Transformation Properties of the Lagrangian and Eulerian Strain Tensors

A coordinate independent derivation of the Eulerian and Lagrangian strain tensors of finite deformation theory is given based on the parallel propagator, the world function, and the displacement vector field as a three-point tensor. The derivation explicitly shows that the Eulerian and Lagrangian strain tensors are two-point tensors, each a function of both the spatial and material coordinates. The Eulerian strain is a two-point tensor that transforms as a second rank tensor under transformation of spatial coordinates and transforms as a scalar under transformation of the material coordinates. The Lagrangian strain is a two-point tensor that transforms as scalar under transformation of spatial coordinates and transforms as a second rank tensor under transformation of the material coordinates. These transformation properties are needed when transforming the strain tensors from one frame of reference to another moving frame.Comment: 35 pages double-space, 3 figure

Clock Synchronization and Navigation in the Vicinity of the Earth

Clock synchronization is the backbone of applications such as high-accuracy satellite navigation, geolocation, space-based interferometry, and cryptographic communication systems. The high accuracy of synchronization needed over satellite-to-ground and satellite-to-satellite distances requires the use of general relativistic concepts. The role of geometrical optics and antenna phase center approximations are discussed in high accuracy work. The clock synchronization problem is explored from a general relativistic point of view, with emphasis on the local measurement process and the use of the tetrad formalism as the correct model of relativistic measurements. The treatment makes use of J. L. Synge's world function of space-time as a basic coordinate independent geometric concept. A metric is used for space-time in the vicinity of the Earth, where coordinate time is proper time on the geoid. The problem of satellite clock syntonization is analyzed by numerically integrating the geodesic equations of motion for low-Earth orbit (LEO), geosynchronous orbit (GEO), and highly elliptical orbit (HEO) satellites. Proper time minus coordinate time is computed for satellites in these orbital regimes. The frequency shift as a function of time is computed for a signal observed on the Earth's geoid from a LEO, GEO, and HEO satellite. Finally, the problem of geolocation in curved space-time is briefly explored using the world function formalism.Comment: 49 pages, 19 figures. To be published in "Progress in General Relativity and Quantum Cosmology Research", Nova Science Publishers, Inc., Hauppauge, New York, December 2004. Paper was replaced due to font problems in the figure

Stress in Rotating Disks and Cylinders

The solution of the classic problem of stress in a rotating elastic disk or cylinder, as solved in standard texts on elasticity theory, has two features: dynamical equations are used that are valid only in an inertial frame of reference, and quadratic terms are dropped in displacement gradient in the definition of the strain. I show that, in an inertial frame of reference where the dynamical equations are valid, it is incorrect to drop the quadratic terms because they are as large as the linear terms that are kept. I provide an alternate formulation of the problem by transforming the dynamical equations to a corotating frame of reference of the disk/cylinder, where dropping the quadratic terms in displacement gradient is justified. The analysis shows that the classic textbook derivation of stress and strain must be interpreted as being carried out in the corotating frame of the medium.Comment: 13 pages. This is a detailed derivation of stress in rotating disks and cylinder

Fermi Coordinates of an Observer Moving in a Circle in Minkowski Space: Apparent Behavior of Clocks

Coordinate transformations are derived from global Minkowski coordinates to the Fermi coordinates of an observer moving in a circle in Minkowski space-time. The metric for the Fermi coordinates is calculated directly from the tensor transformation rule. The behavior of ideal clocks is examined from the observer's reference frame using the Fermi coordinates. A complicated relation exists between Fermi coordinate time and proper time on stationary clocks (in the Fermi frame) and between proper time on satellite clocks that orbit the observer. An orbital Sagnac-like effect exists for portable clocks that orbit the Fermi coordinate origin. The coordinate speed of light is isotropic but varies with Fermi coordinate position and time. The magnitudes of these kinematic effects are computed for parameters that are relevant to the Global Positioning System (GPS) and are found to be small; however, for future high-accuracy time transfer systems, these effects may be of significant magnitude

Quantum Positioning System

A quantum positioning system (QPS) is proposed that can provide a user with all four of his space-time coordinates. The user must carry a corner cube reflector, a good clock, and have a two-way classical channel of communication with the origin of the reference frame. Four pairs of entangled photons (biphotons) are sent through four interferometers: three interferometers are used to determine the user's spatial position, and an additional interferometer is used to synchronize the user's clock to coordinate time in the reference frame. The spatial positioning part of the QPS is similar to a classical time-of-arrival (TOA) system, however, a classical TOA system (such as GPS) must have synchronized clocks that keep coordinate time and therefore the clocks must have long-term stability, whereas in the QPS only a photon coincidence counter is needed and the clocks need only have short-term stability. Several scenarios are considered for a QPS: one is a terrestrial system and another is a space-based-system composed of low-Earth orbit (LEO) satellites. Calculations indicate that for a space-based system, neglecting atmospheric effects, a position accuracy below the 1 cm-level is possible for much of the region near the Earth. The QPS may be used as a primary system to define a global 4-dimensional reference frame.Comment: 20 pages, 10 figure

Interaction of Diatomic Molecules with Photon Angular Momentum

The interaction of a diatomic molecule with photons carrying well-defined angular momentum and parity is investigated to determine whether photon absorption can induce molecular rotational transitions between states having angular momentum $\Delta J >1$. A transformation from laboratory coordinates to coordinates with origin at the center-of-mass of the nuclei is used to obtain the interaction between the photons and the molecule's center-of-mass, electronic, and rotational degrees of freedom. For molecules making transitions between rotational levels, there is a small parameter, $k a \ll 1$, where $k$ is the photon wave vector and $a$ is the size of the molecule, which enters into the $Ej$ and $Mj$ photon absorption probabilities. For electric photons having arbitrary angular momentum $j \hbar$, the probability of absorbing an $E(j+1)$ photon divided by the probability of absorbing an $Ej$ photon, scales as $(k a)^2 /(2 j+1)^2$. The probability of absorbing an $Mj$ photon, divided by the probability of absorbing and $Ej$ photon scales according to the same factor.Comment: 9 pages, no figure

Transfer of Spatial Reference Frame Using Singlet States and Classical Communication

A simple protocol is described for transferring spatial direction from Alice to Bob (two spatially separated observers) up to inversion. The two observers are assumed to share quantum singlet states and classical communication. The protocol assumes that Alice and Bob have complete free will (measurement independence) and is based on maximizing the Shannon mutual information between Alice and Bob's measurement outcomes. Repeated use of this protocol for each spatial axis of Alice allows transfer of a complete 3-dimensional reference frame, up to inversion of each of the axes. The technological complexity of this protocol is similar to that needed for BB84 quantum key distribution, and hence is much simpler to implement than recently proposed schemes for transmission of reference frames. A second protocol based on a Bayesian formalism is also presented.Comment: 6 pages, 3 figure

Force on an Asymmetric Capacitor

When a high voltage (~30 kV) is applied to a capacitor whose electrodes have different physical dimensions, the capacitor experiences a net force toward the smaller electrode (Biefeld-Brown effect). We have verified this effect by building four capacitors of different shapes. The effect may have applications to vehicle propulsion and dielectric pumps. We review the history of this effect briefly through the history of patents by Thomas Townsend Brown. At present, the physical basis for the Biefeld-Brown effect is not understood. The order of magnitude of the net force on the asymmetric capacitor is estimated assuming two different mechanisms of charge conduction between its electrodes: ballistic ionic wind and ionic drift. The calculations indicate that ionic wind is at least three orders of magnitude too small to explain the magnitude of the observed force on the capacitor. The ionic drift transport assumption leads to the correct order of magnitude for the force, however, it is difficult to see how ionic drift enters into the theory. Finally, we present a detailed thermodynamic treatment of the net force on an asymmetric capacitor. In the future, to understand this effect, a detailed theoretical model must be constructed that takes into account plasma effects: ionization of gas (or air) in the high electric field region, charge transport, and resulting dynamic forces on the electrodes. The next series of experiments should determine whether the effect occurs in vacuum, and a careful study should be carried out to determine the dependence of the observed force on gas pressure, gas species and applied voltage.Comment: 25 pages, new version has corrections of typos and wording, and additional references added. This version is to be published as an Army Research Laboratory Technical Report (in press, March 2003

Tradeoff between Efficiency and Melting for a High-Performance Electromagnetic Rail Gun

We estimate the temperature distribution in the rails of an electromagnetic rail gun (EMG) due to the confinement of the current in a narrow surface layer resulting from the skin effect. In order to obtain analytic results, we assume a simple geometry for the rails, an electromagnetic skin effect boundary edge that propagates with the accelerating armature, and a current carrying channel controlled by magnetic field diffusion into the rails. We compute the temperature distribution in the rails at the time that the armature leaves the rails. For the range of exit velocities, from 1500 m/s to 5000 m/s, we find the highest temperatures are near the gun breech. After a single gun firing, the temperature reaches the melting temperature of the metal rails in a layer of finite thickness near the surface of the rails, for rails made of copper or tantalum. We plot the thickness of the melt layer as a function of position along the rails. In all cases, the thickness of the melt layer increases with gun velocity, making damage to the gun rails more likely at higher velocity. We also calculate the efficiency of the EMG as a function of gun velocity and find that the efficiency increases with increasing velocity, but only if the length of the gun is sufficiently long. The thickness of the melted layer also decreases with increasing rail length. Therefore, there is a tradeoff: for rails of sufficient length, the gun efficiency increases with increasing velocity but the melted layer thickness in the rails also increases.Comment: 10 pages, 13 figure

Phase Sensitivity of a Mach-Zehnder Quantum Sensor

We investigate the dependence of the fidelity of a Mach-Zehnder quantum interferometer on the prior information about the phase, for Fock state input and for maximally entangled (N00N) state input. For no prior information, the fidelity for Fock state input is greater than for N00N state input. In the limit of a narrow distribution describing the prior information, we find that both Fock and N00N state inputs lead to nearly equal fidelity.Comment: 4 pages, 4 figure