Tensors Matter

In order to define the energy-momentrun tensor, matter is assruned to be uniformly distributed, as in an ideal gas, with stellar velocities orders of magnitude less than the velocity of light. However, a stable Boltzmann distribution of this type cannot persist in an open system from which entire stars may escape. Einstein therefore proposed a spatially closed continuum of constant curvature. The cosmological constant served to define both the mean density of the equilibrium distribution and a radius of the closed... 

The Ricci tensor that represents the geometry of space is next equated with the so-called energy-momentum (stress) tensor of the matter field that defines the influence of matter and field energy... 

The important result is the obvious symmetry between TM1/ and R u as shown in (42) and (43). Both of these tensors vanish in empty euclidean space and a reciprocal relationship between them is inferred The presence of matter causes space to curl up and curvature of space generates matter. 

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. 

If we understand FM or magnetic properties of quark matter more deeply, we must proceeds to a self-consistent approach, like Hartree-Fock theory, beyond the previous perturbative argument. In ref. [11] we have described how the axial-vector mean field (AV) and the tensor one appear as a consequence of the Fierz transformation within the relativistic mean-field theory for nuclear matter, which is one of the nonperturbative frameworks in many-body theories and corresponds to the Hatree-Fock approximation. We also demonstrated... 

This convention is in conflict with our tensor notation, that we do not want to abandon [8], However, what really matters is only the change of the definition of normal ordering, and this can easily be formulated in our language as well. 

In the preceding sections the optical response of matter has been described by a scalar dielectric function e, which relates the electric field E to the displacement D. More generally, D and E are connected by the tensor constitutive relation (5.46), which we write compactly as D = e0e E. The dielectric tensor is often symmetric, so that a coordinate system can be found in which it is diagonal ... 

Quantification of White Matter Using Diffusion-Tensor Imaging Hae-Jeong Park... 

Puschnig, P. and Ambrosch-Draxel, C. (1999) Density-functional study of the oligomers of poly-para-phenylene Band structures and dielectric tensors. Physical Review. E, Condensed Matter, 60, 7891-8. 

The evolution of the matter density is computed through the so-called conservation equations the stress-energy tensor of the matter species have com-... 

If we neglect the expansion term, we see that the first equation reduces to the usual Poisson equation. The last equation insures that the two Bardeen potentials are similar since in general the anisotropic stresses are small (they are negligible for non relativistic matter as well as for a scalar field). Note that in the absence of any form of matter all the scalar metric perturbations are 0. In addition to the scalar perturbations, there exists one equation for the tensor modes ... 

This corresponds to a damped oscillator, which was expected since the tensor modes correspond to gravitational waves. These can propagate even in the absence of matter. In addition to these equations, one also has to specify the equations governing the evolution of matter, that is the matter conservation and the Euler equations. In the absence of gravitational interactions, they read... 

The discussion is now restricted to matters relevant to the technology of piezoceramics. If it is assumed that the quantities d, g, s, and s are constants and, furthermore, the tensor nature of the various parameters (see below) is not taken into account, then equations of state can be written as follows. 

Moseley ME, Butts K, Yenari MA, de Crespigny A (1995) Clinical aspects of DWI. NMR Biomed 8 387-396 Mukherjee P, Bahn MM, McKinstry RC, Shimony JS, Cull TS, Akbudak E, Snyder AZ, Conturo TE (2000) Differences between gray matter and white matter water diffusion in stroke diffusion-tensor MR imaging in 12 patients. Radiology 215 211-220... 

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