Quasi-Metric Relativity

This is a survey of a new type of relativistic space-time framework; the so-called quasi-metric framework. The basic geometric structure underlying quasi-metric relativity is quasi-metric space-time; this is defined as a 4-dimensional differentiable manifold ${\cal N}$ equipped with two one-parameter families ${\bf {\bar g}}_t$ and ${\bf g}_t$ of Lorentzian 4-metrics parametrized by a global time function $t$. The metric family ${\bf {\bar g}}_t$ is found from field equations, whereas the metric family ${\bf g}_t$ is used to propagate sources and to compare predictions to experiments. A linear and symmetric affine connection compatible with the family ${\bf g}_t$ is defined, giving rise to equations of motion. Furthermore a quasi-metric theory of gravity, including field equations and local conservation laws, is presented. Just as for General Relativity, the field equations accommodate two independent propagating dynamical degrees of freedom. On the other hand, the particular structure of quasi-metric geometry allows only a partial coupling of space-time geometry to the active stress-energy tensor. Besides, the field equations are defined from projections of physical and geometrical tensors with respect to a ``preferred'' foliation of quasi-metric space-time into spatial hypersurfaces. The dynamical nature of this foliation makes the field equations unsuitable for a standard PPN-analysis. This implies that the experimental status of the theory is not completely clear at this point in time. The theory seems to be consistent with a number of cosmological observations and it satisfies all the classical solar system tests, though. Moreover, in its non-metric sector the new theory has experimental support where General Relativity fails or is irrelevant.Comment: 39 pages, no figures, LaTeX; v2: some points clarified; v3: connection changed; v4: extended and local conservation laws changed; v5: major revision; v6: accepted for publication in G&C; v7: must have non-universal gravitational coupling; v8: rewritten with fully coupled theory; v9: major revision (fully coupled theory abandoned

Classical Electrodynamics in Quasi-Metric Space-Time

The quasi-metric manifold $\cal N$ is equipped with two one-parameter families of metric tensors ${\bf {\bar g}}_t$ and ${\bf g}_t$, each parametrized by the global time function $t$. Moreover, in $({\cal N},{\bf {\bar g}}_t)$ one must define two different electromagnetic field tensor families corresponding to the active electromagnetic field tensor family ${\bf {\tilde F}}_t$ and the passive electromagnetic field tensor family ${\bf {\bar F}}_t$, respectively. The active electromagnetic field tensor family ${\bf {\tilde F}}_t$ couples to gravity. By construction, the norm of the passive electromagnetic field tensor family ${\bf {\bar F}}_t$ experiences a secular decrease, defining a global cosmic attenuation (not noticeable locally) of the electromagnetic field. Local conservation laws for passive electromagnetism imply that ${\bf {\bar {\nabla}}}{\cdot} {\bf {\bar F}}_t=0$ in electrovacuum, ensuring that photons move on null geodesics of $({\cal N},{\bf {\bar g}}_t)$. From ${\bf {\bar F}}_t$ one may construct the passive electromagnetic field tensor family ${\bf F}_t$ in $({\cal N},{\bf g}_t)$ in the same way as ${\bf g}_t$ is constructed from ${\bf {\bar g}}_t$. This ensures that photons move on null geodesics of $({\cal N},{\bf g}_t)$ as well. As a simple example, the (exact) quasi-metric counterpart to the Reissner-Nordstr\"{o}m solution in General Relativity is calculated. Besides, it is found that a classical charged test particle electromagnetically bound to a central charge will participate in the cosmic expansion. But since quantum-mechanical states should be unaffected by the expansion, this classic calculation is hardly relevant for quantum-mechanical systems such as atoms, so there is no reason to think that the cosmic expansion should apply to them. Finally, it is shown that the main results of geometric optics hold in quasi-metric space-time.Comment: 23 pages, v2: major revision; v3: errors corrected, exact solution found; v4: accepted for publication in G&C; v5: minor corrections pp. 6,18; v6: must have non-universal gravitational coupling; v7: fully coupled theory implemented; v8: fully coupled theory abandone

The Koszul dual of a weakly Koszul module

We study the so-called weakly Koszul modules and characterise their Koszul duals. We show that the (adjusted) associated graded module of a weakly Koszul module exactly determines the homology modules of the Koszul dual. We give an example of a quasi-Koszul module which is not weakly Koszul.Comment: 20 page

Metrically Stationary, Axially Symmetric, Isolated Systems in Quasi-Metric Gravity

The gravitational field exterior respectively interior to a metrically stationary, axially symmetric, isolated spinning source made of perfect fluid is examined within the quasi-metric framework. (A metrically stationary system is defined as a system which is stationary except for the direct effects of the global cosmic expansion on the space-time geometry.) Field equations are set up and solved approximately for the exterior part. To lowest order in small quantities, the gravitomagnetic part of the found metric family corresponds with the Kerr metric in the metric approximation. On the other hand, the gravitoelectric part of the found metric family includes a tidal term describing the effect of source deformation due to the rotation. This tidal term has a counterpart in Newtonian gravitation, but not in the Kerr metric. Finally, the predicted geodetic effect for a gyroscope in a circular orbit is calculated. There is a correction term, unfortunately barely too small to be detectable by Gravity Probe B, to the standard expression.Comment: 19 pages; v3: somewhat extended; v4: accepted for publication in Acta Physica Polonica B; v5: must have nonuniversal gravitational coupling; v6 minor correction

The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems

We investigate wether three statements in analysis, that can be proved classically, are realizable in the realizability model of extensional continuous functionals induced by Kleene's second model $K_2$. We prove that a formulation of the Riemann Permutation Theorem as well as the statement that all partially Cauchy sequences are Cauchy cannot be realized in this model, while the statement that the product of two anti-Specker spaces is anti-Specker can be realized

Birkhoff's Theorem for Quasi-Metric Gravity

Working within the quasi-metric framework (QMF), it is examined if the gravitational field exterior to an isolated, spherically symmetric body is necessarily metrically static; or equivalently, whether or not Birkhoff's theorem holds for quasi-metric gravity. It is found that it does; however the proof is somewhat different from the general-relativistic case.Comment: 9 page

Discovery of Single Top Quark Production

The first observation of electroweak single top quark production was recently reported by the the D0 and CDF collaborations based on 2.3 and 3.2 fb-1 of ppbar collision data collected at \sqrt{s}=1.96 TeV from the Fermilab Tevatron collider. Several multivariate techniques are used to separate the single top signal from backgrounds, and both collaborations present measurements of the single top cross section and the CKM matrix element |Vtb|.Comment: 4 pages, proceedings for Moriond QCD and High Energy Interactions, March 14th - March 21st 200