From: utzoo!decvax!harpo!seismo!hao!hplabs!sri-unix!gwyn@Brl-Vld.ARPA
Newsgroups: net.physics
Title: Re:  Speed of light
Article-I.D.: sri-arpa.1353
Posted: Wed May 11 16:45:29 1983
Received: Tue May 17 08:14:46 1983

From:      Doug Gwyn (VLD/VMB) <gwyn@Brl-Vld.ARPA>

It is disappointing to see how many respondents fail to understand
the r^ole of measurement unit standards and fundamental constants
in physics.  Perhaps this indicates something about the way the
subject is taught or the effect of lack of a firm epistemological
base for the current state of the science.  In any case, herein
follows a much abbreviated exposition of the nature of the speed
of light.

Fundamental physical laws are not obtained by "curve-fitting"
empirical data, nor are they arbitrarily decreed by a supernatural
agency.  In the case of relativity theory (mostly the special
theory, although insights from generalized field theory are useful
for setting an overall framework for discussion), from a few simple
principles (none of them a curve fit) one can deduce the existence
of a special "fundamental" speed.  Comparison with other knowledge
such as Maxwell's equations leads one to conclude that light travels
with this fundamental speed.

The numerical value of "c", the fundamental speed, is not given by
the theory.  If you measure the actual speed of light in your
favorite system of distance and time units, then you can assign that
approximate value to c USING YOUR UNITS.  According to this approach,
it makes no sense to ask what things would be like were c different
from what you would measure; it can't be!

A more profound understanding of c's relationship to measurement
unit standards can be obtained by considering Minkowski's four-
dimensional "space-time" hyperspace, or its generalization to a
differentiable four-dimensional manifold with metric signature
(+ + + -) in general relativity and beyond.  The idea is that a
local coordinate transformation can reduce the metric tensor to
diagonal form under some circumstances (absence of electromagnetic
fields, etc.); that is, the generalization of Euclidean distance
after this "free-fall" transformation to an "inertial frame" would
be:
	ds^2 = (f dx)^2 + (e dy)^2 + (d dz)^2 - (c dt)^2 ,
where (x,y,z,t) are local Cartesian 3-space coordinates and the
time coordinate measured in arbitrary units.  People (not just
theoretical physicists) are clever enough to use compatible units
of distance for the three 3-space axis directions, so
	f = e = d	by usual convention.
In fact, theoreticians usually choose compatible units for time, so
	f = e = d = c	by theoretical physics convention.
In this case, "ds" units are normal taken to be compatible also, so
	f = e = d = c = 1	is the usual theoretical physics choice.
In the case
	f = e = d = 1	which assigns ds units compatible with distance,
the speed of light is exactly c.  The theoreticians therefore have
chosen units such that the speed of light is precisely 1.

The confusion seems to arise because most people insist on using
incompatible units for distance and time, in which case the numerical
value of c will depend on their choice of unit standards.  It may be
hard to determine an accurate value for c using random distance/time
units, since for example "meters" are related to the circumference of
the Earth and "seconds" are related to the rotational period of the
Earth -- neither of which has any apparent ties to fundamental laws
of physics (and therefore, the constant "c").  The reasons distance and
time are hard to consider as "the same type of thing" lie in the
opposite sign for the time coordinate in the diagonalized metric.

Since many physical field laws (including electromagnetic propagation)
are tied strongly to the metric tensor, the "speed of light" c (for
which read the "quotient of my space/time unit standards") is NOT
a freely adjustable parameter in the laws of physics.  Such ideas as
"the speed of light may change with time and/or distance" are
obviously inconsistent with the r^ole that "c" plays in interpreting
the metric tensor.

An additional note for cosmologists:  Since "ds" is an invariant, the
choice
	f = e = d = 1
is not really free either.  Natural distance units would be tied to
some physical phenomenon (obvious examples are: the "radius of the
universe"; the "size of a nucleon").  The most natural generalization
of general relativity, that is, the one making the fewest physical
assumptions, leads directly to a cosmology with a natural distance
unit.  Anyone who is really curious about this can drop me a note
and I'll send out a copy of my thesis.