Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: SR



I think that defining the angular unit, radian, in
terms of arc/radius is more desirable, in an
introductory course, than defining it in terms
of tan(w), as outlined by JohnD. The traditional
approach is likely to be less confusing to most
students.

Likewise, introducing c as "measured speed
of light" in a vacuum seems to be more desirable
than introducing it as a quantity needed to calibrate
yardsticks. In these cases following traditional
ways of teaching is likely to be more effective.
Ludwik Kowalski

On Monday, May 26, 2003, John S. Denker wrote:

Here is my rant on the subject:

There is a fundamental physical constant "c"
that appears in special relativity. To my
way of thinking, it is a misnomer to call it
the speed of light because "c" exists and is
important in all sorts of situations that
don't involve light or electromagnetism of
any kind.

I like to think of "c" as being the _radian_
of kinematics, i.e. the natural unit for
measuring velocities. Let me explain exactly
what I mean by this.

First, observe that the ordinary trigonometric
tangent function tan(w) is linear for small
angles (w). Next, we get to ask how big
does the angle have to be before the tan()
function becomes significantly nonlinear.
Specifically, let's ask for what angle (xx)
does tan(xx) become equal to twice (xx).
Answer: about 1.166 radians. You can use
this as an operational definition of radian:
just look at the graph of the tan() function
to find (xx) and then define the radian in
terms of (xx).

My operational definition of c proceeds in
strongly analogous fashion: First, observe
that for modest accelerations (a) and modest
times (t), the velocity (v) is a nice linear
function of the product, which is called the
rapidity (rho := a t). Next, we get to ask
how big does the rapidity have to be before
the velocity function becomes significantly
nonlinear. Specifically, let's ask for what
special rapidity (rr) does the velocity
tahn(rr) become equal to half of the rapidity.
Answer: about 1.915 c. You can use this as
an operational definition of "c": just look at
the graph of velocity versus (a t) to find
(rr) and then define "c" in terms of (rr).

Note: For clarity, the foregoing speaks in
terms of large (100%) nonlinearities. If
want to be a little bit clever and look more
closely, you can detect the nonlinearity
when it is quite a bit smaller than that.
Just expand the tan function (for trig)
or the tanh function (for SR) as a Taylor
series. There will be a linear term, and
then the next term will be third order. The
ratio of these terms gives you something
with the dimensions of a radian (trig) or
"c" (SR).

So, just as the radian can be considered the
threshold of nonlinear trigonometry, our friend
"c" can be considered the threshold of nonlinear
kinematics.

My definition of "c" does not involve light.
And it does not involve causality.

========================

It is true that the value of "c" is an officially
defined quantity. It is not a measured quantity.
It is not a measurable quantity.

This is not a particularly profound fact. It is
mostly a nitpicky technicality.

Where the rubber meets the road is in the apparatus
that formerly was used for measuring the speed of
light. The apparatus still exists and is still
physically significant. It has been renamed,
that's all. The apparatus is now called the
calibrate-your-ruler apparatus, because the unit
of length is officially defined in terms of the
unit of time and the official standard numerical
value of the speed of light.

==================

The question of whether real light actually moves
at the speed "c" is a nontrivial question. That
is, the proposition that light always travels
at the speed "c" is a falsifiable proposition,
as Kuhn would put it. Indeed it is sometimes
false, for instance when we consider the
macroscopic description of light moving through
a medium that has a refractive index. Even if
we confine ourselves to light in "the" vacuum,
we get to evaluate the proposition that all
light travels at the same speed, independent
of wavelength, polarization, intensity, et
cetera. Even this proposition is false! There
is such a thing as light-by-light scattering at
high intensities, so the classical Maxwell
description of light cannot be the whole story.

This is how science progresses: it becomes
more robust by adding restrictions: if you
want to get an accurate answer from the
calibrate-your-ruler apparatus (formerly known
as the speed-of-light apparatus), then
-- you need to minimize and/or correct for the
effects of any residual medium (background gas)
in your apparatus, and
-- you need to minimize and/or correct for
nonlinearities (which is easy to do at any
halfway reasonable intensity), and
-- you need to minimize and/or correct for
gravitational (GR) effects (also easy),
-- et cetera blah blah blah.

================

Similarly the proposition that causality and
information-flow are restricted to speeds less
than "c" is falsifiable. All evidence suggests
it is true under normal circumstances, but
there are those who think that there could be
exceptions under very extreme conditions (black
holes et cetera).