Ralph Baierlein

Approximate symmetry groups of inhomogeneous metrics: Examples

By definition, an N‐dimensional positive‐definite inhomogeneous metric is not invariant under any N‐parameter, simply‐transitive continuous group of motions. Nonetheless, it is possible to construct a group (simply‐transitive and of N parameters) that comes closest to leaving the given metric invariant. We call this group the approximate symmetry group of the metric. In an earlier paper, we described a technique for constructing the approximate symmetry group of a given metric. Here, we briefly review that technique and then present some examples of its application. All two‐dimensional metrics are analyzed, and simple criteria are given for determining their approximate symmetry groups. Three three‐dimensional metrics are investigated: the invariant hypersurfaces of the Kantowski–Sachs space–times and two families of hypersurfaces in the Gowdy T3 space–times. The approximate symmetry group of the former is found to be of Bianchi Type I and those of the latter may be I or VI0. Defining, via our technique, ...

Approximate symmetry groups of inhomogeneous metrics

A useful step toward understanding inhomogeneous space–times would be to classify them, perhaps in a fashion analogous to that used for spatially homogeneous space–times. To that end, a technique for determining an approximate simply‐transitive three‐parameter symmetry group of a three‐dimensional positive‐definite Riemannian metric is developed. The technique employs a variational principle to find a set of three orthonormal vectors whose commutation coefficients are as close as possible to a set of structure constants. The Bianchi classification of the structure constants of three‐parameter groups is then used to classify these inhomogeneous metrics. Application of this technique to perturbed homogeneous metrics is discussed in detail. We find that only four types of symmetry groups can be considered generic in the space of all perturbed homogeneous metrics.

On the electromagnetic detection of gravitational waves

Braginsky and Mensky have described a novel gravitational wave detector based on a special “gravitational-electromagnetic resonance” in an annular waveguide. Their analysis is based on geometrical optics. If the configuration is analyzed as a perturbed boundary-value problem, however, no special resonance is evident. Nor does a more general cavity exhibit such a resonance. This paper concludes with a moral: When investigating the interaction of gravity and electromagnetism, one must be circumspect in applying the eikonal approximation.

Experimental determination of some nonlinear elements of third sound dynamics

The frequency shifts of third sound resonances during free decay are observed to have a quadratic dependence on the amplitude of the wave motion. An unambiguous measurement of amplitude allows for quantitative comparisons to predictions for the frequency shifts based on a variety of nonlinear influences acting within the superfluid4He film and resonator. Dispersive terms play a minor role in this analysis. We conclude that nonlinear terms dictated by a straight forward application of classical hydrodynamics are applicable to superfluid helium films.

Stellar clustering as induced by a supernova

Abstract When the shock wave from a supernova expands, it sweeps up not only interstellar matter but also magnetic field. The field is greatly amplified by compression and will provide the dominant pressure during the cool radiative phase of an expanding supernova shell. We examine a hydromagnetic instability in this system (a form of the Parker instability) and find that it will concentrate gas at intervals of the order of parsecs. The length and time scales make the instability promising as an explanation of the stellar clustering that is seen in Canis Major R1.

Small-scale magnetic fields in turbulence: Saffman's approximation revisited

The subject is the small-scale structure of a magnetic field in a turbulent conducting fluid, ‘small scale’ meaning lengths much smaller than the characteristic dissipative length of the turbulence. Philip Saffman developed an approximation to describe this structure and its evolution in time. Its usefulness invites a closer examination of the approximation itself and an attempt to place sharper limits on the numerical parameters that appear in the approximate correlation functions, topics to which the present paper is addressed. A Lagrangian approach is taken, wherein one makes a Fourier decomposition of the magnetic field in a neighbourhood that follows a fluid element. If one construes the viscous-convective range narrowly, by ignoring magnetic dissipation entirely, then results for a magnetic field in two dimensions are consistent with Saffmans approximation, but in three dimensions no steady state could be found. Thus, in three dimensions, turbulent amplification seems to be more effective than Saffmans approximation implies. The cause seems to be a matter of geometry, not of correlation times or relative time scales. Strictly-outward spectral transfer is a characteristic of Saffmans approximation, and this may be an accurate description only when dissipation suppresses the contributions from inwardly directed spectral transfer. In the spectral region where dominance passes from convection to dissipation, one can generate expressions for the parameters that arise in Saffmans approximation. Their numerical evaluation by computer simulation may enable one to sharpen the limits that Saffman had already set for those parameters.

Brownian motion and the superconducting surface sheath

19780123We give an explicit solution for the mean velocity of a Brownian particle that is subject simultaneously to a spatially sinusoidal and a constant force. An application is made to the superconducting surface sheath of lead, wherein the fluxoids are represented by Brownian particles. When Brownian mean velocity curves are fitted to experimental voltage-current curves, we find that the parameter values are physically reasonable provided that interactions among the fluxoids are taken into account. The fluxoids move, in effect, as collections of about 100 fluxoids.