136 research outputs found
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 . 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
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
, where is map-frame position and is clock time in a
chase plane moving such that . 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
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 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
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