Hamiltonian linearization of the rest-frame instant form of tetrad gravity in a completely fixed 3-orthogonal gauge: a radiation gauge for background-independent gravitational waves in a post-Minkowskian Einstein spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy ${\hat E}_{ADM}$, we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of {\it non-harmonic} 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) $r_{\bar a}(\tau ,\vec \sigma)$, $\pi_{\bar a}(\tau ,\vec \sigma)$, $\bar a = 1,2$. We define a Hamiltonian linearization of the theory, i.e. gravitational waves, {\it without introducing any background 4-metric}, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in ${\hat E}_{ADM}$. {\it We solve all the constraints} of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's $r_{\bar a}(\tau ,\vec \sigma)$, which replace the two polarizations of the TT harmonic gauge, and that {\it linearized Einstein's equations are satisfied} . Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.Comment: LaTeX (RevTeX3), 94 pages, 4 figure

Seismic Response of a Platform-Frame System with Steel Columns

Timber platform-frame shear walls are characterized by high ductility and diffuse energy dissipation but limited in-plane shear resistance. A novel lightweight constructive system composed of steel columns braced with oriented strand board (OSB) panels was conceived and tested. Preliminary laboratory tests were performed to study the OSB-to-column connections with self-drilling screws. Then, the seismic response of a shear wall was determined performing a quasi-static cyclic-loading test of a full-scale specimen. Results presented in this work in terms of force-displacement capacity show that this system confers to shear walls high in-plane strength and stiffness with good ductility and dissipative capacity. Therefore, the incorporation of steel columns within OSB bracing panels results in a strong and stiff platform-frame system with high potential for low- and medium-rise buildings in seismic-prone areas

Dragging a string over a step

A simple problem in Newtonian mechanics is considered. The problem consists in finding the maximum value of the length x_{UP} of the portion of string slowly dragged on a step of height h, when the string itself is initially placed to match the vertical profile of the step, the remaining part lying on the ground and the final portion being in static equilibrium during the dragging process. A straightforward analysis is required to find the solution. The problem can be proposed in a lecture or a demonstration in class on the role played by the coefficient of static friction in mechanics

Dynamical bar-mode instability in rotating and magnetized relativistic stars

We present three-dimensional simulations of the dynamical bar-mode instability in magnetized and differentially rotating stars in full general relativity. Our focus is on the effects that magnetic fields have on the dynamics and the onset of the instability. In particular, we perform ideal-magnetohydrodynamics simulations of neutron stars that are known to be either stable or unstable against the purely hydrodynamical instability, but to which a poloidal magnetic field in the range of $10^{14}$--$10^{16}$ G is superimposed initially. As expected, the differential rotation is responsible for the shearing of the poloidal field and the consequent linear growth in time of the toroidal magnetic field. The latter rapidly exceeds in strength the original poloidal one, leading to a magnetic-field amplification in the the stars. Weak initial magnetic fields, i.e. $\lesssim 10^{15}$ G, have negligible effects on the development of the dynamical bar-mode instability, simply braking the stellar configuration via magnetic-field shearing, and over a timescale for which we derived a simple algebraic expression. On the other hand, strong magnetic fields, i.e. $\gtrsim 10^{16}$ G, can suppress the instability completely, with the precise threshold being dependent also on the amount of rotation. As a result, it is unlikely that very highly magnetized neutron stars can be considered as sources of gravitational waves via the dynamical bar-mode instability.Comment: 18 pages, 13 figure

Neutron Star instabilities in full General Relativity using a Γ=2.75\Gamma=2.75 ideal fluid

We present results about the effect of the use of a stiffer equation of state, namely the ideal-fluid $\Gamma=2.75$ ones, on the dynamical bar-mode instability in rapidly rotating polytropic models of neutron stars in full General Relativity. We determine the change on the critical value of the instability parameter $\beta$ for the emergence of the instability when the adiabatic index $\Gamma$ is changed from 2 to 2.75 in order to mimic the behavior of a realistic equation of state. In particular, we show that the threshold for the onset of the bar-mode instability is reduced by this change in the stiffness and give a precise quantification of the change in value of the critical parameter $\beta_c$. We also extend the analysis to lower values of $\beta$ and show that low-beta shear instabilities are present also in the case of matter described by a simple polytropic equation of state.Comment: 16 pages, 16 figure

Minimising movements for the motion of discrete screw dislocations along glide directions

In [3] a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in [8] and predicts motion along the glide directions of the crystal. In this paper, we provide rigorous proofs of the results in [3], and in particular of the passage from the discrete to the continuous dynamics. The proofs are based on $\Gamma$-convergence techniques

Bar-mode instability suppression in magnetized relativistic stars

We show that magnetic fields stronger than about $10^{15}$ G are able to suppress the development of the hydrodynamical bar-mode instability in relativistic stars. The suppression is due to a change in the rest-mass density and angular velocity profiles due to the formation and to the linear growth of a toroidal component that rapidly overcomes the original poloidal one, leading to an amplification of the total magnetic energy. The study is carried out performing three-dimensional ideal-magnetohydrodynamics simulations in full general relativity, superimposing to the initial (matter) equilibrium configurations a purely poloidal magnetic field in the range $10^{14}-10^{16}$ G. When the seed field is a few parts in $10^{15}$ G or above, all the evolved models show the formation of a low-density envelope surrounding the star. For much weaker fields, no effect on the matter evolution is observed, while magnetic fields which are just below the suppression threshold are observed to slow down the growth-rate of the instability.Comment: 6 pages, 4 figures, to appear on the proceedings of the 4th YRM (Trieste 2013

Multiple Regimes in Cross-Region Growth Regressions with Spatial Dependence: A Parametric and a Semi-parametric Approach

This paper studies the distribution dynamics of development across European regions over the period 1975-2000. Regional development is measured in terms of both per capita GDP (Y/P) and its components: labour productivity and employment ratio (that in turn can be decomposed in terms of activity and unemployment rate). The Core/Periphery pattern in the European Union is firstly investigated and a comparative analysis in terms of income, productivity, employment and unemployment rates of the two partitions is carried out. Moreover, for each variable as well as for each partition, a nonparametric beta convergence analysis is applied. Synthetically, the results confirm the lack of regional convergence in per capita incomes, the presence of a negative quasi-linear relationship between growth rates and initial levels of labour productivity and a U-shaped relationship between growth rates and initial levels of unemployment rates. As it is well known, however, b-convergence analysis does not allow any test of multiple equilibria, such as “emerging twin peaks”, in the growth process. Equilibrium multiplicity can be properly assessed by using nonparametric techniques of analysis of the cross-regional distribution. In particular, a way to quantify the intra-distribution dynamics is the multivariate kernel, which estimates the joint density of regional income, productivity and (un)employment distribution at time t0 and t0+t. The results of this analysis suggest that over the period considered the regional growth pattern in Europe has followed a polarisation process rather than a convergence path. This appears particularly true in the case of per capita incomes and unemployment rates. Finally, in order to “explain” polarisation, conditional multivariate kernels are estimated. In particular, the role of spatial contiguity and regional sectoral specialisation is investigated.