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

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

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

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

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

Teaching Newton with anticipation(...)

Care making only clock-specific assertions about elapsed-time, and other ``space-time smart'' strategies from the perspective of a selected inertial map-frame, open doors to an understanding of anyspeed motion via application of the metric equation.Comment: 2 pages (6 figs, 6 refs) RevTeX for broadened readership; http://www.umsl.edu/~fraundor/a1toc.htm

The thermal roots of correlation-based complexity

Bayesian maxent lets one integrate thermal physics and information theory points of view in the quantitative study of complex systems. Since net surprisal (a free energy analog for measuring "departures from expected") allows one to place second law constraints on mutual information (a multi-moment measure of correlations), it makes a quantitative case for the role of reversible thermalization in the natural history of invention, and suggests multiscale strategies to monitor standing crop as well. It prompts one to track evolved complexity starting from live astrophysically-observed processes, rather than only from evidence of past events. Various gradients and boundaries that play a role in availability flow, ranging from the edge of a wave-packet to the boundary between idea-pools, allow one to frame wide-ranging correlations (including that between a phenomenon and its explanation) as delocalized {\em physical} structures.Comment: 8 pages, 3 figures, 43 reference