Quantum Uncertainty Considerations for Gravitational Lens Interferometry

The measurement of the gravitational lens delay time between light paths has relied, to date, on the source having sufficient variability to allow photometric variations from each path to be compared. However, the delay times of many gravitational lenses cannot be measured because the intrinsic source amplitude variations are too small to be detectable. At the fundamental quantum mechanical level, such photometric time stamps allow which-path knowledge, removing the ability to obtain an interference pattern. However, if the two paths can be made equal (zero time delay) then interference can occur. We describe an interferometric approach to measuring gravitational lens delay times using a quantum-eraser/restorer approach, whereby the time travel along the two paths may be rendered measurably equal. Energy and time being non-commuting observables, constraints on the photon energy in the energy-time uncertainty principle, via adjustments of the width of the radio bandpass, dictate the uncertainty of the time delay and therefore whether the path taken along one or the other gravitational lens geodesic is knowable. If one starts with interference, for example, which-path information returns when the bandpass is broadened (constraints on the energy are relaxed) to the point where the uncertainty principle allows a knowledge of the arrival time to better than the gravitational lens delay time itself, at which point the interference will disappear. We discuss the near-term feasibility of such measurements in light of current narrow-band radio detectors and known short time-delay gravitational lenses.Comment: 22 page

Music and time: tempomorphism: nested temporalities in perceived experience of music.

This thesis represents the results of a theoretical and practical investigation of acoustic and electro-acoustic elements of Western music at the start of the twentyfirst century, with specific attention to soundscapes. A commentary on the development of soundscapes is drawn from a multidisciplinary overview of concepts of time, followed by an examination of concepts of time in music. As a response to Jonathan Kramer's concept of `vertical' music (a characteristic aesthetic of which is an absence of conventional harmonic teleology), particular attention is paid to those theories of multiple nested temporalities which have been referred to by Kramer in support of non-teleological musical structures. The survey suggests that new musical concepts, such as vertical music, have emerged from sensibilities resulting from the musical and associated styles of minimalism, and represent an ontological development of aesthetics characteristic of the twentieth century. An original contention of the debate is that innovations in the practice of music as the result of technological developments have led to the possibility of defining a methodology of process in addition to auditive strategies, resulting in a duality defined as 'tempomorphic'. Further observations are supplied, using findings derived from original creative practical research, to define tempomorphic performance, which complete the contribution to knowledge offered by the investigation. Tempomorphism, therefore, is defined as a duality of process and audition: as auditive tool, tempomorphic analysis provides a listening strategy suited to harmonically static music; as a procedural tool, it affords a methodology based primarily on duration

Configuration of the Crucial Set for a Quadratic Rational Map

Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $\varphi(z) \in K(z)$ have degree two. We describe the crucial set of $\varphi$ in terms of the multipliers of $\varphi$ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space $\mathcal{M}_2(K)$ related to the reduction type of $\varphi$. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the non-archimedean Julia set