A one-map two-clock approach to teaching relativity in introductory physics

This paper presents some ideas which might assist teachers incorporating special relativity into an introductory physics curriculum. One can define the proper-time/velocity pair, as well as the coordinate-time/velocity pair, of a traveler using only distances measured with respect to a single ``map'' frame. When this is done, the relativistic equations for momentum, energy, constant acceleration, and force take on forms strikingly similar to their Newtonian counterparts. Thus high-school and college students not ready for Lorentz transforms may solve relativistic versions of any single-frame Newtonian problems they have mastered. We further show that multi-frame calculations (like the velocity-addition rule) acquire simplicity and/or utility not found using coordinate-velocity alone.Comment: 10 pages (1 fig, 3 tables) RevTeX; classroom-focus improved; also http://www.umsl.edu/~fraundor/a1toc.htm

Friendly units for coldness

Measures of temperature that center around human experience get lots of use. Of course thermal physics insights of the last century have shown that reciprocal temperature (1/kT) has applications that temperature addresses less well. In addition to taking on negative absolute values under population inversion (e.g. of magnetic spins), bits and bytes turn 1/kT into an informatic measure of the thermal ambient for developing correlations within any complex system. We show here that, in the human-friendly units of bytes and food Calories, water freezes when 1/kT ~200 ZB/Cal or kT ~5 Cal/YB. Casting familiar benchmarks into these terms shows that habitable human space requires coldness values (part of the time, at least) between 0 and 40 ZB/Cal with respect body temperature ~100 degrees F, a range in kT of ~1 Cal/YB. Insight into these physical quantities underlying thermal equilibration may prove useful for budding scientists, as well as the general public, in years ahead.Comment: 3 pages, 2 figures, 21 refs, RevTeX4 cf. http://www.umsl.edu/~fraundor/ifzx/zbpercal.htm

Heterogeneity, and the secret of the sea

This paper explores tools for modeling and measuring the compositional heterogeneity of a rock, or other solid specimen. Intuitive ``variation per decade'' plots, simple expressions for containment probability, generalization for familiar error-in-the-mean expressions, and a useful dimensionless sample bias coefficient all emerge from the analysis. These calculations have also inspired subsequent work on log-log roughness spectroscopy (with applications to scanning probe microscope data), and on angular correlation mapping of lattice fringe images (with applications in high resolution transmission electron microscopy). It was originally published as Appendix E of a dissertation on ``Microcharacterization of interplanetary dust collected in the earth's stratosphere''.Comment: 9 pages (3 figs, 14 refs) RevTeX, comments http://www.umsl.edu/~fraundor/covariances.htm

Three Self-Consistent Kinematics in (1+1)D Special Relativity

When introducing special relativity, an elegant connection to familiar rules governing Galilean constant acceleration can be made, by describing first the discovery at high speeds that the clocks (as well as odometers) of different travelers may proceed at different rates. One may then show how to parameterize any given interval of constant acceleration with {\em either}: Newtonian (low-velocity approximation) time, inertial relativistic (unaccelerated observer) time, or traveler proper (accelerated observer) time, by defining separate velocities for each of these three kinematics as well. Kinematic invariance remains intact for proper acceleration since $m a_o = dE/dx$. This approach allows students to solve relativistic constant acceleration problems {\em with the Newtonian equations}! It also points up the self-contained and special nature of the accelerated-observer kinematic, with its frame-invariant time, 4-vector velocities which in traveler terms exceed Newtonian values and the speed of light, and of course relativistic momentum conservation.Comment: RevTeX, 3 tables and 1 figure available from the author and in prep for a replacement preprint; also http://newton.umsl.edu/~run

Modernizing Newton, to work at any speed

Modification of three ideas underlying Newton's original world view, with only minor changes in context, might offer two advantages to introductory physics students. First, the students will experience less cognitive dissonance when they encounter relativistic effects. Secondly, the map-based Newtonian tools that they spend so much time learning about can be extended to high speeds, non-inertial frames, and even (locally, of course) to curved-spacetime.Comment: 7 pages (0 figs, 27 refs) RevTeX4; for more see http://www.umsl.edu/~fraundor/a1toc.htm

Localizing periodicity in near-field images

We show that Bayesian inference, like that used in statistical mechanics, can guide the systematic construction of Fourier dark-field methods for localizing periodicity in near-field (e.g. scanning-tunneling and electron-phase-contrast) images. For crystals in an aperiodic field, the Fourier coefficient Ze^{i phi} combines with a prior estimate for background amplitude B to predict background phase (beta) values distributed with a probability p(beta - phi | Z,phi,B) inversely proportional to the amplitude P of the signal of interest, when this latter is treated as an unknown translation scaled to B.Comment: 5 pages (4 figs, 13 refs) RevTeX; apps http://newton.umsl.edu/stei_la

Some minimally-variant map-based rules of motion at any speed

We take J. S. Bell's commendation of ``frame-dependent'' perspectives to the limit here, and consider motion on a ``map'' of landmarks and clocks fixed with respect to a single arbitrary inertial-reference frame. The metric equation connects a traveler-time with map-times, yielding simple integrals of constant proper-acceleration over space (energy), traveler-time (felt impulse), map-time (momentum), and time on the clocks of a chase-plane determined to see Galileo's original equations apply at high speed. Rules follow for applying frame-variant and proper forces in context of one frame. Their usefulness in curved spacetimes via the equivalence principle is maximized by using synchrony-free and/or frame-invariant forms for length, time, velocity, and acceleration. In context of any single system of locally inertial frames, the metric equation thus lets us express electric and magnetic effects with a single frame-invariant but velocity-dependent force, and to contrast such forces with gravity as well.Comment: 9 pages (1 table, 1 fig, 17 refs/context updated) RevTeX, cf. http://www.umsl.edu/~fraundor/a1toc.htm

Heat capacity in bits

Information theory this century has clarified the 19th century work of Gibbs, and has shown that natural units for temperature kT, defined via 1/T=dS/dE, are energy per nat of information uncertainty. This means that (for any system) the total thermal energy E over kT is the log-log derivative of multiplicity with respect to energy, and (for all b) the number of base-b units of information lost about the state of the system per b-fold increase in the amount of thermal energy therein. For ``un-inverted'' (T>0) systems, E/kT is also a temperature-averaged heat capacity, equaling ``degrees-freedom over two'' for the quadratic case. In similar units the work-free differential heat capacity C_v/k is a ``local version'' of this log-log derivative, equal to bits of uncertainty gained per 2-fold increase in temperature. This makes C_v/k (unlike E/kT) independent of the energy zero, explaining in statistical terms its usefulness for detecting both phase changes and quadratic modes.Comment: 7 pages (3 figs, 16 refs) RevTeX; clarify, new plots; comments http://www.umsl.edu/~fraundor/cm971174.htm

Layer-multiplicity as a community order-parameter

A small number of (perhaps only 6) broken-symmetries, marked by the edges of a hierarchical series of physical {\em subsystem-types}, underlie the delicate correlation-based complexity of life on our planet's surface. Order-parameters associated with these broken symmetries might in the future help us broaden our definitions of community health. For instance we show that a model of metazoan attention-focus, on correlation-layers that look in/out from the 3 boundaries of skin, family & culture, predicts that behaviorally-diverse communities require a characteristic task layer-multiplicity {\em per individual} of only about $4 \frac14$ of the six correlation layers that comprise that community. The model may facilitate explorations of task-layer diversity, go beyond GDP & body count in quantifying the impact of policy-changes & disasters, and help manage electronic idea-streams in ways that strengthen community networks. Empirical methods for acquiring task-layer multiplicity data are in their infancy, although experience-sampling via cell-phone button-clicks might be one place to start.Comment: 5 pages (3 figs, 17 refs) RevTeX, cf. http://www.umsl.edu/~fraundorfp/ifzx/taskLayerMultiplicity.htm

Non-coordinate time/velocity pairs in special relativity

Motions with respect to one inertial (or ``map'') frame are often described in terms of the coordinate time/velocity pair (or ``kinematic'') of the map frame itself. Since not all observers experience time in the same way, other time/velocity pairs describe map-frame trajectories as well. Such coexisting kinematics provide alternate variables to describe acceleration. We outline a general strategy to examine these. For example, Galileo's acceleration equations describe unidirectional relativistic motion {\it exactly} if one uses $V=\frac{dx}{dT}$, where $x$ is map-frame position and $T$ is clock time in a chase plane moving such that $\gamma ^{\prime }=\gamma \sqrt{\frac 2{\gamma +1}}$. Velocity in the traveler's kinematic, on the other hand, has dynamical and transformational properties which were lost by coordinate-velocity in the transition to Minkowski space-time. Its repeated appearance with coordinate time, when expressing relationships in simplest form, suggests complementarity between traveler and coordinate kinematic views.Comment: RevTeX, full (3+1)D treatment w/5 tables, 1bw and 1greyscale figure. Mat'l on 1D origins at http://www.umsl.edu/~fraundor/a1toc.htm