensor is... 

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

Cosmological models obtained through the simplifying assumption of an affine-connected manifold must, by definition, be no more than a crude approximation. The one feature in common to all possible models is the balance between a curvature tensor and a stress tensor, as specified by Einstein s field... 

Let us assume that a liquid is incompressible, B oo, and discuss orientational (or torsimial) elasticity of a nematic. In a solid, the stress is caused by a change in the distance between neighbor points in a nematic the stress is caused by the curvature of the director field. Now a curvature tensor dnjdxj plays the role of the strain tensor ,y. Here, indices i,j = 1, 2, 3 and Xj correspond to the Cartesian frame axes. The linear relationship between the curvature and the torsional stress (i.e., Hooke s law) is assumed to be valid. The stress can be caused by boundary conditions, electric or magnetic field, shear, mechanical shot, etc. We are going to write the key expression for the distortion fi-ee energy density gji, related to the director field curvature . To discuss a more general case, we assume that gji t depends not only on quadratic combinations of derivatives dnjdxj, but also on their linear combinations ... 

A system based on the CGS method offers several advantages for curvature measurements for thin films and layered solids. The measurement provides all the normal and shear components of the curvature tensor. It also provides full field information from the entire area of the substrate—film system. The measurement area could also be scaled as necessary from a few millimeters to hundreds of millimeters so that large wafers and flat panels with thin film deposits are tested. The method involves non-contact measurements which are carried out with an adjustable working distance, and performed in-situ and in real time as, for example, during thermal cycling. 

The metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

The final objective is an equation that relates a geometrical object representing the curvature of space-time to a geometrical object representing the source of the gravitational field. The condition that all affine connections must vanish at a euclidean point, defines a tensor [41]... 

Finally, the scalar curvature field R follows from the further contraction of the Ricci tensor (43) with the metric tensor, giving... 

As the presence of gravity (mass) imparts a variable curvature on space the metric tensor is no longer constant. As the summation extends over all values of v and y, the smn consists of 4 X 4 terms, of which 12 are equal in pairs, hence 10 independent functions. The motion of a free material point in this field will follow a curvilinear non-uniform path. 

Relativity theory has equally dramatic implications on the nature of the vacuum, which is shown not to be a void, but a medium that supports wave motion and carries electromagnetic fields. A new perspective on the nature of the vacuum is provided by the principle of equivalence. Space-time curvature can be described mathematically by a Riemann tensor, which the principle implies, should balance the gravitational field, which is sourced in the distribution of matter. This reciprocity indicates that Euclidean space-time is free of matter, which only emerges when curvature sets in. This is interpreted to mean that the homogeneous wave field of Euclidean vacuum generates matter when curved. Like a flat sheet that develops wrinkles when wrapped arormd a curved surface, the wave field generates non-dispersive persistent wave packets in the curved vacuum. 

These fields differ quite substantially in their theoretical description concentrations are scalar variables, orientations are vectors, and differential geometry is at heart a tensor theory but, aU of them are known to mediate interactions. For instance, the fact that proteins might prefer one lipid composition over another and thus aggregate [217-220] is central to an important mechanism attributed to lipid rafts. Tilt-mediated protein interactions have also been studied in multiple contexts [32, 33, 159, 221-223]. It is even possible to describe all these phenomena within a common language [224], using the framework of covariant surface stresses [154, 155, 157-161]. However, in the present review we will restrict the discussion to only two examples, both related to membrane elasticity in Sect. 3.1 we will discuss interactions due to hydrophobic mismatch, and in Sect. 3.2 we will look at interactions mediated by the large-scale curvature deformation of the membrane. 

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